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Theorem chrval 20667
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 = (1r𝑅)
chrval.c 𝐶 = (chr‘𝑅)
Assertion
Ref Expression
chrval (𝑂1 ) = 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 6659 . . . . . 6 (𝑟 = 𝑅 → (od‘𝑟) = (od‘𝑅))
3 chrval.o . . . . . 6 𝑂 = (od‘𝑅)
42, 3syl6eqr 2877 . . . . 5 (𝑟 = 𝑅 → (od‘𝑟) = 𝑂)
5 fveq2 6659 . . . . . 6 (𝑟 = 𝑅 → (1r𝑟) = (1r𝑅))
6 chrval.u . . . . . 6 1 = (1r𝑅)
75, 6syl6eqr 2877 . . . . 5 (𝑟 = 𝑅 → (1r𝑟) = 1 )
84, 7fveq12d 6666 . . . 4 (𝑟 = 𝑅 → ((od‘𝑟)‘(1r𝑟)) = (𝑂1 ))
9 df-chr 20648 . . . 4 chr = (𝑟 ∈ V ↦ ((od‘𝑟)‘(1r𝑟)))
10 fvex 6672 . . . 4 (𝑂1 ) ∈ V
118, 9, 10fvmpt 6757 . . 3 (𝑅 ∈ V → (chr‘𝑅) = (𝑂1 ))
12 fvprc 6652 . . . 4 𝑅 ∈ V → (chr‘𝑅) = ∅)
13 fvprc 6652 . . . . . . 7 𝑅 ∈ V → (od‘𝑅) = ∅)
143, 13syl5eq 2871 . . . . . 6 𝑅 ∈ V → 𝑂 = ∅)
1514fveq1d 6661 . . . . 5 𝑅 ∈ V → (𝑂1 ) = (∅‘ 1 ))
16 0fv 6698 . . . . 5 (∅‘ 1 ) = ∅
1715, 16syl6eq 2875 . . . 4 𝑅 ∈ V → (𝑂1 ) = ∅)
1812, 17eqtr4d 2862 . . 3 𝑅 ∈ V → (chr‘𝑅) = (𝑂1 ))
1911, 18pm2.61i 185 . 2 (chr‘𝑅) = (𝑂1 )
201, 19eqtr2i 2848 1 (𝑂1 ) = 𝐶
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1538  wcel 2115  Vcvv 3480  c0 4276  cfv 6344  odcod 18650  1rcur 19249  chrcchr 20644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-iota 6303  df-fun 6346  df-fv 6352  df-chr 20648
This theorem is referenced by:  chrcl  20668  chrid  20669  chrdvds  20670  chrcong  20671  dvdschrmulg  30885  subrgchr  30892  ofldchr  30914  zrhchr  31244
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