Theorem rngoneglmul 34074

Description: Negation of a product in a ring. (Contributed by Jeff Madsen, 19-Jun-2010.)

Hypotheses

Ref	Expression
ringnegmul.1	⊢ 𝐺 = (1st ‘𝑅)
ringnegmul.2	⊢ 𝐻 = (2nd ‘𝑅)
ringnegmul.3	⊢ 𝑋 = ran 𝐺
ringnegmul.4	⊢ 𝑁 = (inv‘𝐺)

Assertion

Ref	Expression
rngoneglmul	⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝑁‘𝐴)𝐻𝐵))

Proof of Theorem rngoneglmul

Step	Hyp	Ref	Expression
1		ringnegmul.3	. . . . . . 7 ⊢ 𝑋 = ran 𝐺
2		ringnegmul.1	. . . . . . . 8 ⊢ 𝐺 = (1st ‘𝑅)
3	2	rneqi 5508	. . . . . . 7 ⊢ ran 𝐺 = ran (1st ‘𝑅)
4	1, 3	eqtri 2783	. . . . . 6 ⊢ 𝑋 = ran (1st ‘𝑅)
5		ringnegmul.2	. . . . . 6 ⊢ 𝐻 = (2nd ‘𝑅)
6		eqid 2761	. . . . . 6 ⊢ (GId‘𝐻) = (GId‘𝐻)
7	4, 5, 6	rngo1cl 34070	. . . . 5 ⊢ (𝑅 ∈ RingOps → (GId‘𝐻) ∈ 𝑋)
8		ringnegmul.4	. . . . . 6 ⊢ 𝑁 = (inv‘𝐺)
9	2, 1, 8	rngonegcl 34058	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (GId‘𝐻) ∈ 𝑋) → (𝑁‘(GId‘𝐻)) ∈ 𝑋)
10	7, 9	mpdan 705	. . . 4 ⊢ (𝑅 ∈ RingOps → (𝑁‘(GId‘𝐻)) ∈ 𝑋)
11	2, 5, 1	rngoass 34037	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ ((𝑁‘(GId‘𝐻)) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (((𝑁‘(GId‘𝐻))𝐻𝐴)𝐻𝐵) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵)))
12	11	3exp2 1448	. . . 4 ⊢ (𝑅 ∈ RingOps → ((𝑁‘(GId‘𝐻)) ∈ 𝑋 → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → (((𝑁‘(GId‘𝐻))𝐻𝐴)𝐻𝐵) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵))))))
13	10, 12	mpd 15	. . 3 ⊢ (𝑅 ∈ RingOps → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → (((𝑁‘(GId‘𝐻))𝐻𝐴)𝐻𝐵) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵)))))
14	13	3imp 1102	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(GId‘𝐻))𝐻𝐴)𝐻𝐵) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵)))
15	2, 5, 1, 8, 6	rngonegmn1l 34072	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = ((𝑁‘(GId‘𝐻))𝐻𝐴))
16	15	3adant3 1127	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐴) = ((𝑁‘(GId‘𝐻))𝐻𝐴))
17	16	oveq1d 6830	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘𝐴)𝐻𝐵) = (((𝑁‘(GId‘𝐻))𝐻𝐴)𝐻𝐵))
18	2, 5, 1	rngocl 34032	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
19	18	3expb 1114	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
20	2, 5, 1, 8, 6	rngonegmn1l 34072	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴𝐻𝐵) ∈ 𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵)))
21	19, 20	syldan 488	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝑁‘(𝐴𝐻𝐵)) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵)))
22	21	3impb 1108	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵)))
23	14, 17, 22	3eqtr4rd 2806	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝑁‘𝐴)𝐻𝐵))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2140  ran crn 5268  ‘cfv 6050  (class class class)co 6815  1^st c1st 7333  2^nd c2nd 7334  GIdcgi 27675  invcgn 27676  RingOpscrngo 34025

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-riota 6776  df-ov 6818  df-1st 7335  df-2nd 7336  df-grpo 27678  df-gid 27679  df-ginv 27680  df-ablo 27730  df-ass 33974  df-exid 33976  df-mgmOLD 33980  df-sgrOLD 33992  df-mndo 33998  df-rngo 34026

This theorem is referenced by:  rngosubdir  34077