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Theorem chrval 16495
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 5541 . . . . . 6  |-  ( r  =  R  ->  ( od `  r )  =  ( od `  R
) )
3 chrval.o . . . . . 6  |-  O  =  ( od `  R
)
42, 3syl6eqr 2346 . . . . 5  |-  ( r  =  R  ->  ( od `  r )  =  O )
5 fveq2 5541 . . . . . 6  |-  ( r  =  R  ->  ( 1r `  r )  =  ( 1r `  R
) )
6 chrval.u . . . . . 6  |-  .1.  =  ( 1r `  R )
75, 6syl6eqr 2346 . . . . 5  |-  ( r  =  R  ->  ( 1r `  r )  =  .1.  )
84, 7fveq12d 5547 . . . 4  |-  ( r  =  R  ->  (
( od `  r
) `  ( 1r `  r ) )  =  ( O `  .1.  ) )
9 df-chr 16473 . . . 4  |- chr  =  ( r  e.  _V  |->  ( ( od `  r
) `  ( 1r `  r ) ) )
10 fvex 5555 . . . 4  |-  ( O `
 .1.  )  e. 
_V
118, 9, 10fvmpt 5618 . . 3  |-  ( R  e.  _V  ->  (chr `  R )  =  ( O `  .1.  )
)
12 fvprc 5535 . . . 4  |-  ( -.  R  e.  _V  ->  (chr
`  R )  =  (/) )
13 fvprc 5535 . . . . . . 7  |-  ( -.  R  e.  _V  ->  ( od `  R )  =  (/) )
143, 13syl5eq 2340 . . . . . 6  |-  ( -.  R  e.  _V  ->  O  =  (/) )
1514fveq1d 5543 . . . . 5  |-  ( -.  R  e.  _V  ->  ( O `  .1.  )  =  ( (/) `  .1.  ) )
16 fv01 5575 . . . . 5  |-  ( (/) `  .1.  )  =  (/)
1715, 16syl6eq 2344 . . . 4  |-  ( -.  R  e.  _V  ->  ( O `  .1.  )  =  (/) )
1812, 17eqtr4d 2331 . . 3  |-  ( -.  R  e.  _V  ->  (chr
`  R )  =  ( O `  .1.  ) )
1911, 18pm2.61i 156 . 2  |-  (chr `  R )  =  ( O `  .1.  )
201, 19eqtr2i 2317 1  |-  ( O `
 .1.  )  =  C
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468   ` cfv 5271   odcod 14856   1rcur 15355  chrcchr 16469
This theorem is referenced by:  chrcl  16496  chrid  16497  chrdvds  16498  chrcong  16499
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-chr 16473
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