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Theorem chrval 17961
Description: Definition substitution of the ring characteristic. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypotheses
Ref Expression
chrval.o  |-  O  =  ( od `  R
)
chrval.u  |-  .1.  =  ( 1r `  R )
chrval.c  |-  C  =  (chr `  R )
Assertion
Ref Expression
chrval  |-  ( O `
 .1.  )  =  C

Proof of Theorem chrval
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 chrval.c . 2  |-  C  =  (chr `  R )
2 fveq2 5696 . . . . . 6  |-  ( r  =  R  ->  ( od `  r )  =  ( od `  R
) )
3 chrval.o . . . . . 6  |-  O  =  ( od `  R
)
42, 3syl6eqr 2493 . . . . 5  |-  ( r  =  R  ->  ( od `  r )  =  O )
5 fveq2 5696 . . . . . 6  |-  ( r  =  R  ->  ( 1r `  r )  =  ( 1r `  R
) )
6 chrval.u . . . . . 6  |-  .1.  =  ( 1r `  R )
75, 6syl6eqr 2493 . . . . 5  |-  ( r  =  R  ->  ( 1r `  r )  =  .1.  )
84, 7fveq12d 5702 . . . 4  |-  ( r  =  R  ->  (
( od `  r
) `  ( 1r `  r ) )  =  ( O `  .1.  ) )
9 df-chr 17942 . . . 4  |- chr  =  ( r  e.  _V  |->  ( ( od `  r
) `  ( 1r `  r ) ) )
10 fvex 5706 . . . 4  |-  ( O `
 .1.  )  e. 
_V
118, 9, 10fvmpt 5779 . . 3  |-  ( R  e.  _V  ->  (chr `  R )  =  ( O `  .1.  )
)
12 fvprc 5690 . . . 4  |-  ( -.  R  e.  _V  ->  (chr
`  R )  =  (/) )
13 fvprc 5690 . . . . . . 7  |-  ( -.  R  e.  _V  ->  ( od `  R )  =  (/) )
143, 13syl5eq 2487 . . . . . 6  |-  ( -.  R  e.  _V  ->  O  =  (/) )
1514fveq1d 5698 . . . . 5  |-  ( -.  R  e.  _V  ->  ( O `  .1.  )  =  ( (/) `  .1.  ) )
16 0fv 5728 . . . . 5  |-  ( (/) `  .1.  )  =  (/)
1715, 16syl6eq 2491 . . . 4  |-  ( -.  R  e.  _V  ->  ( O `  .1.  )  =  (/) )
1812, 17eqtr4d 2478 . . 3  |-  ( -.  R  e.  _V  ->  (chr
`  R )  =  ( O `  .1.  ) )
1911, 18pm2.61i 164 . 2  |-  (chr `  R )  =  ( O `  .1.  )
201, 19eqtr2i 2464 1  |-  ( O `
 .1.  )  =  C
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1369    e. wcel 1756   _Vcvv 2977   (/)c0 3642   ` cfv 5423   odcod 16033   1rcur 16608  chrcchr 17938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5386  df-fun 5425  df-fv 5431  df-chr 17942
This theorem is referenced by:  chrcl  17962  chrid  17963  chrdvds  17964  chrcong  17965  subrgchr  26267  ofldchr  26287  zrhchr  26410
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