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Theorem rngonegmn1l 35100
Description: Negation in a ring is the same as left multiplication by -1. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
ringneg.1 𝐺 = (1st𝑅)
ringneg.2 𝐻 = (2nd𝑅)
ringneg.3 𝑋 = ran 𝐺
ringneg.4 𝑁 = (inv‘𝐺)
ringneg.5 𝑈 = (GId‘𝐻)
Assertion
Ref Expression
rngonegmn1l ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑁𝐴) = ((𝑁𝑈)𝐻𝐴))

Proof of Theorem rngonegmn1l
StepHypRef Expression
1 ringneg.3 . . . . . . 7 𝑋 = ran 𝐺
2 ringneg.1 . . . . . . . 8 𝐺 = (1st𝑅)
32rneqi 5800 . . . . . . 7 ran 𝐺 = ran (1st𝑅)
41, 3eqtri 2841 . . . . . 6 𝑋 = ran (1st𝑅)
5 ringneg.2 . . . . . 6 𝐻 = (2nd𝑅)
6 ringneg.5 . . . . . 6 𝑈 = (GId‘𝐻)
74, 5, 6rngo1cl 35098 . . . . 5 (𝑅 ∈ RingOps → 𝑈𝑋)
8 ringneg.4 . . . . . . 7 𝑁 = (inv‘𝐺)
92, 1, 8rngonegcl 35086 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑈𝑋) → (𝑁𝑈) ∈ 𝑋)
107, 9mpdan 683 . . . . 5 (𝑅 ∈ RingOps → (𝑁𝑈) ∈ 𝑋)
117, 10jca 512 . . . 4 (𝑅 ∈ RingOps → (𝑈𝑋 ∧ (𝑁𝑈) ∈ 𝑋))
122, 5, 1rngodir 35064 . . . . . . 7 ((𝑅 ∈ RingOps ∧ (𝑈𝑋 ∧ (𝑁𝑈) ∈ 𝑋𝐴𝑋)) → ((𝑈𝐺(𝑁𝑈))𝐻𝐴) = ((𝑈𝐻𝐴)𝐺((𝑁𝑈)𝐻𝐴)))
13123exp2 1346 . . . . . 6 (𝑅 ∈ RingOps → (𝑈𝑋 → ((𝑁𝑈) ∈ 𝑋 → (𝐴𝑋 → ((𝑈𝐺(𝑁𝑈))𝐻𝐴) = ((𝑈𝐻𝐴)𝐺((𝑁𝑈)𝐻𝐴))))))
1413imp42 427 . . . . 5 (((𝑅 ∈ RingOps ∧ (𝑈𝑋 ∧ (𝑁𝑈) ∈ 𝑋)) ∧ 𝐴𝑋) → ((𝑈𝐺(𝑁𝑈))𝐻𝐴) = ((𝑈𝐻𝐴)𝐺((𝑁𝑈)𝐻𝐴)))
1514an32s 648 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ (𝑈𝑋 ∧ (𝑁𝑈) ∈ 𝑋)) → ((𝑈𝐺(𝑁𝑈))𝐻𝐴) = ((𝑈𝐻𝐴)𝐺((𝑁𝑈)𝐻𝐴)))
1611, 15mpidan 685 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑈𝐺(𝑁𝑈))𝐻𝐴) = ((𝑈𝐻𝐴)𝐺((𝑁𝑈)𝐻𝐴)))
17 eqid 2818 . . . . . . . 8 (GId‘𝐺) = (GId‘𝐺)
182, 1, 8, 17rngoaddneg1 35087 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑈𝑋) → (𝑈𝐺(𝑁𝑈)) = (GId‘𝐺))
197, 18mpdan 683 . . . . . 6 (𝑅 ∈ RingOps → (𝑈𝐺(𝑁𝑈)) = (GId‘𝐺))
2019adantr 481 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑈𝐺(𝑁𝑈)) = (GId‘𝐺))
2120oveq1d 7160 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑈𝐺(𝑁𝑈))𝐻𝐴) = ((GId‘𝐺)𝐻𝐴))
2217, 1, 2, 5rngolz 35081 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((GId‘𝐺)𝐻𝐴) = (GId‘𝐺))
2321, 22eqtrd 2853 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑈𝐺(𝑁𝑈))𝐻𝐴) = (GId‘𝐺))
245, 4, 6rngolidm 35096 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑈𝐻𝐴) = 𝐴)
2524oveq1d 7160 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑈𝐻𝐴)𝐺((𝑁𝑈)𝐻𝐴)) = (𝐴𝐺((𝑁𝑈)𝐻𝐴)))
2616, 23, 253eqtr3rd 2862 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐺((𝑁𝑈)𝐻𝐴)) = (GId‘𝐺))
272, 5, 1rngocl 35060 . . . . . 6 ((𝑅 ∈ RingOps ∧ (𝑁𝑈) ∈ 𝑋𝐴𝑋) → ((𝑁𝑈)𝐻𝐴) ∈ 𝑋)
28273expa 1110 . . . . 5 (((𝑅 ∈ RingOps ∧ (𝑁𝑈) ∈ 𝑋) ∧ 𝐴𝑋) → ((𝑁𝑈)𝐻𝐴) ∈ 𝑋)
2928an32s 648 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐴𝑋) ∧ (𝑁𝑈) ∈ 𝑋) → ((𝑁𝑈)𝐻𝐴) ∈ 𝑋)
3010, 29mpidan 685 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑁𝑈)𝐻𝐴) ∈ 𝑋)
312rngogrpo 35069 . . . 4 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
321, 17, 8grpoinvid1 28232 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋 ∧ ((𝑁𝑈)𝐻𝐴) ∈ 𝑋) → ((𝑁𝐴) = ((𝑁𝑈)𝐻𝐴) ↔ (𝐴𝐺((𝑁𝑈)𝐻𝐴)) = (GId‘𝐺)))
3331, 32syl3an1 1155 . . 3 ((𝑅 ∈ RingOps ∧ 𝐴𝑋 ∧ ((𝑁𝑈)𝐻𝐴) ∈ 𝑋) → ((𝑁𝐴) = ((𝑁𝑈)𝐻𝐴) ↔ (𝐴𝐺((𝑁𝑈)𝐻𝐴)) = (GId‘𝐺)))
3430, 33mpd3an3 1453 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑁𝐴) = ((𝑁𝑈)𝐻𝐴) ↔ (𝐴𝐺((𝑁𝑈)𝐻𝐴)) = (GId‘𝐺)))
3526, 34mpbird 258 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑁𝐴) = ((𝑁𝑈)𝐻𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  ran crn 5549  cfv 6348  (class class class)co 7145  1st c1st 7676  2nd c2nd 7677  GrpOpcgr 28193  GIdcgi 28194  invcgn 28195  RingOpscrngo 35053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-1st 7678  df-2nd 7679  df-grpo 28197  df-gid 28198  df-ginv 28199  df-ablo 28249  df-ass 35002  df-exid 35004  df-mgmOLD 35008  df-sgrOLD 35020  df-mndo 35026  df-rngo 35054
This theorem is referenced by:  rngoneglmul  35102  idlnegcl  35181
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