Theorem oppr0g 13713

Description: Additive identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)

Hypotheses

Ref	Expression
opprbas.1	⊢ 𝑂 = (oppr‘𝑅)
oppr0.2	⊢ 0 = (0g‘𝑅)

Assertion

Ref	Expression
oppr0g	⊢ (𝑅 ∈ 𝑉 → 0 = (0g‘𝑂))

Proof of Theorem oppr0g

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

Step	Hyp	Ref	Expression
1		opprbas.1	. . . . . 6 ⊢ 𝑂 = (oppr‘𝑅)
2		eqid 2196	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
3	1, 2	opprbasg 13707	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑂))
4	3	eleq2d 2266	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑦 ∈ (Base‘𝑅) ↔ 𝑦 ∈ (Base‘𝑂)))
5		eqid 2196	. . . . . . . . 9 ⊢ (+g‘𝑅) = (+g‘𝑅)
6	1, 5	oppraddg 13708	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘𝑂))
7	6	oveqd 5942	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (𝑦(+g‘𝑅)𝑥) = (𝑦(+g‘𝑂)𝑥))
8	7	eqeq1d 2205	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → ((𝑦(+g‘𝑅)𝑥) = 𝑥 ↔ (𝑦(+g‘𝑂)𝑥) = 𝑥))
9	6	oveqd 5942	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑂)𝑦))
10	9	eqeq1d 2205	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → ((𝑥(+g‘𝑅)𝑦) = 𝑥 ↔ (𝑥(+g‘𝑂)𝑦) = 𝑥))
11	8, 10	anbi12d 473	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (((𝑦(+g‘𝑅)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑅)𝑦) = 𝑥) ↔ ((𝑦(+g‘𝑂)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑂)𝑦) = 𝑥)))
12	3, 11	raleqbidv 2709	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (∀𝑥 ∈ (Base‘𝑅)((𝑦(+g‘𝑅)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑅)𝑦) = 𝑥) ↔ ∀𝑥 ∈ (Base‘𝑂)((𝑦(+g‘𝑂)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑂)𝑦) = 𝑥)))
13	4, 12	anbi12d 473	. . 3 ⊢ (𝑅 ∈ 𝑉 → ((𝑦 ∈ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)((𝑦(+g‘𝑅)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑅)𝑦) = 𝑥)) ↔ (𝑦 ∈ (Base‘𝑂) ∧ ∀𝑥 ∈ (Base‘𝑂)((𝑦(+g‘𝑂)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑂)𝑦) = 𝑥))))
14	13	iotabidv 5242	. 2 ⊢ (𝑅 ∈ 𝑉 → (℩𝑦(𝑦 ∈ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)((𝑦(+g‘𝑅)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑅)𝑦) = 𝑥))) = (℩𝑦(𝑦 ∈ (Base‘𝑂) ∧ ∀𝑥 ∈ (Base‘𝑂)((𝑦(+g‘𝑂)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑂)𝑦) = 𝑥))))
15		oppr0.2	. . 3 ⊢ 0 = (0g‘𝑅)
16	2, 5, 15	grpidvalg 13075	. 2 ⊢ (𝑅 ∈ 𝑉 → 0 = (℩𝑦(𝑦 ∈ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)((𝑦(+g‘𝑅)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑅)𝑦) = 𝑥))))
17	1	opprex 13705	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑂 ∈ V)
18		eqid 2196	. . . 4 ⊢ (Base‘𝑂) = (Base‘𝑂)
19		eqid 2196	. . . 4 ⊢ (+g‘𝑂) = (+g‘𝑂)
20		eqid 2196	. . . 4 ⊢ (0g‘𝑂) = (0g‘𝑂)
21	18, 19, 20	grpidvalg 13075	. . 3 ⊢ (𝑂 ∈ V → (0g‘𝑂) = (℩𝑦(𝑦 ∈ (Base‘𝑂) ∧ ∀𝑥 ∈ (Base‘𝑂)((𝑦(+g‘𝑂)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑂)𝑦) = 𝑥))))
22	17, 21	syl 14	. 2 ⊢ (𝑅 ∈ 𝑉 → (0g‘𝑂) = (℩𝑦(𝑦 ∈ (Base‘𝑂) ∧ ∀𝑥 ∈ (Base‘𝑂)((𝑦(+g‘𝑂)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑂)𝑦) = 𝑥))))
23	14, 16, 22	3eqtr4d 2239	1 ⊢ (𝑅 ∈ 𝑉 → 0 = (0g‘𝑂))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167  ∀wral 2475  Vcvv 2763  ℩cio 5218  ‘cfv 5259  (class class class)co 5925  Basecbs 12703  +_gcplusg 12780  0_gc0g 12958  opp_rcoppr 13699

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-pre-ltirr 8008  ax-pre-lttrn 8010  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-tpos 6312  df-pnf 8080  df-mnf 8081  df-ltxr 8083  df-inn 9008  df-2 9066  df-3 9067  df-ndx 12706  df-slot 12707  df-base 12709  df-sets 12710  df-plusg 12793  df-mulr 12794  df-0g 12960  df-oppr 13700

This theorem is referenced by:  opprnegg  13715  opprnzrbg  13817  opprdomnbg  13906  ridl0  14142  2idlcpblrng  14155