Theorem opprdomnbg 14412

Description: A class is a domain if and only if its opposite is a domain, biconditional form of opprdomn 14413. (Contributed by SN, 15-Jun-2015.)

Hypothesis

Ref	Expression
opprdomn.1	⊢ 𝑂 = (oppr‘𝑅)

Assertion

Ref	Expression
opprdomnbg	⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Domn ↔ 𝑂 ∈ Domn))

Proof of Theorem opprdomnbg

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

Step	Hyp	Ref	Expression
1		opprdomn.1	. . . 4 ⊢ 𝑂 = (oppr‘𝑅)
2	1	opprnzrbg 14322	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ NzRing ↔ 𝑂 ∈ NzRing))
3		eqid 2232	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
4	1, 3	opprbasg 14211	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑂))
5		vex 2815	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
6		vex 2815	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
7		eqid 2232	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
8		eqid 2232	. . . . . . . . . . 11 ⊢ (.r‘𝑂) = (.r‘𝑂)
9	3, 7, 1, 8	opprmulg 14207	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦(.r‘𝑂)𝑥) = (𝑥(.r‘𝑅)𝑦))
10	5, 6, 9	mp3an23 1366	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (𝑦(.r‘𝑂)𝑥) = (𝑥(.r‘𝑅)𝑦))
11	10	eqcomd 2238	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑥(.r‘𝑅)𝑦) = (𝑦(.r‘𝑂)𝑥))
12		eqid 2232	. . . . . . . . 9 ⊢ (0g‘𝑅) = (0g‘𝑅)
13	1, 12	oppr0g 14217	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (0g‘𝑅) = (0g‘𝑂))
14	11, 13	eqeq12d 2247	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → ((𝑥(.r‘𝑅)𝑦) = (0g‘𝑅) ↔ (𝑦(.r‘𝑂)𝑥) = (0g‘𝑂)))
15	13	eqeq2d 2244	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (𝑥 = (0g‘𝑅) ↔ 𝑥 = (0g‘𝑂)))
16	13	eqeq2d 2244	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (𝑦 = (0g‘𝑅) ↔ 𝑦 = (0g‘𝑂)))
17	15, 16	orbi12d 801	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → ((𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅)) ↔ (𝑥 = (0g‘𝑂) ∨ 𝑦 = (0g‘𝑂))))
18		orcom 736	. . . . . . . 8 ⊢ ((𝑥 = (0g‘𝑂) ∨ 𝑦 = (0g‘𝑂)) ↔ (𝑦 = (0g‘𝑂) ∨ 𝑥 = (0g‘𝑂)))
19	17, 18	bitrdi 196	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → ((𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅)) ↔ (𝑦 = (0g‘𝑂) ∨ 𝑥 = (0g‘𝑂))))
20	14, 19	imbi12d 234	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (((𝑥(.r‘𝑅)𝑦) = (0g‘𝑅) → (𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅))) ↔ ((𝑦(.r‘𝑂)𝑥) = (0g‘𝑂) → (𝑦 = (0g‘𝑂) ∨ 𝑥 = (0g‘𝑂)))))
21	4, 20	raleqbidv 2756	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) = (0g‘𝑅) → (𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅))) ↔ ∀𝑦 ∈ (Base‘𝑂)((𝑦(.r‘𝑂)𝑥) = (0g‘𝑂) → (𝑦 = (0g‘𝑂) ∨ 𝑥 = (0g‘𝑂)))))
22	4, 21	raleqbidv 2756	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) = (0g‘𝑅) → (𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅))) ↔ ∀𝑥 ∈ (Base‘𝑂)∀𝑦 ∈ (Base‘𝑂)((𝑦(.r‘𝑂)𝑥) = (0g‘𝑂) → (𝑦 = (0g‘𝑂) ∨ 𝑥 = (0g‘𝑂)))))
23		ralcom 2706	. . . 4 ⊢ (∀𝑥 ∈ (Base‘𝑂)∀𝑦 ∈ (Base‘𝑂)((𝑦(.r‘𝑂)𝑥) = (0g‘𝑂) → (𝑦 = (0g‘𝑂) ∨ 𝑥 = (0g‘𝑂))) ↔ ∀𝑦 ∈ (Base‘𝑂)∀𝑥 ∈ (Base‘𝑂)((𝑦(.r‘𝑂)𝑥) = (0g‘𝑂) → (𝑦 = (0g‘𝑂) ∨ 𝑥 = (0g‘𝑂))))
24	22, 23	bitrdi 196	. . 3 ⊢ (𝑅 ∈ 𝑉 → (∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) = (0g‘𝑅) → (𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅))) ↔ ∀𝑦 ∈ (Base‘𝑂)∀𝑥 ∈ (Base‘𝑂)((𝑦(.r‘𝑂)𝑥) = (0g‘𝑂) → (𝑦 = (0g‘𝑂) ∨ 𝑥 = (0g‘𝑂)))))
25	2, 24	anbi12d 473	. 2 ⊢ (𝑅 ∈ 𝑉 → ((𝑅 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) = (0g‘𝑅) → (𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅)))) ↔ (𝑂 ∈ NzRing ∧ ∀𝑦 ∈ (Base‘𝑂)∀𝑥 ∈ (Base‘𝑂)((𝑦(.r‘𝑂)𝑥) = (0g‘𝑂) → (𝑦 = (0g‘𝑂) ∨ 𝑥 = (0g‘𝑂))))))
26	3, 7, 12	isdomn 14407	. 2 ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) = (0g‘𝑅) → (𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅)))))
27		eqid 2232	. . 3 ⊢ (Base‘𝑂) = (Base‘𝑂)
28		eqid 2232	. . 3 ⊢ (0g‘𝑂) = (0g‘𝑂)
29	27, 8, 28	isdomn 14407	. 2 ⊢ (𝑂 ∈ Domn ↔ (𝑂 ∈ NzRing ∧ ∀𝑦 ∈ (Base‘𝑂)∀𝑥 ∈ (Base‘𝑂)((𝑦(.r‘𝑂)𝑥) = (0g‘𝑂) → (𝑦 = (0g‘𝑂) ∨ 𝑥 = (0g‘𝑂)))))
30	25, 26, 29	3bitr4g 223	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Domn ↔ 𝑂 ∈ Domn))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716   = wceq 1398   ∈ wcel 2203  ∀wral 2520  Vcvv 2812  ‘cfv 5351  (class class class)co 6049  Basecbs 13204  ._rcmulr 13283  0_gc0g 13461  opp_rcoppr 14203  NzRingcnzr 14316  Domncdomn 14393

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-addcom 8226  ax-addass 8228  ax-i2m1 8231  ax-0lt1 8232  ax-0id 8234  ax-rnegex 8235  ax-pre-ltirr 8238  ax-pre-lttrn 8240  ax-pre-ltadd 8242

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-tpos 6475  df-pnf 8309  df-mnf 8310  df-ltxr 8312  df-inn 9237  df-2 9295  df-3 9296  df-ndx 13207  df-slot 13208  df-base 13210  df-sets 13211  df-plusg 13295  df-mulr 13296  df-0g 13463  df-mgm 13561  df-sgrp 13607  df-mnd 13622  df-grp 13708  df-mgp 14057  df-ur 14096  df-ring 14134  df-oppr 14204  df-nzr 14317  df-domn 14396

This theorem is referenced by:  opprdomn  14413