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Theorem rngonegrmul 35354
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 5795 . . . . . . 7 ran 𝐺 = ran (1st𝑅)
41, 3eqtri 2847 . . . . . 6 𝑋 = ran (1st𝑅)
5 ringnegmul.2 . . . . . 6 𝐻 = (2nd𝑅)
6 eqid 2824 . . . . . 6 (GId‘𝐻) = (GId‘𝐻)
74, 5, 6rngo1cl 35349 . . . . 5 (𝑅 ∈ RingOps → (GId‘𝐻) ∈ 𝑋)
8 ringnegmul.4 . . . . . 6 𝑁 = (inv‘𝐺)
92, 1, 8rngonegcl 35337 . . . . 5 ((𝑅 ∈ RingOps ∧ (GId‘𝐻) ∈ 𝑋) → (𝑁‘(GId‘𝐻)) ∈ 𝑋)
107, 9mpdan 686 . . . 4 (𝑅 ∈ RingOps → (𝑁‘(GId‘𝐻)) ∈ 𝑋)
112, 5, 1rngoass 35316 . . . . . . 7 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋 ∧ (𝑁‘(GId‘𝐻)) ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐻(𝑁‘(GId‘𝐻))) = (𝐴𝐻(𝐵𝐻(𝑁‘(GId‘𝐻)))))
12113exp2 1351 . . . . . 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 1108 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐻𝐵)𝐻(𝑁‘(GId‘𝐻))) = (𝐴𝐻(𝐵𝐻(𝑁‘(GId‘𝐻)))))
172, 5, 1rngocl 35311 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
18173expb 1117 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
192, 5, 1, 8, 6rngonegmn1r 35352 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴𝐻𝐵) ∈ 𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝐴𝐻𝐵)𝐻(𝑁‘(GId‘𝐻))))
2018, 19syldan 594 . . 3 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋)) → (𝑁‘(𝐴𝐻𝐵)) = ((𝐴𝐻𝐵)𝐻(𝑁‘(GId‘𝐻))))
21203impb 1112 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝐴𝐻𝐵)𝐻(𝑁‘(GId‘𝐻))))
222, 5, 1, 8, 6rngonegmn1r 35352 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐵𝑋) → (𝑁𝐵) = (𝐵𝐻(𝑁‘(GId‘𝐻))))
23223adant2 1128 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝑁𝐵) = (𝐵𝐻(𝑁‘(GId‘𝐻))))
2423oveq2d 7167 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐻(𝑁𝐵)) = (𝐴𝐻(𝐵𝐻(𝑁‘(GId‘𝐻)))))
2516, 21, 243eqtr4d 2869 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐻𝐵)) = (𝐴𝐻(𝑁𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  ran crn 5544  cfv 6345  (class class class)co 7151  1st c1st 7684  2nd c2nd 7685  GIdcgi 28282  invcgn 28283  RingOpscrngo 35304
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-riota 7109  df-ov 7154  df-1st 7686  df-2nd 7687  df-grpo 28285  df-gid 28286  df-ginv 28287  df-ablo 28337  df-ass 35253  df-exid 35255  df-mgmOLD 35259  df-sgrOLD 35271  df-mndo 35277  df-rngo 35305
This theorem is referenced by:  rngosubdi  35355
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