Theorem rngoneglmul 37903

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 5962	. . . . . . 7 ⊢ ran 𝐺 = ran (1st ‘𝑅)
4	1, 3	eqtri 2768	. . . . . 6 ⊢ 𝑋 = ran (1st ‘𝑅)
5		ringnegmul.2	. . . . . 6 ⊢ 𝐻 = (2nd ‘𝑅)
6		eqid 2740	. . . . . 6 ⊢ (GId‘𝐻) = (GId‘𝐻)
7	4, 5, 6	rngo1cl 37899	. . . . 5 ⊢ (𝑅 ∈ RingOps → (GId‘𝐻) ∈ 𝑋)
8		ringnegmul.4	. . . . . 6 ⊢ 𝑁 = (inv‘𝐺)
9	2, 1, 8	rngonegcl 37887	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (GId‘𝐻) ∈ 𝑋) → (𝑁‘(GId‘𝐻)) ∈ 𝑋)
10	7, 9	mpdan 686	. . . 4 ⊢ (𝑅 ∈ RingOps → (𝑁‘(GId‘𝐻)) ∈ 𝑋)
11	2, 5, 1	rngoass 37866	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ ((𝑁‘(GId‘𝐻)) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (((𝑁‘(GId‘𝐻))𝐻𝐴)𝐻𝐵) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵)))
12	11	3exp2 1354	. . . 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 1111	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(GId‘𝐻))𝐻𝐴)𝐻𝐵) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵)))
15	2, 5, 1, 8, 6	rngonegmn1l 37901	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = ((𝑁‘(GId‘𝐻))𝐻𝐴))
16	15	3adant3 1132	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐴) = ((𝑁‘(GId‘𝐻))𝐻𝐴))
17	16	oveq1d 7463	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘𝐴)𝐻𝐵) = (((𝑁‘(GId‘𝐻))𝐻𝐴)𝐻𝐵))
18	2, 5, 1	rngocl 37861	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
19	18	3expb 1120	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
20	2, 5, 1, 8, 6	rngonegmn1l 37901	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴𝐻𝐵) ∈ 𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵)))
21	19, 20	syldan 590	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝑁‘(𝐴𝐻𝐵)) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵)))
22	21	3impb 1115	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵)))
23	14, 17, 22	3eqtr4rd 2791	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝑁‘𝐴)𝐻𝐵))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ran crn 5701  ‘cfv 6573  (class class class)co 7448  1^st c1st 8028  2^nd c2nd 8029  GIdcgi 30522  invcgn 30523  RingOpscrngo 37854

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-1st 8030  df-2nd 8031  df-grpo 30525  df-gid 30526  df-ginv 30527  df-ablo 30577  df-ass 37803  df-exid 37805  df-mgmOLD 37809  df-sgrOLD 37821  df-mndo 37827  df-rngo 37855

This theorem is referenced by:  rngosubdir  37906