Theorem oppr0g 14084

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 2229	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
3	1, 2	opprbasg 14078	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑂))
4	3	eleq2d 2299	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑦 ∈ (Base‘𝑅) ↔ 𝑦 ∈ (Base‘𝑂)))
5		eqid 2229	. . . . . . . . 9 ⊢ (+g‘𝑅) = (+g‘𝑅)
6	1, 5	oppraddg 14079	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘𝑂))
7	6	oveqd 6030	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (𝑦(+g‘𝑅)𝑥) = (𝑦(+g‘𝑂)𝑥))
8	7	eqeq1d 2238	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → ((𝑦(+g‘𝑅)𝑥) = 𝑥 ↔ (𝑦(+g‘𝑂)𝑥) = 𝑥))
9	6	oveqd 6030	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑂)𝑦))
10	9	eqeq1d 2238	. . . . . 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 2744	. . . 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 5307	. 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 13446	. 2 ⊢ (𝑅 ∈ 𝑉 → 0 = (℩𝑦(𝑦 ∈ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)((𝑦(+g‘𝑅)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑅)𝑦) = 𝑥))))
17	1	opprex 14076	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑂 ∈ V)
18		eqid 2229	. . . 4 ⊢ (Base‘𝑂) = (Base‘𝑂)
19		eqid 2229	. . . 4 ⊢ (+g‘𝑂) = (+g‘𝑂)
20		eqid 2229	. . . 4 ⊢ (0g‘𝑂) = (0g‘𝑂)
21	18, 19, 20	grpidvalg 13446	. . 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 2272	1 ⊢ (𝑅 ∈ 𝑉 → 0 = (0g‘𝑂))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  ∀wral 2508  Vcvv 2800  ℩cio 5282  ‘cfv 5324  (class class class)co 6013  Basecbs 13072  +_gcplusg 13150  0_gc0g 13329  opp_rcoppr 14070

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-pre-ltirr 8134  ax-pre-lttrn 8136  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-tpos 6406  df-pnf 8206  df-mnf 8207  df-ltxr 8209  df-inn 9134  df-2 9192  df-3 9193  df-ndx 13075  df-slot 13076  df-base 13078  df-sets 13079  df-plusg 13163  df-mulr 13164  df-0g 13331  df-oppr 14071

This theorem is referenced by:  opprnegg  14086  opprnzrbg  14189  opprdomnbg  14278  ridl0  14514  2idlcpblrng  14527