Theorem oddpwdcv 28119

Description: Lemma for eulerpart 28146: value of the F function. (Contributed by Thierry Arnoux, 9-Sep-2017.)

Hypotheses

Ref	Expression
oddpwdc.j	|- J = { z e. NN | -. 2 || z }
oddpwdc.f	|- F = ( x e. J , y e. NN0 |-> ( ( 2 ^ y ) x. x ) )

Assertion

Ref	Expression
oddpwdcv	|- ( W e. ( J X. NN0 ) -> ( F ` W ) = ( ( 2 ^ ( 2nd ` W ) ) x. ( 1st ` W ) ) )

Distinct variable groups:   x, y, z   x, J, y   x, W, y

Allowed substitution hints:   F( x, y, z)   J( z)   W( z)

Proof of Theorem oddpwdcv

Step	Hyp	Ref	Expression
1		1st2nd2 6832	. . 3 |- ( W e. ( J X. NN0 ) -> W = <. ( 1st ` W ) , ( 2nd ` W ) >. )
2	1	fveq2d 5876	. 2 |- ( W e. ( J X. NN0 ) -> ( F ` W ) = ( F ` <. ( 1st ` W ) , ( 2nd ` W ) >. ) )
3		df-ov 6298	. . 3 |- ( ( 1st ` W ) F ( 2nd ` W ) ) = ( F ` <. ( 1st ` W ) , ( 2nd ` W ) >. )
4	3	a1i 11	. 2 |- ( W e. ( J X. NN0 ) -> ( ( 1st ` W ) F ( 2nd ` W ) ) = ( F ` <. ( 1st ` W ) , ( 2nd ` W ) >. ) )
5		elxp6 6827	. . . 4 |- ( W e. ( J X. NN0 ) <-> ( W = <. ( 1st ` W ) , ( 2nd ` W ) >. /\ ( ( 1st ` W ) e. J /\ ( 2nd ` W ) e. NN0 ) ) )
6	5	simprbi 464	. . 3 |- ( W e. ( J X. NN0 ) -> ( ( 1st ` W ) e. J /\ ( 2nd ` W ) e. NN0 ) )
7		oveq2 6303	. . . 4 |- ( x = ( 1st ` W ) -> ( ( 2 ^ y ) x. x ) = ( ( 2 ^ y ) x. ( 1st ` W ) ) )
8		oveq2 6303	. . . . 5 |- ( y = ( 2nd ` W ) -> ( 2 ^ y ) = ( 2 ^ ( 2nd ` W ) ) )
9	8	oveq1d 6310	. . . 4 |- ( y = ( 2nd ` W ) -> ( ( 2 ^ y ) x. ( 1st ` W ) ) = ( ( 2 ^ ( 2nd ` W ) ) x. ( 1st ` W ) ) )
10		oddpwdc.f	. . . 4 |- F = ( x e. J , y e. NN0 |-> ( ( 2 ^ y ) x. x ) )
11		ovex 6320	. . . 4 |- ( ( 2 ^ ( 2nd ` W ) ) x. ( 1st ` W ) ) e. _V
12	7, 9, 10, 11	ovmpt2 6433	. . 3 |- ( ( ( 1st ` W ) e. J /\ ( 2nd ` W ) e. NN0 ) -> ( ( 1st ` W ) F ( 2nd ` W ) ) = ( ( 2 ^ ( 2nd ` W ) ) x. ( 1st ` W ) ) )
13	6, 12	syl 16	. 2 |- ( W e. ( J X. NN0 ) -> ( ( 1st ` W ) F ( 2nd ` W ) ) = ( ( 2 ^ ( 2nd ` W ) ) x. ( 1st ` W ) ) )
14	2, 4, 13	3eqtr2d 2514	1 |- ( W e. ( J X. NN0 ) -> ( F ` W ) = ( ( 2 ^ ( 2nd ` W ) ) x. ( 1st ` W ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1379   e. wcel 1767   {crab 2821   <.cop 4039   class class class wbr 4453   X. cxp 5003   ` cfv 5594  (class class class)co 6295   |-> cmpt2 6297   1stc1st 6793   2ndc2nd 6794   x. cmul 9509   NNcn 10548   2c2 10597   NN0cn0 10807   ^cexp 12146   || cdivides 13864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796

This theorem is referenced by:  eulerpartlemgvv  28140  eulerpartlemgh  28142  eulerpartlemgs2  28144