Theorem chrval 21296

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 6890	. . . . . 6 ⊢ (𝑟 = 𝑅 → (od‘𝑟) = (od‘𝑅))
3		chrval.o	. . . . . 6 ⊢ 𝑂 = (od‘𝑅)
4	2, 3	eqtr4di 2788	. . . . 5 ⊢ (𝑟 = 𝑅 → (od‘𝑟) = 𝑂)
5		fveq2 6890	. . . . . 6 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅))
6		chrval.u	. . . . . 6 ⊢ 1 = (1r‘𝑅)
7	5, 6	eqtr4di 2788	. . . . 5 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 )
8	4, 7	fveq12d 6897	. . . 4 ⊢ (𝑟 = 𝑅 → ((od‘𝑟)‘(1r‘𝑟)) = (𝑂‘ 1 ))
9		df-chr 21274	. . . 4 ⊢ chr = (𝑟 ∈ V ↦ ((od‘𝑟)‘(1r‘𝑟)))
10		fvex 6903	. . . 4 ⊢ (𝑂‘ 1 ) ∈ V
11	8, 9, 10	fvmpt 6997	. . 3 ⊢ (𝑅 ∈ V → (chr‘𝑅) = (𝑂‘ 1 ))
12		fvprc 6882	. . . 4 ⊢ (¬ 𝑅 ∈ V → (chr‘𝑅) = ∅)
13		fvprc 6882	. . . . . . 7 ⊢ (¬ 𝑅 ∈ V → (od‘𝑅) = ∅)
14	3, 13	eqtrid 2782	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → 𝑂 = ∅)
15	14	fveq1d 6892	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (𝑂‘ 1 ) = (∅‘ 1 ))
16		0fv 6934	. . . . 5 ⊢ (∅‘ 1 ) = ∅
17	15, 16	eqtrdi 2786	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑂‘ 1 ) = ∅)
18	12, 17	eqtr4d 2773	. . 3 ⊢ (¬ 𝑅 ∈ V → (chr‘𝑅) = (𝑂‘ 1 ))
19	11, 18	pm2.61i 182	. 2 ⊢ (chr‘𝑅) = (𝑂‘ 1 )
20	1, 19	eqtr2i 2759	1 ⊢ (𝑂‘ 1 ) = 𝐶

Syntax hints:  ¬ wn 3   = wceq 1539   ∈ wcel 2104  Vcvv 3472  ∅c0 4321  ‘cfv 6542  odcod 19433  1_rcur 20075  chrcchr 21270

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-chr 21274

This theorem is referenced by:  chrcl  21297  chrid  21298  chrdvds  21299  chrcong  21300  dvdschrmulg  32650  subrgchr  32656  ofldchr  32702  ply1chr  32935  zrhchr  33254