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Theorem chrval 21453
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 6897 . . . . . 6 (𝑟 = 𝑅 → (od‘𝑟) = (od‘𝑅))
3 chrval.o . . . . . 6 𝑂 = (od‘𝑅)
42, 3eqtr4di 2786 . . . . 5 (𝑟 = 𝑅 → (od‘𝑟) = 𝑂)
5 fveq2 6897 . . . . . 6 (𝑟 = 𝑅 → (1r𝑟) = (1r𝑅))
6 chrval.u . . . . . 6 1 = (1r𝑅)
75, 6eqtr4di 2786 . . . . 5 (𝑟 = 𝑅 → (1r𝑟) = 1 )
84, 7fveq12d 6904 . . . 4 (𝑟 = 𝑅 → ((od‘𝑟)‘(1r𝑟)) = (𝑂1 ))
9 df-chr 21431 . . . 4 chr = (𝑟 ∈ V ↦ ((od‘𝑟)‘(1r𝑟)))
10 fvex 6910 . . . 4 (𝑂1 ) ∈ V
118, 9, 10fvmpt 7005 . . 3 (𝑅 ∈ V → (chr‘𝑅) = (𝑂1 ))
12 fvprc 6889 . . . 4 𝑅 ∈ V → (chr‘𝑅) = ∅)
13 fvprc 6889 . . . . . . 7 𝑅 ∈ V → (od‘𝑅) = ∅)
143, 13eqtrid 2780 . . . . . 6 𝑅 ∈ V → 𝑂 = ∅)
1514fveq1d 6899 . . . . 5 𝑅 ∈ V → (𝑂1 ) = (∅‘ 1 ))
16 0fv 6941 . . . . 5 (∅‘ 1 ) = ∅
1715, 16eqtrdi 2784 . . . 4 𝑅 ∈ V → (𝑂1 ) = ∅)
1812, 17eqtr4d 2771 . . 3 𝑅 ∈ V → (chr‘𝑅) = (𝑂1 ))
1911, 18pm2.61i 182 . 2 (chr‘𝑅) = (𝑂1 )
201, 19eqtr2i 2757 1 (𝑂1 ) = 𝐶
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1534  wcel 2099  Vcvv 3471  c0 4323  cfv 6548  odcod 19479  1rcur 20121  chrcchr 21427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-chr 21431
This theorem is referenced by:  chrcl  21454  chrid  21455  chrdvds  21456  chrcong  21457  dvdschrmulg  21458  ply1chr  22225  subrgchr  32958  ofldchr  33042  zrhchr  33577
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