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Theorem grpoinvid2 28083
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
grpoinvid2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴) = 𝐵 ↔ (𝐵𝐺𝐴) = 𝑈))

Proof of Theorem grpoinvid2
StepHypRef Expression
1 oveq1 6983 . . . 4 ((𝑁𝐴) = 𝐵 → ((𝑁𝐴)𝐺𝐴) = (𝐵𝐺𝐴))
21adantl 474 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑁𝐴) = 𝐵) → ((𝑁𝐴)𝐺𝐴) = (𝐵𝐺𝐴))
3 grpinv.1 . . . . . 6 𝑋 = ran 𝐺
4 grpinv.2 . . . . . 6 𝑈 = (GId‘𝐺)
5 grpinv.3 . . . . . 6 𝑁 = (inv‘𝐺)
63, 4, 5grpolinv 28080 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁𝐴)𝐺𝐴) = 𝑈)
763adant3 1112 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴)𝐺𝐴) = 𝑈)
87adantr 473 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑁𝐴) = 𝐵) → ((𝑁𝐴)𝐺𝐴) = 𝑈)
92, 8eqtr3d 2816 . 2 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝑁𝐴) = 𝐵) → (𝐵𝐺𝐴) = 𝑈)
103, 5grpoinvcl 28078 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)
113, 4grpolid 28070 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝑁𝐴) ∈ 𝑋) → (𝑈𝐺(𝑁𝐴)) = (𝑁𝐴))
1210, 11syldan 582 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑈𝐺(𝑁𝐴)) = (𝑁𝐴))
13123adant3 1112 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑈𝐺(𝑁𝐴)) = (𝑁𝐴))
1413eqcomd 2784 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁𝐴) = (𝑈𝐺(𝑁𝐴)))
1514adantr 473 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐵𝐺𝐴) = 𝑈) → (𝑁𝐴) = (𝑈𝐺(𝑁𝐴)))
16 oveq1 6983 . . . 4 ((𝐵𝐺𝐴) = 𝑈 → ((𝐵𝐺𝐴)𝐺(𝑁𝐴)) = (𝑈𝐺(𝑁𝐴)))
1716adantl 474 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐵𝐺𝐴) = 𝑈) → ((𝐵𝐺𝐴)𝐺(𝑁𝐴)) = (𝑈𝐺(𝑁𝐴)))
18 simprr 760 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → 𝐵𝑋)
19 simprl 758 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → 𝐴𝑋)
2010adantrr 704 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → (𝑁𝐴) ∈ 𝑋)
2118, 19, 203jca 1108 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → (𝐵𝑋𝐴𝑋 ∧ (𝑁𝐴) ∈ 𝑋))
223grpoass 28057 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐴𝑋 ∧ (𝑁𝐴) ∈ 𝑋)) → ((𝐵𝐺𝐴)𝐺(𝑁𝐴)) = (𝐵𝐺(𝐴𝐺(𝑁𝐴))))
2321, 22syldan 582 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋)) → ((𝐵𝐺𝐴)𝐺(𝑁𝐴)) = (𝐵𝐺(𝐴𝐺(𝑁𝐴))))
24233impb 1095 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐵𝐺𝐴)𝐺(𝑁𝐴)) = (𝐵𝐺(𝐴𝐺(𝑁𝐴))))
253, 4, 5grporinv 28081 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺(𝑁𝐴)) = 𝑈)
2625oveq2d 6992 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐵𝐺(𝐴𝐺(𝑁𝐴))) = (𝐵𝐺𝑈))
27263adant3 1112 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐵𝐺(𝐴𝐺(𝑁𝐴))) = (𝐵𝐺𝑈))
283, 4grporid 28071 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋) → (𝐵𝐺𝑈) = 𝐵)
29283adant2 1111 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐵𝐺𝑈) = 𝐵)
3024, 27, 293eqtrd 2818 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐵𝐺𝐴)𝐺(𝑁𝐴)) = 𝐵)
3130adantr 473 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐵𝐺𝐴) = 𝑈) → ((𝐵𝐺𝐴)𝐺(𝑁𝐴)) = 𝐵)
3215, 17, 313eqtr2d 2820 . 2 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) ∧ (𝐵𝐺𝐴) = 𝑈) → (𝑁𝐴) = 𝐵)
339, 32impbida 788 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴) = 𝐵 ↔ (𝐵𝐺𝐴) = 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 2750  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 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  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:  rngonegmn1r  34668
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