Theorem chrval 21809

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 6877	. . . . . 6 ⊢ (𝑟 = 𝑅 → (od‘𝑟) = (od‘𝑅))
3		chrval.o	. . . . . 6 ⊢ 𝑂 = (od‘𝑅)
4	2, 3	eqtr4di 2814	. . . . 5 ⊢ (𝑟 = 𝑅 → (od‘𝑟) = 𝑂)
5		fveq2 6877	. . . . . 6 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅))
6		chrval.u	. . . . . 6 ⊢ 1 = (1r‘𝑅)
7	5, 6	eqtr4di 2814	. . . . 5 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 )
8	4, 7	fveq12d 6884	. . . 4 ⊢ (𝑟 = 𝑅 → ((od‘𝑟)‘(1r‘𝑟)) = (𝑂‘ 1 ))
9		df-chr 21791	. . . 4 ⊢ chr = (𝑟 ∈ V ↦ ((od‘𝑟)‘(1r‘𝑟)))
10		fvex 6890	. . . 4 ⊢ (𝑂‘ 1 ) ∈ V
11	8, 9, 10	fvmpt 6985	. . 3 ⊢ (𝑅 ∈ V → (chr‘𝑅) = (𝑂‘ 1 ))
12		fvprc 6869	. . . 4 ⊢ (¬ 𝑅 ∈ V → (chr‘𝑅) = ∅)
13		fvprc 6869	. . . . . . 7 ⊢ (¬ 𝑅 ∈ V → (od‘𝑅) = ∅)
14	3, 13	eqtrid 2808	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → 𝑂 = ∅)
15	14	fveq1d 6879	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (𝑂‘ 1 ) = (∅‘ 1 ))
16		0fv 6918	. . . . 5 ⊢ (∅‘ 1 ) = ∅
17	15, 16	eqtrdi 2812	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑂‘ 1 ) = ∅)
18	12, 17	eqtr4d 2799	. . 3 ⊢ (¬ 𝑅 ∈ V → (chr‘𝑅) = (𝑂‘ 1 ))
19	11, 18	pm2.61i 184	. 2 ⊢ (chr‘𝑅) = (𝑂‘ 1 )
20	1, 19	eqtr2i 2785	1 ⊢ (𝑂‘ 1 ) = 𝐶

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2145  Vcvv 3451  ∅c0 4279  ‘cfv 6531  odcod 19718  1_rcur 20387  chrcchr 21787

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6487  df-fun 6533  df-fv 6539  df-chr 21791

This theorem is used by:  chrcl  21810  chrid  21811  chrdvds  21812  chrcong  21813  dvdschrmulg  21814  ofldchr  21862  ply1chr  22604  subrgchr  33779  zrhchr  34588