Theorem rngonegrmul 37945

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 5904	. . . . . . 7 ⊢ ran 𝐺 = ran (1st ‘𝑅)
4	1, 3	eqtri 2753	. . . . . 6 ⊢ 𝑋 = ran (1st ‘𝑅)
5		ringnegmul.2	. . . . . 6 ⊢ 𝐻 = (2nd ‘𝑅)
6		eqid 2730	. . . . . 6 ⊢ (GId‘𝐻) = (GId‘𝐻)
7	4, 5, 6	rngo1cl 37940	. . . . 5 ⊢ (𝑅 ∈ RingOps → (GId‘𝐻) ∈ 𝑋)
8		ringnegmul.4	. . . . . 6 ⊢ 𝑁 = (inv‘𝐺)
9	2, 1, 8	rngonegcl 37928	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (GId‘𝐻) ∈ 𝑋) → (𝑁‘(GId‘𝐻)) ∈ 𝑋)
10	7, 9	mpdan 687	. . . 4 ⊢ (𝑅 ∈ RingOps → (𝑁‘(GId‘𝐻)) ∈ 𝑋)
11	2, 5, 1	rngoass 37907	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝑁‘(GId‘𝐻)) ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐻(𝑁‘(GId‘𝐻))) = (𝐴𝐻(𝐵𝐻(𝑁‘(GId‘𝐻)))))
12	11	3exp2 1355	. . . . . 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 1110	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐻𝐵)𝐻(𝑁‘(GId‘𝐻))) = (𝐴𝐻(𝐵𝐻(𝑁‘(GId‘𝐻)))))
17	2, 5, 1	rngocl 37902	. . . . 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 37943	. . . 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 1114	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝐴𝐻𝐵)𝐻(𝑁‘(GId‘𝐻))))
22	2, 5, 1, 8, 6	rngonegmn1r 37943	. . . 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 7406	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻(𝑁‘𝐵)) = (𝐴𝐻(𝐵𝐻(𝑁‘(GId‘𝐻)))))
25	16, 21, 24	3eqtr4d 2775	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐻𝐵)) = (𝐴𝐻(𝑁‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ran crn 5642  ‘cfv 6514  (class class class)co 7390  1^st c1st 7969  2^nd c2nd 7970  GIdcgi 30426  invcgn 30427  RingOpscrngo 37895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-1st 7971  df-2nd 7972  df-grpo 30429  df-gid 30430  df-ginv 30431  df-ablo 30481  df-ass 37844  df-exid 37846  df-mgmOLD 37850  df-sgrOLD 37862  df-mndo 37868  df-rngo 37896

This theorem is referenced by:  rngosubdi  37946