Theorem oddpwdcv 26905

Description: Lemma for eulerpart 26932: 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 6726	. . 3 |- ( W e. ( J X. NN0 ) -> W = <. ( 1st ` W ) , ( 2nd ` W ) >. )
2	1	fveq2d 5806	. 2 |- ( W e. ( J X. NN0 ) -> ( F ` W ) = ( F ` <. ( 1st ` W ) , ( 2nd ` W ) >. ) )
3		df-ov 6206	. . 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 6721	. . . 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 6211	. . . 4 |- ( x = ( 1st ` W ) -> ( ( 2 ^ y ) x. x ) = ( ( 2 ^ y ) x. ( 1st ` W ) ) )
8		oveq2 6211	. . . . 5 |- ( y = ( 2nd ` W ) -> ( 2 ^ y ) = ( 2 ^ ( 2nd ` W ) ) )
9	8	oveq1d 6218	. . . 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 6228	. . . 4 |- ( ( 2 ^ ( 2nd ` W ) ) x. ( 1st ` W ) ) e. _V
12	7, 9, 10, 11	ovmpt2 6339	. . 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 2501	1 |- ( W e. ( J X. NN0 ) -> ( F ` W ) = ( ( 2 ^ ( 2nd ` W ) ) x. ( 1st ` W ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1370   e. wcel 1758   {crab 2803   <.cop 3994   class class class wbr 4403   X. cxp 4949   ` cfv 5529  (class class class)co 6203   |-> cmpt2 6205   1stc1st 6688   2ndc2nd 6689   x. cmul 9402   NNcn 10437   2c2 10486   NN0cn0 10694   ^cexp 11986   || cdivides 13657

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-iota 5492  df-fun 5531  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691

This theorem is referenced by:  eulerpartlemgvv  26926  eulerpartlemgh  26928  eulerpartlemgs2  26930