Theorem oppr0g 14093

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 2231	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
3	1, 2	opprbasg 14087	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑂))
4	3	eleq2d 2301	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑦 ∈ (Base‘𝑅) ↔ 𝑦 ∈ (Base‘𝑂)))
5		eqid 2231	. . . . . . . . 9 ⊢ (+g‘𝑅) = (+g‘𝑅)
6	1, 5	oppraddg 14088	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘𝑂))
7	6	oveqd 6034	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (𝑦(+g‘𝑅)𝑥) = (𝑦(+g‘𝑂)𝑥))
8	7	eqeq1d 2240	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → ((𝑦(+g‘𝑅)𝑥) = 𝑥 ↔ (𝑦(+g‘𝑂)𝑥) = 𝑥))
9	6	oveqd 6034	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑂)𝑦))
10	9	eqeq1d 2240	. . . . . 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 2746	. . . 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 5309	. 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 13455	. 2 ⊢ (𝑅 ∈ 𝑉 → 0 = (℩𝑦(𝑦 ∈ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)((𝑦(+g‘𝑅)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑅)𝑦) = 𝑥))))
17	1	opprex 14085	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑂 ∈ V)
18		eqid 2231	. . . 4 ⊢ (Base‘𝑂) = (Base‘𝑂)
19		eqid 2231	. . . 4 ⊢ (+g‘𝑂) = (+g‘𝑂)
20		eqid 2231	. . . 4 ⊢ (0g‘𝑂) = (0g‘𝑂)
21	18, 19, 20	grpidvalg 13455	. . 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 2274	1 ⊢ (𝑅 ∈ 𝑉 → 0 = (0g‘𝑂))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2202  ∀wral 2510  Vcvv 2802  ℩cio 5284  ‘cfv 5326  (class class class)co 6017  Basecbs 13081  +_gcplusg 13159  0_gc0g 13338  opp_rcoppr 14079

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-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-lttrn 8145  ax-pre-ltadd 8147

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-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-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-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-tpos 6410  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-plusg 13172  df-mulr 13173  df-0g 13340  df-oppr 14080

This theorem is referenced by:  opprnegg  14095  opprnzrbg  14198  opprdomnbg  14287  ridl0  14523  2idlcpblrng  14536