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Theorem rngoneglmul 34074
Description: Negation of a product in a ring. (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
ringnegmul.1 𝐺 = (1st𝑅)
ringnegmul.2 𝐻 = (2nd𝑅)
ringnegmul.3 𝑋 = ran 𝐺
ringnegmul.4 𝑁 = (inv‘𝐺)
Assertion
Ref Expression
rngoneglmul ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝑁𝐴)𝐻𝐵))

Proof of Theorem rngoneglmul
StepHypRef Expression
1 ringnegmul.3 . . . . . . 7 𝑋 = ran 𝐺
2 ringnegmul.1 . . . . . . . 8 𝐺 = (1st𝑅)
32rneqi 5508 . . . . . . 7 ran 𝐺 = ran (1st𝑅)
41, 3eqtri 2783 . . . . . 6 𝑋 = ran (1st𝑅)
5 ringnegmul.2 . . . . . 6 𝐻 = (2nd𝑅)
6 eqid 2761 . . . . . 6 (GId‘𝐻) = (GId‘𝐻)
74, 5, 6rngo1cl 34070 . . . . 5 (𝑅 ∈ RingOps → (GId‘𝐻) ∈ 𝑋)
8 ringnegmul.4 . . . . . 6 𝑁 = (inv‘𝐺)
92, 1, 8rngonegcl 34058 . . . . 5 ((𝑅 ∈ RingOps ∧ (GId‘𝐻) ∈ 𝑋) → (𝑁‘(GId‘𝐻)) ∈ 𝑋)
107, 9mpdan 705 . . . 4 (𝑅 ∈ RingOps → (𝑁‘(GId‘𝐻)) ∈ 𝑋)
112, 5, 1rngoass 34037 . . . . 5 ((𝑅 ∈ RingOps ∧ ((𝑁‘(GId‘𝐻)) ∈ 𝑋𝐴𝑋𝐵𝑋)) → (((𝑁‘(GId‘𝐻))𝐻𝐴)𝐻𝐵) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵)))
12113exp2 1448 . . . 4 (𝑅 ∈ RingOps → ((𝑁‘(GId‘𝐻)) ∈ 𝑋 → (𝐴𝑋 → (𝐵𝑋 → (((𝑁‘(GId‘𝐻))𝐻𝐴)𝐻𝐵) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵))))))
1310, 12mpd 15 . . 3 (𝑅 ∈ RingOps → (𝐴𝑋 → (𝐵𝑋 → (((𝑁‘(GId‘𝐻))𝐻𝐴)𝐻𝐵) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵)))))
14133imp 1102 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (((𝑁‘(GId‘𝐻))𝐻𝐴)𝐻𝐵) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵)))
152, 5, 1, 8, 6rngonegmn1l 34072 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑁𝐴) = ((𝑁‘(GId‘𝐻))𝐻𝐴))
16153adant3 1127 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝑁𝐴) = ((𝑁‘(GId‘𝐻))𝐻𝐴))
1716oveq1d 6830 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴)𝐻𝐵) = (((𝑁‘(GId‘𝐻))𝐻𝐴)𝐻𝐵))
182, 5, 1rngocl 34032 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
19183expb 1114 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
202, 5, 1, 8, 6rngonegmn1l 34072 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴𝐻𝐵) ∈ 𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵)))
2119, 20syldan 488 . . 3 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋)) → (𝑁‘(𝐴𝐻𝐵)) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵)))
22213impb 1108 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵)))
2314, 17, 223eqtr4rd 2806 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝑁𝐴)𝐻𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2140  ran crn 5268  cfv 6050  (class class class)co 6815  1st c1st 7333  2nd 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
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