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Theorem rngoneglmul 36811
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
rngoneglmul ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (π‘β€˜(𝐴𝐻𝐡)) = ((π‘β€˜π΄)𝐻𝐡))

Proof of Theorem rngoneglmul
StepHypRef Expression
1 ringnegmul.3 . . . . . . 7 𝑋 = ran 𝐺
2 ringnegmul.1 . . . . . . . 8 𝐺 = (1st β€˜π‘…)
32rneqi 5937 . . . . . . 7 ran 𝐺 = ran (1st β€˜π‘…)
41, 3eqtri 2761 . . . . . 6 𝑋 = ran (1st β€˜π‘…)
5 ringnegmul.2 . . . . . 6 𝐻 = (2nd β€˜π‘…)
6 eqid 2733 . . . . . 6 (GIdβ€˜π») = (GIdβ€˜π»)
74, 5, 6rngo1cl 36807 . . . . 5 (𝑅 ∈ RingOps β†’ (GIdβ€˜π») ∈ 𝑋)
8 ringnegmul.4 . . . . . 6 𝑁 = (invβ€˜πΊ)
92, 1, 8rngonegcl 36795 . . . . 5 ((𝑅 ∈ RingOps ∧ (GIdβ€˜π») ∈ 𝑋) β†’ (π‘β€˜(GIdβ€˜π»)) ∈ 𝑋)
107, 9mpdan 686 . . . 4 (𝑅 ∈ RingOps β†’ (π‘β€˜(GIdβ€˜π»)) ∈ 𝑋)
112, 5, 1rngoass 36774 . . . . 5 ((𝑅 ∈ RingOps ∧ ((π‘β€˜(GIdβ€˜π»)) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (((π‘β€˜(GIdβ€˜π»))𝐻𝐴)𝐻𝐡) = ((π‘β€˜(GIdβ€˜π»))𝐻(𝐴𝐻𝐡)))
12113exp2 1355 . . . 4 (𝑅 ∈ RingOps β†’ ((π‘β€˜(GIdβ€˜π»)) ∈ 𝑋 β†’ (𝐴 ∈ 𝑋 β†’ (𝐡 ∈ 𝑋 β†’ (((π‘β€˜(GIdβ€˜π»))𝐻𝐴)𝐻𝐡) = ((π‘β€˜(GIdβ€˜π»))𝐻(𝐴𝐻𝐡))))))
1310, 12mpd 15 . . 3 (𝑅 ∈ RingOps β†’ (𝐴 ∈ 𝑋 β†’ (𝐡 ∈ 𝑋 β†’ (((π‘β€˜(GIdβ€˜π»))𝐻𝐴)𝐻𝐡) = ((π‘β€˜(GIdβ€˜π»))𝐻(𝐴𝐻𝐡)))))
14133imp 1112 . 2 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (((π‘β€˜(GIdβ€˜π»))𝐻𝐴)𝐻𝐡) = ((π‘β€˜(GIdβ€˜π»))𝐻(𝐴𝐻𝐡)))
152, 5, 1, 8, 6rngonegmn1l 36809 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π΄) = ((π‘β€˜(GIdβ€˜π»))𝐻𝐴))
16153adant3 1133 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (π‘β€˜π΄) = ((π‘β€˜(GIdβ€˜π»))𝐻𝐴))
1716oveq1d 7424 . 2 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((π‘β€˜π΄)𝐻𝐡) = (((π‘β€˜(GIdβ€˜π»))𝐻𝐴)𝐻𝐡))
182, 5, 1rngocl 36769 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐻𝐡) ∈ 𝑋)
19183expb 1121 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (𝐴𝐻𝐡) ∈ 𝑋)
202, 5, 1, 8, 6rngonegmn1l 36809 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴𝐻𝐡) ∈ 𝑋) β†’ (π‘β€˜(𝐴𝐻𝐡)) = ((π‘β€˜(GIdβ€˜π»))𝐻(𝐴𝐻𝐡)))
2119, 20syldan 592 . . 3 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (π‘β€˜(𝐴𝐻𝐡)) = ((π‘β€˜(GIdβ€˜π»))𝐻(𝐴𝐻𝐡)))
22213impb 1116 . 2 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (π‘β€˜(𝐴𝐻𝐡)) = ((π‘β€˜(GIdβ€˜π»))𝐻(𝐴𝐻𝐡)))
2314, 17, 223eqtr4rd 2784 1 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (π‘β€˜(𝐴𝐻𝐡)) = ((π‘β€˜π΄)𝐻𝐡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ran crn 5678  β€˜cfv 6544  (class class class)co 7409  1st c1st 7973  2nd c2nd 7974  GIdcgi 29743  invcgn 29744  RingOpscrngo 36762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-1st 7975  df-2nd 7976  df-grpo 29746  df-gid 29747  df-ginv 29748  df-ablo 29798  df-ass 36711  df-exid 36713  df-mgmOLD 36717  df-sgrOLD 36729  df-mndo 36735  df-rngo 36763
This theorem is referenced by:  rngosubdir  36814
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