Theorem chrval 21561

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 6920	. . . . . 6 ⊢ (𝑟 = 𝑅 → (od‘𝑟) = (od‘𝑅))
3		chrval.o	. . . . . 6 ⊢ 𝑂 = (od‘𝑅)
4	2, 3	eqtr4di 2798	. . . . 5 ⊢ (𝑟 = 𝑅 → (od‘𝑟) = 𝑂)
5		fveq2 6920	. . . . . 6 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅))
6		chrval.u	. . . . . 6 ⊢ 1 = (1r‘𝑅)
7	5, 6	eqtr4di 2798	. . . . 5 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 )
8	4, 7	fveq12d 6927	. . . 4 ⊢ (𝑟 = 𝑅 → ((od‘𝑟)‘(1r‘𝑟)) = (𝑂‘ 1 ))
9		df-chr 21539	. . . 4 ⊢ chr = (𝑟 ∈ V ↦ ((od‘𝑟)‘(1r‘𝑟)))
10		fvex 6933	. . . 4 ⊢ (𝑂‘ 1 ) ∈ V
11	8, 9, 10	fvmpt 7029	. . 3 ⊢ (𝑅 ∈ V → (chr‘𝑅) = (𝑂‘ 1 ))
12		fvprc 6912	. . . 4 ⊢ (¬ 𝑅 ∈ V → (chr‘𝑅) = ∅)
13		fvprc 6912	. . . . . . 7 ⊢ (¬ 𝑅 ∈ V → (od‘𝑅) = ∅)
14	3, 13	eqtrid 2792	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → 𝑂 = ∅)
15	14	fveq1d 6922	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (𝑂‘ 1 ) = (∅‘ 1 ))
16		0fv 6964	. . . . 5 ⊢ (∅‘ 1 ) = ∅
17	15, 16	eqtrdi 2796	. . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑂‘ 1 ) = ∅)
18	12, 17	eqtr4d 2783	. . 3 ⊢ (¬ 𝑅 ∈ V → (chr‘𝑅) = (𝑂‘ 1 ))
19	11, 18	pm2.61i 182	. 2 ⊢ (chr‘𝑅) = (𝑂‘ 1 )
20	1, 19	eqtr2i 2769	1 ⊢ (𝑂‘ 1 ) = 𝐶

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2108  Vcvv 3488  ∅c0 4352  ‘cfv 6573  odcod 19566  1_rcur 20208  chrcchr 21535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-chr 21539

This theorem is referenced by:  chrcl  21562  chrid  21563  chrdvds  21564  chrcong  21565  dvdschrmulg  21566  ply1chr  22331  subrgchr  33217  ofldchr  33309  zrhchr  33922