Theorem rngonegrmul 36807

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
rngonegrmul	⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐻𝐵)) = (𝐴𝐻(𝑁‘𝐵)))

Proof of Theorem rngonegrmul

Step	Hyp	Ref	Expression
1		ringnegmul.3	. . . . . . 7 ⊢ 𝑋 = ran 𝐺
2		ringnegmul.1	. . . . . . . 8 ⊢ 𝐺 = (1st ‘𝑅)
3	2	rneqi 5936	. . . . . . 7 ⊢ ran 𝐺 = ran (1st ‘𝑅)
4	1, 3	eqtri 2760	. . . . . 6 ⊢ 𝑋 = ran (1st ‘𝑅)
5		ringnegmul.2	. . . . . 6 ⊢ 𝐻 = (2nd ‘𝑅)
6		eqid 2732	. . . . . 6 ⊢ (GId‘𝐻) = (GId‘𝐻)
7	4, 5, 6	rngo1cl 36802	. . . . 5 ⊢ (𝑅 ∈ RingOps → (GId‘𝐻) ∈ 𝑋)
8		ringnegmul.4	. . . . . 6 ⊢ 𝑁 = (inv‘𝐺)
9	2, 1, 8	rngonegcl 36790	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (GId‘𝐻) ∈ 𝑋) → (𝑁‘(GId‘𝐻)) ∈ 𝑋)
10	7, 9	mpdan 685	. . . 4 ⊢ (𝑅 ∈ RingOps → (𝑁‘(GId‘𝐻)) ∈ 𝑋)
11	2, 5, 1	rngoass 36769	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝑁‘(GId‘𝐻)) ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐻(𝑁‘(GId‘𝐻))) = (𝐴𝐻(𝐵𝐻(𝑁‘(GId‘𝐻)))))
12	11	3exp2 1354	. . . . . 6 ⊢ (𝑅 ∈ RingOps → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → ((𝑁‘(GId‘𝐻)) ∈ 𝑋 → ((𝐴𝐻𝐵)𝐻(𝑁‘(GId‘𝐻))) = (𝐴𝐻(𝐵𝐻(𝑁‘(GId‘𝐻))))))))
13	12	com24 95	. . . . 5 ⊢ (𝑅 ∈ RingOps → ((𝑁‘(GId‘𝐻)) ∈ 𝑋 → (𝐵 ∈ 𝑋 → (𝐴 ∈ 𝑋 → ((𝐴𝐻𝐵)𝐻(𝑁‘(GId‘𝐻))) = (𝐴𝐻(𝐵𝐻(𝑁‘(GId‘𝐻))))))))
14	13	com34 91	. . . 4 ⊢ (𝑅 ∈ RingOps → ((𝑁‘(GId‘𝐻)) ∈ 𝑋 → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → ((𝐴𝐻𝐵)𝐻(𝑁‘(GId‘𝐻))) = (𝐴𝐻(𝐵𝐻(𝑁‘(GId‘𝐻))))))))
15	10, 14	mpd 15	. . 3 ⊢ (𝑅 ∈ RingOps → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → ((𝐴𝐻𝐵)𝐻(𝑁‘(GId‘𝐻))) = (𝐴𝐻(𝐵𝐻(𝑁‘(GId‘𝐻)))))))
16	15	3imp 1111	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐻𝐵)𝐻(𝑁‘(GId‘𝐻))) = (𝐴𝐻(𝐵𝐻(𝑁‘(GId‘𝐻)))))
17	2, 5, 1	rngocl 36764	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
18	17	3expb 1120	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
19	2, 5, 1, 8, 6	rngonegmn1r 36805	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴𝐻𝐵) ∈ 𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝐴𝐻𝐵)𝐻(𝑁‘(GId‘𝐻))))
20	18, 19	syldan 591	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝑁‘(𝐴𝐻𝐵)) = ((𝐴𝐻𝐵)𝐻(𝑁‘(GId‘𝐻))))
21	20	3impb 1115	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝐴𝐻𝐵)𝐻(𝑁‘(GId‘𝐻))))
22	2, 5, 1, 8, 6	rngonegmn1r 36805	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) = (𝐵𝐻(𝑁‘(GId‘𝐻))))
23	22	3adant2 1131	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) = (𝐵𝐻(𝑁‘(GId‘𝐻))))
24	23	oveq2d 7424	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻(𝑁‘𝐵)) = (𝐴𝐻(𝐵𝐻(𝑁‘(GId‘𝐻)))))
25	16, 21, 24	3eqtr4d 2782	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐻𝐵)) = (𝐴𝐻(𝑁‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ran crn 5677  ‘cfv 6543  (class class class)co 7408  1^st c1st 7972  2^nd c2nd 7973  GIdcgi 29738  invcgn 29739  RingOpscrngo 36757

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-1st 7974  df-2nd 7975  df-grpo 29741  df-gid 29742  df-ginv 29743  df-ablo 29793  df-ass 36706  df-exid 36708  df-mgmOLD 36712  df-sgrOLD 36724  df-mndo 36730  df-rngo 36758

This theorem is referenced by:  rngosubdi  36808