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Theorem chrval 16806
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 5728 . . . . . 6  |-  ( r  =  R  ->  ( od `  r )  =  ( od `  R
) )
3 chrval.o . . . . . 6  |-  O  =  ( od `  R
)
42, 3syl6eqr 2486 . . . . 5  |-  ( r  =  R  ->  ( od `  r )  =  O )
5 fveq2 5728 . . . . . 6  |-  ( r  =  R  ->  ( 1r `  r )  =  ( 1r `  R
) )
6 chrval.u . . . . . 6  |-  .1.  =  ( 1r `  R )
75, 6syl6eqr 2486 . . . . 5  |-  ( r  =  R  ->  ( 1r `  r )  =  .1.  )
84, 7fveq12d 5734 . . . 4  |-  ( r  =  R  ->  (
( od `  r
) `  ( 1r `  r ) )  =  ( O `  .1.  ) )
9 df-chr 16784 . . . 4  |- chr  =  ( r  e.  _V  |->  ( ( od `  r
) `  ( 1r `  r ) ) )
10 fvex 5742 . . . 4  |-  ( O `
 .1.  )  e. 
_V
118, 9, 10fvmpt 5806 . . 3  |-  ( R  e.  _V  ->  (chr `  R )  =  ( O `  .1.  )
)
12 fvprc 5722 . . . 4  |-  ( -.  R  e.  _V  ->  (chr
`  R )  =  (/) )
13 fvprc 5722 . . . . . . 7  |-  ( -.  R  e.  _V  ->  ( od `  R )  =  (/) )
143, 13syl5eq 2480 . . . . . 6  |-  ( -.  R  e.  _V  ->  O  =  (/) )
1514fveq1d 5730 . . . . 5  |-  ( -.  R  e.  _V  ->  ( O `  .1.  )  =  ( (/) `  .1.  ) )
16 fv01 5763 . . . . 5  |-  ( (/) `  .1.  )  =  (/)
1715, 16syl6eq 2484 . . . 4  |-  ( -.  R  e.  _V  ->  ( O `  .1.  )  =  (/) )
1812, 17eqtr4d 2471 . . 3  |-  ( -.  R  e.  _V  ->  (chr
`  R )  =  ( O `  .1.  ) )
1911, 18pm2.61i 158 . 2  |-  (chr `  R )  =  ( O `  .1.  )
201, 19eqtr2i 2457 1  |-  ( O `
 .1.  )  =  C
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1652    e. wcel 1725   _Vcvv 2956   (/)c0 3628   ` cfv 5454   odcod 15163   1rcur 15662  chrcchr 16780
This theorem is referenced by:  chrcl  16807  chrid  16808  chrdvds  16809  chrcong  16810  subrgchr  24230  ofldchr  24244  zrhchr  24360
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-chr 16784
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