Theorem oppr0g 13843

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 2205	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
3	1, 2	opprbasg 13837	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑂))
4	3	eleq2d 2275	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑦 ∈ (Base‘𝑅) ↔ 𝑦 ∈ (Base‘𝑂)))
5		eqid 2205	. . . . . . . . 9 ⊢ (+g‘𝑅) = (+g‘𝑅)
6	1, 5	oppraddg 13838	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘𝑂))
7	6	oveqd 5961	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (𝑦(+g‘𝑅)𝑥) = (𝑦(+g‘𝑂)𝑥))
8	7	eqeq1d 2214	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → ((𝑦(+g‘𝑅)𝑥) = 𝑥 ↔ (𝑦(+g‘𝑂)𝑥) = 𝑥))
9	6	oveqd 5961	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑂)𝑦))
10	9	eqeq1d 2214	. . . . . 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 2718	. . . 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 5254	. 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 13205	. 2 ⊢ (𝑅 ∈ 𝑉 → 0 = (℩𝑦(𝑦 ∈ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)((𝑦(+g‘𝑅)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑅)𝑦) = 𝑥))))
17	1	opprex 13835	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑂 ∈ V)
18		eqid 2205	. . . 4 ⊢ (Base‘𝑂) = (Base‘𝑂)
19		eqid 2205	. . . 4 ⊢ (+g‘𝑂) = (+g‘𝑂)
20		eqid 2205	. . . 4 ⊢ (0g‘𝑂) = (0g‘𝑂)
21	18, 19, 20	grpidvalg 13205	. . 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 2248	1 ⊢ (𝑅 ∈ 𝑉 → 0 = (0g‘𝑂))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2176  ∀wral 2484  Vcvv 2772  ℩cio 5230  ‘cfv 5271  (class class class)co 5944  Basecbs 12832  +_gcplusg 12909  0_gc0g 13088  opp_rcoppr 13829

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-pre-ltirr 8037  ax-pre-lttrn 8039  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-tpos 6331  df-pnf 8109  df-mnf 8110  df-ltxr 8112  df-inn 9037  df-2 9095  df-3 9096  df-ndx 12835  df-slot 12836  df-base 12838  df-sets 12839  df-plusg 12922  df-mulr 12923  df-0g 13090  df-oppr 13830

This theorem is referenced by:  opprnegg  13845  opprnzrbg  13947  opprdomnbg  14036  ridl0  14272  2idlcpblrng  14285