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Theorem rngoneglmul 37950
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 5948 . . . . . . 7 ran 𝐺 = ran (1st𝑅)
41, 3eqtri 2765 . . . . . 6 𝑋 = ran (1st𝑅)
5 ringnegmul.2 . . . . . 6 𝐻 = (2nd𝑅)
6 eqid 2737 . . . . . 6 (GId‘𝐻) = (GId‘𝐻)
74, 5, 6rngo1cl 37946 . . . . 5 (𝑅 ∈ RingOps → (GId‘𝐻) ∈ 𝑋)
8 ringnegmul.4 . . . . . 6 𝑁 = (inv‘𝐺)
92, 1, 8rngonegcl 37934 . . . . 5 ((𝑅 ∈ RingOps ∧ (GId‘𝐻) ∈ 𝑋) → (𝑁‘(GId‘𝐻)) ∈ 𝑋)
107, 9mpdan 687 . . . 4 (𝑅 ∈ RingOps → (𝑁‘(GId‘𝐻)) ∈ 𝑋)
112, 5, 1rngoass 37913 . . . . 5 ((𝑅 ∈ RingOps ∧ ((𝑁‘(GId‘𝐻)) ∈ 𝑋𝐴𝑋𝐵𝑋)) → (((𝑁‘(GId‘𝐻))𝐻𝐴)𝐻𝐵) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵)))
12113exp2 1355 . . . 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 1111 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (((𝑁‘(GId‘𝐻))𝐻𝐴)𝐻𝐵) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵)))
152, 5, 1, 8, 6rngonegmn1l 37948 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑁𝐴) = ((𝑁‘(GId‘𝐻))𝐻𝐴))
16153adant3 1133 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝑁𝐴) = ((𝑁‘(GId‘𝐻))𝐻𝐴))
1716oveq1d 7446 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴)𝐻𝐵) = (((𝑁‘(GId‘𝐻))𝐻𝐴)𝐻𝐵))
182, 5, 1rngocl 37908 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
19183expb 1121 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
202, 5, 1, 8, 6rngonegmn1l 37948 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴𝐻𝐵) ∈ 𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵)))
2119, 20syldan 591 . . 3 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋)) → (𝑁‘(𝐴𝐻𝐵)) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵)))
22213impb 1115 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝑁‘(GId‘𝐻))𝐻(𝐴𝐻𝐵)))
2314, 17, 223eqtr4rd 2788 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝑁𝐴)𝐻𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  ran crn 5686  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  GIdcgi 30509  invcgn 30510  RingOpscrngo 37901
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-1st 8014  df-2nd 8015  df-grpo 30512  df-gid 30513  df-ginv 30514  df-ablo 30564  df-ass 37850  df-exid 37852  df-mgmOLD 37856  df-sgrOLD 37868  df-mndo 37874  df-rngo 37902
This theorem is referenced by:  rngosubdir  37953
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