Theorem oddpwdcv 30417

Description: Lemma for eulerpart 30444: 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 7205	. . 3 |- ( W e. ( J X. NN0 ) -> W = <. ( 1st ` W ) , ( 2nd ` W ) >. )
2	1	fveq2d 6195	. 2 |- ( W e. ( J X. NN0 ) -> ( F ` W ) = ( F ` <. ( 1st ` W ) , ( 2nd ` W ) >. ) )
3		df-ov 6653	. . 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 7200	. . . 4 |- ( W e. ( J X. NN0 ) <-> ( W = <. ( 1st ` W ) , ( 2nd ` W ) >. /\ ( ( 1st ` W ) e. J /\ ( 2nd ` W ) e. NN0 ) ) )
6	5	simprbi 480	. . 3 |- ( W e. ( J X. NN0 ) -> ( ( 1st ` W ) e. J /\ ( 2nd ` W ) e. NN0 ) )
7		oveq2 6658	. . . 4 |- ( x = ( 1st ` W ) -> ( ( 2 ^ y ) x. x ) = ( ( 2 ^ y ) x. ( 1st ` W ) ) )
8		oveq2 6658	. . . . 5 |- ( y = ( 2nd ` W ) -> ( 2 ^ y ) = ( 2 ^ ( 2nd ` W ) ) )
9	8	oveq1d 6665	. . . 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 6678	. . . 4 |- ( ( 2 ^ ( 2nd ` W ) ) x. ( 1st ` W ) ) e. _V
12	7, 9, 10, 11	ovmpt2 6796	. . 3 |- ( ( ( 1st ` W ) e. J /\ ( 2nd ` W ) e. NN0 ) -> ( ( 1st ` W ) F ( 2nd ` W ) ) = ( ( 2 ^ ( 2nd ` W ) ) x. ( 1st ` W ) ) )
13	6, 12	syl 17	. 2 |- ( W e. ( J X. NN0 ) -> ( ( 1st ` W ) F ( 2nd ` W ) ) = ( ( 2 ^ ( 2nd ` W ) ) x. ( 1st ` W ) ) )
14	2, 4, 13	3eqtr2d 2662	1 |- ( W e. ( J X. NN0 ) -> ( F ` W ) = ( ( 2 ^ ( 2nd ` W ) ) x. ( 1st ` W ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   <.cop 4183   class class class wbr 4653   X. cxp 5112   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   x. cmul 9941   NNcn 11020   2c2 11070   NN0cn0 11292   ^cexp 12860   || cdvds 14983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169

This theorem is referenced by:  eulerpartlemgvv  30438  eulerpartlemgh  30440  eulerpartlemgs2  30442