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Theorem grpoinvid1 30461
Description: The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpinv.1 𝑋 = ran 𝐺
grpinv.2 𝑈 = (GId‘𝐺)
grpinv.3 𝑁 = (inv‘𝐺)
Assertion
Ref Expression
grpoinvid1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴) = 𝐵 ↔ (𝐴𝐺𝐵) = 𝑈))

Proof of Theorem grpoinvid1
StepHypRef Expression
1 oveq2 7432 . . . 4 ((𝑁𝐴) = 𝐵 → (𝐴𝐺(𝑁𝐴)) = (𝐴𝐺𝐵))
21adantl 480 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑁𝐴) = 𝐵) → (𝐴𝐺(𝑁𝐴)) = (𝐴𝐺𝐵))
3 grpinv.1 . . . . . 6 𝑋 = ran 𝐺
4 grpinv.2 . . . . . 6 𝑈 = (GId‘𝐺)
5 grpinv.3 . . . . . 6 𝑁 = (inv‘𝐺)
63, 4, 5grporinv 30460 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺(𝑁𝐴)) = 𝑈)
763adant3 1129 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺(𝑁𝐴)) = 𝑈)
87adantr 479 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑁𝐴) = 𝐵) → (𝐴𝐺(𝑁𝐴)) = 𝑈)
92, 8eqtr3d 2768 . 2 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑁𝐴) = 𝐵) → (𝐴𝐺𝐵) = 𝑈)
10 oveq2 7432 . . . 4 ((𝐴𝐺𝐵) = 𝑈 → ((𝑁𝐴)𝐺(𝐴𝐺𝐵)) = ((𝑁𝐴)𝐺𝑈))
1110adantl 480 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐺𝐵) = 𝑈) → ((𝑁𝐴)𝐺(𝐴𝐺𝐵)) = ((𝑁𝐴)𝐺𝑈))
123, 4, 5grpolinv 30459 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁𝐴)𝐺𝐴) = 𝑈)
1312oveq1d 7439 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑁𝐴)𝐺𝐴)𝐺𝐵) = (𝑈𝐺𝐵))
14133adant3 1129 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (((𝑁𝐴)𝐺𝐴)𝐺𝐵) = (𝑈𝐺𝐵))
153, 5grpoinvcl 30457 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)
1615adantrr 715 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → (𝑁𝐴) ∈ 𝑋)
17 simprl 769 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → 𝐴𝑋)
18 simprr 771 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → 𝐵𝑋)
1916, 17, 183jca 1125 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → ((𝑁𝐴) ∈ 𝑋𝐴𝑋𝐵𝑋))
203grpoass 30436 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ ((𝑁𝐴) ∈ 𝑋𝐴𝑋𝐵𝑋)) → (((𝑁𝐴)𝐺𝐴)𝐺𝐵) = ((𝑁𝐴)𝐺(𝐴𝐺𝐵)))
2119, 20syldan 589 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → (((𝑁𝐴)𝐺𝐴)𝐺𝐵) = ((𝑁𝐴)𝐺(𝐴𝐺𝐵)))
22213impb 1112 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (((𝑁𝐴)𝐺𝐴)𝐺𝐵) = ((𝑁𝐴)𝐺(𝐴𝐺𝐵)))
2314, 22eqtr3d 2768 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑈𝐺𝐵) = ((𝑁𝐴)𝐺(𝐴𝐺𝐵)))
243, 4grpolid 30449 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝑈𝐺𝐵) = 𝐵)
25243adant2 1128 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑈𝐺𝐵) = 𝐵)
2623, 25eqtr3d 2768 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴)𝐺(𝐴𝐺𝐵)) = 𝐵)
2726adantr 479 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐺𝐵) = 𝑈) → ((𝑁𝐴)𝐺(𝐴𝐺𝐵)) = 𝐵)
283, 4grporid 30450 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝑁𝐴) ∈ 𝑋) → ((𝑁𝐴)𝐺𝑈) = (𝑁𝐴))
2915, 28syldan 589 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁𝐴)𝐺𝑈) = (𝑁𝐴))
30293adant3 1129 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴)𝐺𝑈) = (𝑁𝐴))
3130adantr 479 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐺𝐵) = 𝑈) → ((𝑁𝐴)𝐺𝑈) = (𝑁𝐴))
3211, 27, 313eqtr3rd 2775 . 2 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐺𝐵) = 𝑈) → (𝑁𝐴) = 𝐵)
339, 32impbida 799 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴) = 𝐵 ↔ (𝐴𝐺𝐵) = 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1534  wcel 2099  ran crn 5683  cfv 6554  (class class class)co 7424  GrpOpcgr 30422  GIdcgi 30423  invcgn 30424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-grpo 30426  df-gid 30427  df-ginv 30428
This theorem is referenced by:  grpoinvop  30466  rngonegmn1l  37642
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