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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
StepHypRef Expression
1 1st2nd2 6832 . . 3  |-  ( W  e.  ( J  X.  NN0 )  ->  W  = 
<. ( 1st `  W
) ,  ( 2nd `  W ) >. )
21fveq2d 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
) >. )
43a1i 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 ) ) )
65simprbi 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 ) ) )
98oveq1d 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
127, 9, 10, 11ovmpt2 6433 . . 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 2514 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 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
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