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Theorem chrval 21538
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 6906 . . . . . 6 (𝑟 = 𝑅 → (od‘𝑟) = (od‘𝑅))
3 chrval.o . . . . . 6 𝑂 = (od‘𝑅)
42, 3eqtr4di 2795 . . . . 5 (𝑟 = 𝑅 → (od‘𝑟) = 𝑂)
5 fveq2 6906 . . . . . 6 (𝑟 = 𝑅 → (1r𝑟) = (1r𝑅))
6 chrval.u . . . . . 6 1 = (1r𝑅)
75, 6eqtr4di 2795 . . . . 5 (𝑟 = 𝑅 → (1r𝑟) = 1 )
84, 7fveq12d 6913 . . . 4 (𝑟 = 𝑅 → ((od‘𝑟)‘(1r𝑟)) = (𝑂1 ))
9 df-chr 21516 . . . 4 chr = (𝑟 ∈ V ↦ ((od‘𝑟)‘(1r𝑟)))
10 fvex 6919 . . . 4 (𝑂1 ) ∈ V
118, 9, 10fvmpt 7016 . . 3 (𝑅 ∈ V → (chr‘𝑅) = (𝑂1 ))
12 fvprc 6898 . . . 4 𝑅 ∈ V → (chr‘𝑅) = ∅)
13 fvprc 6898 . . . . . . 7 𝑅 ∈ V → (od‘𝑅) = ∅)
143, 13eqtrid 2789 . . . . . 6 𝑅 ∈ V → 𝑂 = ∅)
1514fveq1d 6908 . . . . 5 𝑅 ∈ V → (𝑂1 ) = (∅‘ 1 ))
16 0fv 6950 . . . . 5 (∅‘ 1 ) = ∅
1715, 16eqtrdi 2793 . . . 4 𝑅 ∈ V → (𝑂1 ) = ∅)
1812, 17eqtr4d 2780 . . 3 𝑅 ∈ V → (chr‘𝑅) = (𝑂1 ))
1911, 18pm2.61i 182 . 2 (chr‘𝑅) = (𝑂1 )
201, 19eqtr2i 2766 1 (𝑂1 ) = 𝐶
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333  cfv 6561  odcod 19542  1rcur 20178  chrcchr 21512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-chr 21516
This theorem is referenced by:  chrcl  21539  chrid  21540  chrdvds  21541  chrcong  21542  dvdschrmulg  21543  ply1chr  22310  subrgchr  33241  ofldchr  33344  zrhchr  33975
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