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Theorem grpoinvid1 28082
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 6984 . . . 4 ((𝑁𝐴) = 𝐵 → (𝐴𝐺(𝑁𝐴)) = (𝐴𝐺𝐵))
21adantl 474 . . 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 28081 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺(𝑁𝐴)) = 𝑈)
763adant3 1112 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺(𝑁𝐴)) = 𝑈)
87adantr 473 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑁𝐴) = 𝐵) → (𝐴𝐺(𝑁𝐴)) = 𝑈)
92, 8eqtr3d 2817 . 2 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑁𝐴) = 𝐵) → (𝐴𝐺𝐵) = 𝑈)
10 oveq2 6984 . . . 4 ((𝐴𝐺𝐵) = 𝑈 → ((𝑁𝐴)𝐺(𝐴𝐺𝐵)) = ((𝑁𝐴)𝐺𝑈))
1110adantl 474 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐺𝐵) = 𝑈) → ((𝑁𝐴)𝐺(𝐴𝐺𝐵)) = ((𝑁𝐴)𝐺𝑈))
123, 4, 5grpolinv 28080 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁𝐴)𝐺𝐴) = 𝑈)
1312oveq1d 6991 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑁𝐴)𝐺𝐴)𝐺𝐵) = (𝑈𝐺𝐵))
14133adant3 1112 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (((𝑁𝐴)𝐺𝐴)𝐺𝐵) = (𝑈𝐺𝐵))
153, 5grpoinvcl 28078 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)
1615adantrr 704 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → (𝑁𝐴) ∈ 𝑋)
17 simprl 758 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → 𝐴𝑋)
18 simprr 760 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → 𝐵𝑋)
1916, 17, 183jca 1108 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → ((𝑁𝐴) ∈ 𝑋𝐴𝑋𝐵𝑋))
203grpoass 28057 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ ((𝑁𝐴) ∈ 𝑋𝐴𝑋𝐵𝑋)) → (((𝑁𝐴)𝐺𝐴)𝐺𝐵) = ((𝑁𝐴)𝐺(𝐴𝐺𝐵)))
2119, 20syldan 582 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → (((𝑁𝐴)𝐺𝐴)𝐺𝐵) = ((𝑁𝐴)𝐺(𝐴𝐺𝐵)))
22213impb 1095 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (((𝑁𝐴)𝐺𝐴)𝐺𝐵) = ((𝑁𝐴)𝐺(𝐴𝐺𝐵)))
2314, 22eqtr3d 2817 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑈𝐺𝐵) = ((𝑁𝐴)𝐺(𝐴𝐺𝐵)))
243, 4grpolid 28070 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝑈𝐺𝐵) = 𝐵)
25243adant2 1111 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑈𝐺𝐵) = 𝐵)
2623, 25eqtr3d 2817 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴)𝐺(𝐴𝐺𝐵)) = 𝐵)
2726adantr 473 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐺𝐵) = 𝑈) → ((𝑁𝐴)𝐺(𝐴𝐺𝐵)) = 𝐵)
283, 4grporid 28071 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝑁𝐴) ∈ 𝑋) → ((𝑁𝐴)𝐺𝑈) = (𝑁𝐴))
2915, 28syldan 582 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁𝐴)𝐺𝑈) = (𝑁𝐴))
30293adant3 1112 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴)𝐺𝑈) = (𝑁𝐴))
3130adantr 473 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐺𝐵) = 𝑈) → ((𝑁𝐴)𝐺𝑈) = (𝑁𝐴))
3211, 27, 313eqtr3rd 2824 . 2 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐴𝐺𝐵) = 𝑈) → (𝑁𝐴) = 𝐵)
339, 32impbida 788 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴) = 𝐵 ↔ (𝐴𝐺𝐵) = 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1068   = wceq 1507  wcel 2050  ran crn 5408  cfv 6188  (class class class)co 6976  GrpOpcgr 28043  GIdcgi 28044  invcgn 28045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-riota 6937  df-ov 6979  df-grpo 28047  df-gid 28048  df-ginv 28049
This theorem is referenced by:  grpoinvop  28087  rngonegmn1l  34658
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