Theorem chrval 20067

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 6344	. . . . . 6 ⊢ (𝑟 = 𝑅 → (od‘𝑟) = (od‘𝑅))
3		chrval.o	. . . . . 6 ⊢ 𝑂 = (od‘𝑅)
4	2, 3	syl6eqr 2804	. . . . 5 ⊢ (𝑟 = 𝑅 → (od‘𝑟) = 𝑂)
5		fveq2 6344	. . . . . 6 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅))
6		chrval.u	. . . . . 6 ⊢ 1 = (1r‘𝑅)
7	5, 6	syl6eqr 2804	. . . . 5 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 )
8	4, 7	fveq12d 6350	. . . 4 ⊢ (𝑟 = 𝑅 → ((od‘𝑟)‘(1r‘𝑟)) = (𝑂‘ 1 ))
9		df-chr 20048	. . . 4 ⊢ chr = (𝑟 ∈ V ↦ ((od‘𝑟)‘(1r‘𝑟)))
10		fvex 6354	. . . 4 ⊢ (𝑂‘ 1 ) ∈ V
11	8, 9, 10	fvmpt 6436	. . 3 ⊢ (𝑅 ∈ V → (chr‘𝑅) = (𝑂‘ 1 ))
12		fvprc 6338	. . . 4 ⊢ (¬ 𝑅 ∈ V → (chr‘𝑅) = ∅)
13		fvprc 6338	. . . . . . 7 ⊢ (¬ 𝑅 ∈ V → (od‘𝑅) = ∅)
14	3, 13	syl5eq 2798	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → 𝑂 = ∅)
15	14	fveq1d 6346	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (𝑂‘ 1 ) = (∅‘ 1 ))
16		0fv 6380	. . . . 5 ⊢ (∅‘ 1 ) = ∅
17	15, 16	syl6eq 2802	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑂‘ 1 ) = ∅)
18	12, 17	eqtr4d 2789	. . 3 ⊢ (¬ 𝑅 ∈ V → (chr‘𝑅) = (𝑂‘ 1 ))
19	11, 18	pm2.61i 176	. 2 ⊢ (chr‘𝑅) = (𝑂‘ 1 )
20	1, 19	eqtr2i 2775	1 ⊢ (𝑂‘ 1 ) = 𝐶

Syntax hints:  ¬ wn 3   = wceq 1624   ∈ wcel 2131  Vcvv 3332  ∅c0 4050  ‘cfv 6041  odcod 18136  1_rcur 18693  chrcchr 20044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-iota 6004  df-fun 6043  df-fv 6049  df-chr 20048

This theorem is referenced by:  chrcl  20068  chrid  20069  chrdvds  20070  chrcong  20071  subrgchr  30095  ofldchr  30115  zrhchr  30321