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Theorem grpoinvid1 28311
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 7143 . . . 4 ((𝑁𝐴) = 𝐵 → (𝐴𝐺(𝑁𝐴)) = (𝐴𝐺𝐵))
21adantl 485 . . 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 28310 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺(𝑁𝐴)) = 𝑈)
763adant3 1129 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺(𝑁𝐴)) = 𝑈)
87adantr 484 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑁𝐴) = 𝐵) → (𝐴𝐺(𝑁𝐴)) = 𝑈)
92, 8eqtr3d 2835 . 2 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑁𝐴) = 𝐵) → (𝐴𝐺𝐵) = 𝑈)
10 oveq2 7143 . . . 4 ((𝐴𝐺𝐵) = 𝑈 → ((𝑁𝐴)𝐺(𝐴𝐺𝐵)) = ((𝑁𝐴)𝐺𝑈))
1110adantl 485 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐺𝐵) = 𝑈) → ((𝑁𝐴)𝐺(𝐴𝐺𝐵)) = ((𝑁𝐴)𝐺𝑈))
123, 4, 5grpolinv 28309 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁𝐴)𝐺𝐴) = 𝑈)
1312oveq1d 7150 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑁𝐴)𝐺𝐴)𝐺𝐵) = (𝑈𝐺𝐵))
14133adant3 1129 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (((𝑁𝐴)𝐺𝐴)𝐺𝐵) = (𝑈𝐺𝐵))
153, 5grpoinvcl 28307 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)
1615adantrr 716 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → (𝑁𝐴) ∈ 𝑋)
17 simprl 770 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → 𝐴𝑋)
18 simprr 772 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → 𝐵𝑋)
1916, 17, 183jca 1125 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → ((𝑁𝐴) ∈ 𝑋𝐴𝑋𝐵𝑋))
203grpoass 28286 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ ((𝑁𝐴) ∈ 𝑋𝐴𝑋𝐵𝑋)) → (((𝑁𝐴)𝐺𝐴)𝐺𝐵) = ((𝑁𝐴)𝐺(𝐴𝐺𝐵)))
2119, 20syldan 594 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → (((𝑁𝐴)𝐺𝐴)𝐺𝐵) = ((𝑁𝐴)𝐺(𝐴𝐺𝐵)))
22213impb 1112 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (((𝑁𝐴)𝐺𝐴)𝐺𝐵) = ((𝑁𝐴)𝐺(𝐴𝐺𝐵)))
2314, 22eqtr3d 2835 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑈𝐺𝐵) = ((𝑁𝐴)𝐺(𝐴𝐺𝐵)))
243, 4grpolid 28299 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝑈𝐺𝐵) = 𝐵)
25243adant2 1128 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑈𝐺𝐵) = 𝐵)
2623, 25eqtr3d 2835 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴)𝐺(𝐴𝐺𝐵)) = 𝐵)
2726adantr 484 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐺𝐵) = 𝑈) → ((𝑁𝐴)𝐺(𝐴𝐺𝐵)) = 𝐵)
283, 4grporid 28300 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝑁𝐴) ∈ 𝑋) → ((𝑁𝐴)𝐺𝑈) = (𝑁𝐴))
2915, 28syldan 594 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁𝐴)𝐺𝑈) = (𝑁𝐴))
30293adant3 1129 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴)𝐺𝑈) = (𝑁𝐴))
3130adantr 484 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐺𝐵) = 𝑈) → ((𝑁𝐴)𝐺𝑈) = (𝑁𝐴))
3211, 27, 313eqtr3rd 2842 . 2 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐺𝐵) = 𝑈) → (𝑁𝐴) = 𝐵)
339, 32impbida 800 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴) = 𝐵 ↔ (𝐴𝐺𝐵) = 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2111  ran crn 5520  cfv 6324  (class class class)co 7135  GrpOpcgr 28272  GIdcgi 28273  invcgn 28274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-grpo 28276  df-gid 28277  df-ginv 28278
This theorem is referenced by:  grpoinvop  28316  rngonegmn1l  35379
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