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Theorem rngonegrmul 37115
Description: Negation of a product in a ring. (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
ringnegmul.1 𝐺 = (1st β€˜π‘…)
ringnegmul.2 𝐻 = (2nd β€˜π‘…)
ringnegmul.3 𝑋 = ran 𝐺
ringnegmul.4 𝑁 = (invβ€˜πΊ)
Assertion
Ref Expression
rngonegrmul ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (π‘β€˜(𝐴𝐻𝐡)) = (𝐴𝐻(π‘β€˜π΅)))

Proof of Theorem rngonegrmul
StepHypRef Expression
1 ringnegmul.3 . . . . . . 7 𝑋 = ran 𝐺
2 ringnegmul.1 . . . . . . . 8 𝐺 = (1st β€˜π‘…)
32rneqi 5935 . . . . . . 7 ran 𝐺 = ran (1st β€˜π‘…)
41, 3eqtri 2758 . . . . . 6 𝑋 = ran (1st β€˜π‘…)
5 ringnegmul.2 . . . . . 6 𝐻 = (2nd β€˜π‘…)
6 eqid 2730 . . . . . 6 (GIdβ€˜π») = (GIdβ€˜π»)
74, 5, 6rngo1cl 37110 . . . . 5 (𝑅 ∈ RingOps β†’ (GIdβ€˜π») ∈ 𝑋)
8 ringnegmul.4 . . . . . 6 𝑁 = (invβ€˜πΊ)
92, 1, 8rngonegcl 37098 . . . . 5 ((𝑅 ∈ RingOps ∧ (GIdβ€˜π») ∈ 𝑋) β†’ (π‘β€˜(GIdβ€˜π»)) ∈ 𝑋)
107, 9mpdan 683 . . . 4 (𝑅 ∈ RingOps β†’ (π‘β€˜(GIdβ€˜π»)) ∈ 𝑋)
112, 5, 1rngoass 37077 . . . . . . 7 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ (π‘β€˜(GIdβ€˜π»)) ∈ 𝑋)) β†’ ((𝐴𝐻𝐡)𝐻(π‘β€˜(GIdβ€˜π»))) = (𝐴𝐻(𝐡𝐻(π‘β€˜(GIdβ€˜π»)))))
12113exp2 1352 . . . . . 6 (𝑅 ∈ RingOps β†’ (𝐴 ∈ 𝑋 β†’ (𝐡 ∈ 𝑋 β†’ ((π‘β€˜(GIdβ€˜π»)) ∈ 𝑋 β†’ ((𝐴𝐻𝐡)𝐻(π‘β€˜(GIdβ€˜π»))) = (𝐴𝐻(𝐡𝐻(π‘β€˜(GIdβ€˜π»))))))))
1312com24 95 . . . . 5 (𝑅 ∈ RingOps β†’ ((π‘β€˜(GIdβ€˜π»)) ∈ 𝑋 β†’ (𝐡 ∈ 𝑋 β†’ (𝐴 ∈ 𝑋 β†’ ((𝐴𝐻𝐡)𝐻(π‘β€˜(GIdβ€˜π»))) = (𝐴𝐻(𝐡𝐻(π‘β€˜(GIdβ€˜π»))))))))
1413com34 91 . . . 4 (𝑅 ∈ RingOps β†’ ((π‘β€˜(GIdβ€˜π»)) ∈ 𝑋 β†’ (𝐴 ∈ 𝑋 β†’ (𝐡 ∈ 𝑋 β†’ ((𝐴𝐻𝐡)𝐻(π‘β€˜(GIdβ€˜π»))) = (𝐴𝐻(𝐡𝐻(π‘β€˜(GIdβ€˜π»))))))))
1510, 14mpd 15 . . 3 (𝑅 ∈ RingOps β†’ (𝐴 ∈ 𝑋 β†’ (𝐡 ∈ 𝑋 β†’ ((𝐴𝐻𝐡)𝐻(π‘β€˜(GIdβ€˜π»))) = (𝐴𝐻(𝐡𝐻(π‘β€˜(GIdβ€˜π»)))))))
16153imp 1109 . 2 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((𝐴𝐻𝐡)𝐻(π‘β€˜(GIdβ€˜π»))) = (𝐴𝐻(𝐡𝐻(π‘β€˜(GIdβ€˜π»)))))
172, 5, 1rngocl 37072 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐻𝐡) ∈ 𝑋)
18173expb 1118 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (𝐴𝐻𝐡) ∈ 𝑋)
192, 5, 1, 8, 6rngonegmn1r 37113 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴𝐻𝐡) ∈ 𝑋) β†’ (π‘β€˜(𝐴𝐻𝐡)) = ((𝐴𝐻𝐡)𝐻(π‘β€˜(GIdβ€˜π»))))
2018, 19syldan 589 . . 3 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (π‘β€˜(𝐴𝐻𝐡)) = ((𝐴𝐻𝐡)𝐻(π‘β€˜(GIdβ€˜π»))))
21203impb 1113 . 2 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (π‘β€˜(𝐴𝐻𝐡)) = ((𝐴𝐻𝐡)𝐻(π‘β€˜(GIdβ€˜π»))))
222, 5, 1, 8, 6rngonegmn1r 37113 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐡 ∈ 𝑋) β†’ (π‘β€˜π΅) = (𝐡𝐻(π‘β€˜(GIdβ€˜π»))))
23223adant2 1129 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (π‘β€˜π΅) = (𝐡𝐻(π‘β€˜(GIdβ€˜π»))))
2423oveq2d 7427 . 2 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐻(π‘β€˜π΅)) = (𝐴𝐻(𝐡𝐻(π‘β€˜(GIdβ€˜π»)))))
2516, 21, 243eqtr4d 2780 1 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (π‘β€˜(𝐴𝐻𝐡)) = (𝐴𝐻(π‘β€˜π΅)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  ran crn 5676  β€˜cfv 6542  (class class class)co 7411  1st c1st 7975  2nd c2nd 7976  GIdcgi 30010  invcgn 30011  RingOpscrngo 37065
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-1st 7977  df-2nd 7978  df-grpo 30013  df-gid 30014  df-ginv 30015  df-ablo 30065  df-ass 37014  df-exid 37016  df-mgmOLD 37020  df-sgrOLD 37032  df-mndo 37038  df-rngo 37066
This theorem is referenced by:  rngosubdi  37116
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