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Theorem chrval 18864
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 5851 . . . . . 6  |-  ( r  =  R  ->  ( od `  r )  =  ( od `  R
) )
3 chrval.o . . . . . 6  |-  O  =  ( od `  R
)
42, 3syl6eqr 2463 . . . . 5  |-  ( r  =  R  ->  ( od `  r )  =  O )
5 fveq2 5851 . . . . . 6  |-  ( r  =  R  ->  ( 1r `  r )  =  ( 1r `  R
) )
6 chrval.u . . . . . 6  |-  .1.  =  ( 1r `  R )
75, 6syl6eqr 2463 . . . . 5  |-  ( r  =  R  ->  ( 1r `  r )  =  .1.  )
84, 7fveq12d 5857 . . . 4  |-  ( r  =  R  ->  (
( od `  r
) `  ( 1r `  r ) )  =  ( O `  .1.  ) )
9 df-chr 18845 . . . 4  |- chr  =  ( r  e.  _V  |->  ( ( od `  r
) `  ( 1r `  r ) ) )
10 fvex 5861 . . . 4  |-  ( O `
 .1.  )  e. 
_V
118, 9, 10fvmpt 5934 . . 3  |-  ( R  e.  _V  ->  (chr `  R )  =  ( O `  .1.  )
)
12 fvprc 5845 . . . 4  |-  ( -.  R  e.  _V  ->  (chr
`  R )  =  (/) )
13 fvprc 5845 . . . . . . 7  |-  ( -.  R  e.  _V  ->  ( od `  R )  =  (/) )
143, 13syl5eq 2457 . . . . . 6  |-  ( -.  R  e.  _V  ->  O  =  (/) )
1514fveq1d 5853 . . . . 5  |-  ( -.  R  e.  _V  ->  ( O `  .1.  )  =  ( (/) `  .1.  ) )
16 0fv 5884 . . . . 5  |-  ( (/) `  .1.  )  =  (/)
1715, 16syl6eq 2461 . . . 4  |-  ( -.  R  e.  _V  ->  ( O `  .1.  )  =  (/) )
1812, 17eqtr4d 2448 . . 3  |-  ( -.  R  e.  _V  ->  (chr
`  R )  =  ( O `  .1.  ) )
1911, 18pm2.61i 166 . 2  |-  (chr `  R )  =  ( O `  .1.  )
201, 19eqtr2i 2434 1  |-  ( O `
 .1.  )  =  C
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1407    e. wcel 1844   _Vcvv 3061   (/)c0 3740   ` cfv 5571   odcod 16875   1rcur 17475  chrcchr 18841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-iota 5535  df-fun 5573  df-fv 5579  df-chr 18845
This theorem is referenced by:  chrcl  18865  chrid  18866  chrdvds  18867  chrcong  18868  subrgchr  28250  ofldchr  28270  zrhchr  28422
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