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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
StepHypRef Expression
1 1st2nd2 6726 . . 3  |-  ( W  e.  ( J  X.  NN0 )  ->  W  = 
<. ( 1st `  W
) ,  ( 2nd `  W ) >. )
21fveq2d 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
) >. )
43a1i 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 ) ) )
65simprbi 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 ) ) )
98oveq1d 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
127, 9, 10, 11ovmpt2 6339 . . 3  |-  ( ( ( 1st `  W
)  e.  J  /\  ( 2nd `  W )  e.  NN0 )  -> 
( ( 1st `  W
) F ( 2nd `  W ) )  =  ( ( 2 ^ ( 2nd `  W
) )  x.  ( 1st `  W ) ) )
136, 12syl 16 . 2  |-  ( W  e.  ( J  X.  NN0 )  ->  ( ( 1st `  W ) F ( 2nd `  W
) )  =  ( ( 2 ^ ( 2nd `  W ) )  x.  ( 1st `  W
) ) )
142, 4, 133eqtr2d 2501 1  |-  ( W  e.  ( J  X.  NN0 )  ->  ( F `
 W )  =  ( ( 2 ^ ( 2nd `  W
) )  x.  ( 1st `  W ) ) )
Colors of variables: wff setvar class
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
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