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Theorem rngonegrmul 34670
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
rngonegrmul ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐻𝐵)) = (𝐴𝐻(𝑁𝐵)))

Proof of Theorem rngonegrmul
StepHypRef Expression
1 ringnegmul.3 . . . . . . 7 𝑋 = ran 𝐺
2 ringnegmul.1 . . . . . . . 8 𝐺 = (1st𝑅)
32rneqi 5650 . . . . . . 7 ran 𝐺 = ran (1st𝑅)
41, 3eqtri 2802 . . . . . 6 𝑋 = ran (1st𝑅)
5 ringnegmul.2 . . . . . 6 𝐻 = (2nd𝑅)
6 eqid 2778 . . . . . 6 (GId‘𝐻) = (GId‘𝐻)
74, 5, 6rngo1cl 34665 . . . . 5 (𝑅 ∈ RingOps → (GId‘𝐻) ∈ 𝑋)
8 ringnegmul.4 . . . . . 6 𝑁 = (inv‘𝐺)
92, 1, 8rngonegcl 34653 . . . . 5 ((𝑅 ∈ RingOps ∧ (GId‘𝐻) ∈ 𝑋) → (𝑁‘(GId‘𝐻)) ∈ 𝑋)
107, 9mpdan 674 . . . 4 (𝑅 ∈ RingOps → (𝑁‘(GId‘𝐻)) ∈ 𝑋)
112, 5, 1rngoass 34632 . . . . . . 7 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋 ∧ (𝑁‘(GId‘𝐻)) ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐻(𝑁‘(GId‘𝐻))) = (𝐴𝐻(𝐵𝐻(𝑁‘(GId‘𝐻)))))
12113exp2 1334 . . . . . 6 (𝑅 ∈ RingOps → (𝐴𝑋 → (𝐵𝑋 → ((𝑁‘(GId‘𝐻)) ∈ 𝑋 → ((𝐴𝐻𝐵)𝐻(𝑁‘(GId‘𝐻))) = (𝐴𝐻(𝐵𝐻(𝑁‘(GId‘𝐻))))))))
1312com24 95 . . . . 5 (𝑅 ∈ RingOps → ((𝑁‘(GId‘𝐻)) ∈ 𝑋 → (𝐵𝑋 → (𝐴𝑋 → ((𝐴𝐻𝐵)𝐻(𝑁‘(GId‘𝐻))) = (𝐴𝐻(𝐵𝐻(𝑁‘(GId‘𝐻))))))))
1413com34 91 . . . 4 (𝑅 ∈ RingOps → ((𝑁‘(GId‘𝐻)) ∈ 𝑋 → (𝐴𝑋 → (𝐵𝑋 → ((𝐴𝐻𝐵)𝐻(𝑁‘(GId‘𝐻))) = (𝐴𝐻(𝐵𝐻(𝑁‘(GId‘𝐻))))))))
1510, 14mpd 15 . . 3 (𝑅 ∈ RingOps → (𝐴𝑋 → (𝐵𝑋 → ((𝐴𝐻𝐵)𝐻(𝑁‘(GId‘𝐻))) = (𝐴𝐻(𝐵𝐻(𝑁‘(GId‘𝐻)))))))
16153imp 1091 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐻𝐵)𝐻(𝑁‘(GId‘𝐻))) = (𝐴𝐻(𝐵𝐻(𝑁‘(GId‘𝐻)))))
172, 5, 1rngocl 34627 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
18173expb 1100 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
192, 5, 1, 8, 6rngonegmn1r 34668 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴𝐻𝐵) ∈ 𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝐴𝐻𝐵)𝐻(𝑁‘(GId‘𝐻))))
2018, 19syldan 582 . . 3 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋)) → (𝑁‘(𝐴𝐻𝐵)) = ((𝐴𝐻𝐵)𝐻(𝑁‘(GId‘𝐻))))
21203impb 1095 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝐴𝐻𝐵)𝐻(𝑁‘(GId‘𝐻))))
222, 5, 1, 8, 6rngonegmn1r 34668 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐵𝑋) → (𝑁𝐵) = (𝐵𝐻(𝑁‘(GId‘𝐻))))
23223adant2 1111 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝑁𝐵) = (𝐵𝐻(𝑁‘(GId‘𝐻))))
2423oveq2d 6992 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐻(𝑁𝐵)) = (𝐴𝐻(𝐵𝐻(𝑁‘(GId‘𝐻)))))
2516, 21, 243eqtr4d 2824 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐻𝐵)) = (𝐴𝐻(𝑁𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  w3a 1068   = wceq 1507  wcel 2050  ran crn 5408  cfv 6188  (class class class)co 6976  1st c1st 7499  2nd c2nd 7500  GIdcgi 28044  invcgn 28045  RingOpscrngo 34620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-riota 6937  df-ov 6979  df-1st 7501  df-2nd 7502  df-grpo 28047  df-gid 28048  df-ginv 28049  df-ablo 28099  df-ass 34569  df-exid 34571  df-mgmOLD 34575  df-sgrOLD 34587  df-mndo 34593  df-rngo 34621
This theorem is referenced by:  rngosubdi  34671
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