Theorem opprnegg 14099

Description: The negative function in an opposite ring. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)

Hypotheses

Ref	Expression
opprbas.1	⊢ 𝑂 = (oppr‘𝑅)
opprneg.2	⊢ 𝑁 = (invg‘𝑅)

Assertion

Ref	Expression
opprnegg	⊢ (𝑅 ∈ 𝑉 → 𝑁 = (invg‘𝑂))

Proof of Theorem opprnegg

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opprbas.1	. . . 4 ⊢ 𝑂 = (oppr‘𝑅)
2		eqid 2231	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
3	1, 2	opprbasg 14091	. . 3 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑂))
4		eqid 2231	. . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅)
5	1, 4	oppraddg 14092	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘𝑂))
6	5	oveqd 6035	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (𝑦(+g‘𝑅)𝑥) = (𝑦(+g‘𝑂)𝑥))
7		eqid 2231	. . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅)
8	1, 7	oppr0g 14097	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (0g‘𝑅) = (0g‘𝑂))
9	6, 8	eqeq12d 2246	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ((𝑦(+g‘𝑅)𝑥) = (0g‘𝑅) ↔ (𝑦(+g‘𝑂)𝑥) = (0g‘𝑂)))
10	3, 9	riotaeqbidv 5974	. . 3 ⊢ (𝑅 ∈ 𝑉 → (℩𝑦 ∈ (Base‘𝑅)(𝑦(+g‘𝑅)𝑥) = (0g‘𝑅)) = (℩𝑦 ∈ (Base‘𝑂)(𝑦(+g‘𝑂)𝑥) = (0g‘𝑂)))
11	3, 10	mpteq12dv 4171	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ (Base‘𝑅) ↦ (℩𝑦 ∈ (Base‘𝑅)(𝑦(+g‘𝑅)𝑥) = (0g‘𝑅))) = (𝑥 ∈ (Base‘𝑂) ↦ (℩𝑦 ∈ (Base‘𝑂)(𝑦(+g‘𝑂)𝑥) = (0g‘𝑂))))
12		opprneg.2	. . 3 ⊢ 𝑁 = (invg‘𝑅)
13	2, 4, 7, 12	grpinvfvalg 13627	. 2 ⊢ (𝑅 ∈ 𝑉 → 𝑁 = (𝑥 ∈ (Base‘𝑅) ↦ (℩𝑦 ∈ (Base‘𝑅)(𝑦(+g‘𝑅)𝑥) = (0g‘𝑅))))
14	1	opprex 14089	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑂 ∈ V)
15		eqid 2231	. . . 4 ⊢ (Base‘𝑂) = (Base‘𝑂)
16		eqid 2231	. . . 4 ⊢ (+g‘𝑂) = (+g‘𝑂)
17		eqid 2231	. . . 4 ⊢ (0g‘𝑂) = (0g‘𝑂)
18		eqid 2231	. . . 4 ⊢ (invg‘𝑂) = (invg‘𝑂)
19	15, 16, 17, 18	grpinvfvalg 13627	. . 3 ⊢ (𝑂 ∈ V → (invg‘𝑂) = (𝑥 ∈ (Base‘𝑂) ↦ (℩𝑦 ∈ (Base‘𝑂)(𝑦(+g‘𝑂)𝑥) = (0g‘𝑂))))
20	14, 19	syl 14	. 2 ⊢ (𝑅 ∈ 𝑉 → (invg‘𝑂) = (𝑥 ∈ (Base‘𝑂) ↦ (℩𝑦 ∈ (Base‘𝑂)(𝑦(+g‘𝑂)𝑥) = (0g‘𝑂))))
21	11, 13, 20	3eqtr4d 2274	1 ⊢ (𝑅 ∈ 𝑉 → 𝑁 = (invg‘𝑂))

Syntax hints:   → wi 4   = wceq 1397   ∈ wcel 2202  Vcvv 2802   ↦ cmpt 4150  ‘cfv 5326  ℩crio 5970  (class class class)co 6018  Basecbs 13084  +_gcplusg 13162  0_gc0g 13341  inv_gcminusg 13586  opp_rcoppr 14083

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-pre-ltirr 8144  ax-pre-lttrn 8146  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-tpos 6411  df-pnf 8216  df-mnf 8217  df-ltxr 8219  df-inn 9144  df-2 9202  df-3 9203  df-ndx 13087  df-slot 13088  df-base 13090  df-sets 13091  df-plusg 13175  df-mulr 13176  df-0g 13343  df-minusg 13589  df-oppr 14084

This theorem is referenced by:  unitnegcl  14147