If your jurisdiction uses the International Plumbing Code or a derivative, there are several issues with this set-up.
First - 1002.1 would require the washer to have its own trap (none of the exceptions are applicable), and the trap needs to be within 30" of the standpipe:
1002.1 Fixture traps.
Each plumbing fixture shall be separately trapped by a liquid-seal
trap, except as otherwise permitted by this code. The vertical
distance from the fixture outlet to the trap weir shall not exceed 24
inches (610 mm), and the horizontal distance shall not exceed 30
inches (610 mm) measured from the centerline of the fixture outlet to
the centerline of the inlet of the trap.
Second - 709.1 (see linked table) and 406.2 require that residential clothes washers have a 2" trap:
406.2 Waste connection. The waste from an automatic clothes washer shall discharge through an air break into a standpipe in accordance
with Section 802.4 or into a laundry sink. The trap and fixture drain
for an automatic clothes washer standpipe shall be not less than 2
inches (51 mm) in diameter.
Third - Even assuming that you were putting in a 2" trap for the washer per 1002.1, 909.1 and 909.2 would require that the vent be within 8' of the washer's trap (which per the above has to be within 30" of the washer for a maximum of 10',6" from standpipe to vent stack):
909.2 Venting of fixture drains. The total fall in a fixture drain due to pipe slope shall not exceed the diameter of the fixture drain, nor
shall the vent connection to a fixture drain, except for water
closets, be below the weir of the trap.
If you can't vent the washer (and assuming the sink drains into a 3" stack), you could wye into the main stack and run a 3" drain instead of 1½", but I'd guess that isn't as practical.
If the drain is accessible below the kitchen, you'd likely be better off running a separate 2" vent for the washer (either re-venting it above or running it all the way outside by itself) and then branching into the drain below the kitchen. If you have the access, this would probably be easier than trying to run 12' of pipe through cabinetry.