Theorem opprnegg 13206

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 2177	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
3	1, 2	opprbasg 13200	. . 3 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑂))
4		eqid 2177	. . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅)
5	1, 4	oppraddg 13201	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘𝑂))
6	5	oveqd 5891	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (𝑦(+g‘𝑅)𝑥) = (𝑦(+g‘𝑂)𝑥))
7		eqid 2177	. . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅)
8	1, 7	oppr0g 13204	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (0g‘𝑅) = (0g‘𝑂))
9	6, 8	eqeq12d 2192	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ((𝑦(+g‘𝑅)𝑥) = (0g‘𝑅) ↔ (𝑦(+g‘𝑂)𝑥) = (0g‘𝑂)))
10	3, 9	riotaeqbidv 5833	. . 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 4085	. 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 12869	. 2 ⊢ (𝑅 ∈ 𝑉 → 𝑁 = (𝑥 ∈ (Base‘𝑅) ↦ (℩𝑦 ∈ (Base‘𝑅)(𝑦(+g‘𝑅)𝑥) = (0g‘𝑅))))
14	1	opprex 13198	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑂 ∈ V)
15		eqid 2177	. . . 4 ⊢ (Base‘𝑂) = (Base‘𝑂)
16		eqid 2177	. . . 4 ⊢ (+g‘𝑂) = (+g‘𝑂)
17		eqid 2177	. . . 4 ⊢ (0g‘𝑂) = (0g‘𝑂)
18		eqid 2177	. . . 4 ⊢ (invg‘𝑂) = (invg‘𝑂)
19	15, 16, 17, 18	grpinvfvalg 12869	. . 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 2220	1 ⊢ (𝑅 ∈ 𝑉 → 𝑁 = (invg‘𝑂))

Syntax hints:   → wi 4   = wceq 1353   ∈ wcel 2148  Vcvv 2737   ↦ cmpt 4064  ‘cfv 5216  ℩crio 5829  (class class class)co 5874  Basecbs 12456  +_gcplusg 12530  0_gc0g 12695  inv_gcminusg 12832  opp_rcoppr 13192

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-pre-ltirr 7922  ax-pre-lttrn 7924  ax-pre-ltadd 7926

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-tpos 6245  df-pnf 7992  df-mnf 7993  df-ltxr 7995  df-inn 8918  df-2 8976  df-3 8977  df-ndx 12459  df-slot 12460  df-base 12462  df-sets 12463  df-plusg 12543  df-mulr 12544  df-0g 12697  df-minusg 12835  df-oppr 13193

This theorem is referenced by:  unitnegcl  13252