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Theorem chrval 21556
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 6907 . . . . . 6 (𝑟 = 𝑅 → (od‘𝑟) = (od‘𝑅))
3 chrval.o . . . . . 6 𝑂 = (od‘𝑅)
42, 3eqtr4di 2793 . . . . 5 (𝑟 = 𝑅 → (od‘𝑟) = 𝑂)
5 fveq2 6907 . . . . . 6 (𝑟 = 𝑅 → (1r𝑟) = (1r𝑅))
6 chrval.u . . . . . 6 1 = (1r𝑅)
75, 6eqtr4di 2793 . . . . 5 (𝑟 = 𝑅 → (1r𝑟) = 1 )
84, 7fveq12d 6914 . . . 4 (𝑟 = 𝑅 → ((od‘𝑟)‘(1r𝑟)) = (𝑂1 ))
9 df-chr 21534 . . . 4 chr = (𝑟 ∈ V ↦ ((od‘𝑟)‘(1r𝑟)))
10 fvex 6920 . . . 4 (𝑂1 ) ∈ V
118, 9, 10fvmpt 7016 . . 3 (𝑅 ∈ V → (chr‘𝑅) = (𝑂1 ))
12 fvprc 6899 . . . 4 𝑅 ∈ V → (chr‘𝑅) = ∅)
13 fvprc 6899 . . . . . . 7 𝑅 ∈ V → (od‘𝑅) = ∅)
143, 13eqtrid 2787 . . . . . 6 𝑅 ∈ V → 𝑂 = ∅)
1514fveq1d 6909 . . . . 5 𝑅 ∈ V → (𝑂1 ) = (∅‘ 1 ))
16 0fv 6951 . . . . 5 (∅‘ 1 ) = ∅
1715, 16eqtrdi 2791 . . . 4 𝑅 ∈ V → (𝑂1 ) = ∅)
1812, 17eqtr4d 2778 . . 3 𝑅 ∈ V → (chr‘𝑅) = (𝑂1 ))
1911, 18pm2.61i 182 . 2 (chr‘𝑅) = (𝑂1 )
201, 19eqtr2i 2764 1 (𝑂1 ) = 𝐶
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339  cfv 6563  odcod 19557  1rcur 20199  chrcchr 21530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-chr 21534
This theorem is referenced by:  chrcl  21557  chrid  21558  chrdvds  21559  chrcong  21560  dvdschrmulg  21561  ply1chr  22326  subrgchr  33227  ofldchr  33324  zrhchr  33937
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