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Theorem rngonegmn1r 36447
Description: Negation in a ring is the same as right multiplication by -1. (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
ringneg.1 𝐺 = (1st β€˜π‘…)
ringneg.2 𝐻 = (2nd β€˜π‘…)
ringneg.3 𝑋 = ran 𝐺
ringneg.4 𝑁 = (invβ€˜πΊ)
ringneg.5 π‘ˆ = (GIdβ€˜π»)
Assertion
Ref Expression
rngonegmn1r ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π΄) = (𝐴𝐻(π‘β€˜π‘ˆ)))

Proof of Theorem rngonegmn1r
StepHypRef Expression
1 ringneg.3 . . . . . . . . 9 𝑋 = ran 𝐺
2 ringneg.1 . . . . . . . . . 10 𝐺 = (1st β€˜π‘…)
32rneqi 5893 . . . . . . . . 9 ran 𝐺 = ran (1st β€˜π‘…)
41, 3eqtri 2761 . . . . . . . 8 𝑋 = ran (1st β€˜π‘…)
5 ringneg.2 . . . . . . . 8 𝐻 = (2nd β€˜π‘…)
6 ringneg.5 . . . . . . . 8 π‘ˆ = (GIdβ€˜π»)
74, 5, 6rngo1cl 36444 . . . . . . 7 (𝑅 ∈ RingOps β†’ π‘ˆ ∈ 𝑋)
8 ringneg.4 . . . . . . . 8 𝑁 = (invβ€˜πΊ)
92, 1, 8rngonegcl 36432 . . . . . . 7 ((𝑅 ∈ RingOps ∧ π‘ˆ ∈ 𝑋) β†’ (π‘β€˜π‘ˆ) ∈ 𝑋)
107, 9mpdan 686 . . . . . 6 (𝑅 ∈ RingOps β†’ (π‘β€˜π‘ˆ) ∈ 𝑋)
1110adantr 482 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π‘ˆ) ∈ 𝑋)
127adantr 482 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ π‘ˆ ∈ 𝑋)
1311, 12jca 513 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ ((π‘β€˜π‘ˆ) ∈ 𝑋 ∧ π‘ˆ ∈ 𝑋))
142, 5, 1rngodi 36409 . . . . . 6 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ (π‘β€˜π‘ˆ) ∈ 𝑋 ∧ π‘ˆ ∈ 𝑋)) β†’ (𝐴𝐻((π‘β€˜π‘ˆ)πΊπ‘ˆ)) = ((𝐴𝐻(π‘β€˜π‘ˆ))𝐺(π΄π»π‘ˆ)))
15143exp2 1355 . . . . 5 (𝑅 ∈ RingOps β†’ (𝐴 ∈ 𝑋 β†’ ((π‘β€˜π‘ˆ) ∈ 𝑋 β†’ (π‘ˆ ∈ 𝑋 β†’ (𝐴𝐻((π‘β€˜π‘ˆ)πΊπ‘ˆ)) = ((𝐴𝐻(π‘β€˜π‘ˆ))𝐺(π΄π»π‘ˆ))))))
1615imp43 429 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ ((π‘β€˜π‘ˆ) ∈ 𝑋 ∧ π‘ˆ ∈ 𝑋)) β†’ (𝐴𝐻((π‘β€˜π‘ˆ)πΊπ‘ˆ)) = ((𝐴𝐻(π‘β€˜π‘ˆ))𝐺(π΄π»π‘ˆ)))
1713, 16mpdan 686 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (𝐴𝐻((π‘β€˜π‘ˆ)πΊπ‘ˆ)) = ((𝐴𝐻(π‘β€˜π‘ˆ))𝐺(π΄π»π‘ˆ)))
18 eqid 2733 . . . . . . . 8 (GIdβ€˜πΊ) = (GIdβ€˜πΊ)
192, 1, 8, 18rngoaddneg2 36434 . . . . . . 7 ((𝑅 ∈ RingOps ∧ π‘ˆ ∈ 𝑋) β†’ ((π‘β€˜π‘ˆ)πΊπ‘ˆ) = (GIdβ€˜πΊ))
207, 19mpdan 686 . . . . . 6 (𝑅 ∈ RingOps β†’ ((π‘β€˜π‘ˆ)πΊπ‘ˆ) = (GIdβ€˜πΊ))
2120adantr 482 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ ((π‘β€˜π‘ˆ)πΊπ‘ˆ) = (GIdβ€˜πΊ))
2221oveq2d 7374 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (𝐴𝐻((π‘β€˜π‘ˆ)πΊπ‘ˆ)) = (𝐴𝐻(GIdβ€˜πΊ)))
2318, 1, 2, 5rngorz 36428 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (𝐴𝐻(GIdβ€˜πΊ)) = (GIdβ€˜πΊ))
2422, 23eqtrd 2773 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (𝐴𝐻((π‘β€˜π‘ˆ)πΊπ‘ˆ)) = (GIdβ€˜πΊ))
255, 4, 6rngoridm 36443 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (π΄π»π‘ˆ) = 𝐴)
2625oveq2d 7374 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ ((𝐴𝐻(π‘β€˜π‘ˆ))𝐺(π΄π»π‘ˆ)) = ((𝐴𝐻(π‘β€˜π‘ˆ))𝐺𝐴))
2717, 24, 263eqtr3rd 2782 . 2 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ ((𝐴𝐻(π‘β€˜π‘ˆ))𝐺𝐴) = (GIdβ€˜πΊ))
282, 5, 1rngocl 36406 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ (π‘β€˜π‘ˆ) ∈ 𝑋) β†’ (𝐴𝐻(π‘β€˜π‘ˆ)) ∈ 𝑋)
2911, 28mpd3an3 1463 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (𝐴𝐻(π‘β€˜π‘ˆ)) ∈ 𝑋)
302rngogrpo 36415 . . . 4 (𝑅 ∈ RingOps β†’ 𝐺 ∈ GrpOp)
311, 18, 8grpoinvid2 29513 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ (𝐴𝐻(π‘β€˜π‘ˆ)) ∈ 𝑋) β†’ ((π‘β€˜π΄) = (𝐴𝐻(π‘β€˜π‘ˆ)) ↔ ((𝐴𝐻(π‘β€˜π‘ˆ))𝐺𝐴) = (GIdβ€˜πΊ)))
3230, 31syl3an1 1164 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ (𝐴𝐻(π‘β€˜π‘ˆ)) ∈ 𝑋) β†’ ((π‘β€˜π΄) = (𝐴𝐻(π‘β€˜π‘ˆ)) ↔ ((𝐴𝐻(π‘β€˜π‘ˆ))𝐺𝐴) = (GIdβ€˜πΊ)))
3329, 32mpd3an3 1463 . 2 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ ((π‘β€˜π΄) = (𝐴𝐻(π‘β€˜π‘ˆ)) ↔ ((𝐴𝐻(π‘β€˜π‘ˆ))𝐺𝐴) = (GIdβ€˜πΊ)))
3427, 33mpbird 257 1 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π΄) = (𝐴𝐻(π‘β€˜π‘ˆ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ran crn 5635  β€˜cfv 6497  (class class class)co 7358  1st c1st 7920  2nd c2nd 7921  GrpOpcgr 29473  GIdcgi 29474  invcgn 29475  RingOpscrngo 36399
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 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
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 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-1st 7922  df-2nd 7923  df-grpo 29477  df-gid 29478  df-ginv 29479  df-ablo 29529  df-ass 36348  df-exid 36350  df-mgmOLD 36354  df-sgrOLD 36366  df-mndo 36372  df-rngo 36400
This theorem is referenced by:  rngonegrmul  36449
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