Theorem chrval 19692

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.

Step	Hyp	Ref	Expression
1		chrval.c	. 2 ⊢ 𝐶 = (chr‘𝑅)
2		fveq2 6103	. . . . . 6 ⊢ (𝑟 = 𝑅 → (od‘𝑟) = (od‘𝑅))
3		chrval.o	. . . . . 6 ⊢ 𝑂 = (od‘𝑅)
4	2, 3	syl6eqr 2662	. . . . 5 ⊢ (𝑟 = 𝑅 → (od‘𝑟) = 𝑂)
5		fveq2 6103	. . . . . 6 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅))
6		chrval.u	. . . . . 6 ⊢ 1 = (1r‘𝑅)
7	5, 6	syl6eqr 2662	. . . . 5 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 )
8	4, 7	fveq12d 6109	. . . 4 ⊢ (𝑟 = 𝑅 → ((od‘𝑟)‘(1r‘𝑟)) = (𝑂‘ 1 ))
9		df-chr 19673	. . . 4 ⊢ chr = (𝑟 ∈ V ↦ ((od‘𝑟)‘(1r‘𝑟)))
10		fvex 6113	. . . 4 ⊢ (𝑂‘ 1 ) ∈ V
11	8, 9, 10	fvmpt 6191	. . 3 ⊢ (𝑅 ∈ V → (chr‘𝑅) = (𝑂‘ 1 ))
12		fvprc 6097	. . . 4 ⊢ (¬ 𝑅 ∈ V → (chr‘𝑅) = ∅)
13		fvprc 6097	. . . . . . 7 ⊢ (¬ 𝑅 ∈ V → (od‘𝑅) = ∅)
14	3, 13	syl5eq 2656	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → 𝑂 = ∅)
15	14	fveq1d 6105	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (𝑂‘ 1 ) = (∅‘ 1 ))
16		0fv 6137	. . . . 5 ⊢ (∅‘ 1 ) = ∅
17	15, 16	syl6eq 2660	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑂‘ 1 ) = ∅)
18	12, 17	eqtr4d 2647	. . 3 ⊢ (¬ 𝑅 ∈ V → (chr‘𝑅) = (𝑂‘ 1 ))
19	11, 18	pm2.61i 175	. 2 ⊢ (chr‘𝑅) = (𝑂‘ 1 )
20	1, 19	eqtr2i 2633	1 ⊢ (𝑂‘ 1 ) = 𝐶

Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  ‘cfv 5804  odcod 17767  1_rcur 18324  chrcchr 19669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-chr 19673

This theorem is referenced by:  chrcl  19693  chrid  19694  chrdvds  19695  chrcong  19696  subrgchr  29125  ofldchr  29145  zrhchr  29348