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Theorem rngonegrmul 34280
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 5588 . . . . . . 7 ran 𝐺 = ran (1st𝑅)
41, 3eqtri 2849 . . . . . 6 𝑋 = ran (1st𝑅)
5 ringnegmul.2 . . . . . 6 𝐻 = (2nd𝑅)
6 eqid 2825 . . . . . 6 (GId‘𝐻) = (GId‘𝐻)
74, 5, 6rngo1cl 34275 . . . . 5 (𝑅 ∈ RingOps → (GId‘𝐻) ∈ 𝑋)
8 ringnegmul.4 . . . . . 6 𝑁 = (inv‘𝐺)
92, 1, 8rngonegcl 34263 . . . . 5 ((𝑅 ∈ RingOps ∧ (GId‘𝐻) ∈ 𝑋) → (𝑁‘(GId‘𝐻)) ∈ 𝑋)
107, 9mpdan 678 . . . 4 (𝑅 ∈ RingOps → (𝑁‘(GId‘𝐻)) ∈ 𝑋)
112, 5, 1rngoass 34242 . . . . . . 7 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋 ∧ (𝑁‘(GId‘𝐻)) ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐻(𝑁‘(GId‘𝐻))) = (𝐴𝐻(𝐵𝐻(𝑁‘(GId‘𝐻)))))
12113exp2 1467 . . . . . 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 1141 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐻𝐵)𝐻(𝑁‘(GId‘𝐻))) = (𝐴𝐻(𝐵𝐻(𝑁‘(GId‘𝐻)))))
172, 5, 1rngocl 34237 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
18173expb 1153 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
192, 5, 1, 8, 6rngonegmn1r 34278 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴𝐻𝐵) ∈ 𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝐴𝐻𝐵)𝐻(𝑁‘(GId‘𝐻))))
2018, 19syldan 585 . . 3 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋)) → (𝑁‘(𝐴𝐻𝐵)) = ((𝐴𝐻𝐵)𝐻(𝑁‘(GId‘𝐻))))
21203impb 1147 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝐴𝐻𝐵)𝐻(𝑁‘(GId‘𝐻))))
222, 5, 1, 8, 6rngonegmn1r 34278 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐵𝑋) → (𝑁𝐵) = (𝐵𝐻(𝑁‘(GId‘𝐻))))
23223adant2 1165 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝑁𝐵) = (𝐵𝐻(𝑁‘(GId‘𝐻))))
2423oveq2d 6926 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐻(𝑁𝐵)) = (𝐴𝐻(𝐵𝐻(𝑁‘(GId‘𝐻)))))
2516, 21, 243eqtr4d 2871 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐻𝐵)) = (𝐴𝐻(𝑁𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1111   = wceq 1656  wcel 2164  ran crn 5347  cfv 6127  (class class class)co 6910  1st c1st 7431  2nd c2nd 7432  GIdcgi 27896  invcgn 27897  RingOpscrngo 34230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-1st 7433  df-2nd 7434  df-grpo 27899  df-gid 27900  df-ginv 27901  df-ablo 27951  df-ass 34179  df-exid 34181  df-mgmOLD 34185  df-sgrOLD 34197  df-mndo 34203  df-rngo 34231
This theorem is referenced by:  rngosubdi  34281
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