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Theorem grpoinv 27897
Description: The properties of a group element's inverse. (Contributed by NM, 27-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (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
grpoinv ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑁𝐴)𝐺𝐴) = 𝑈 ∧ (𝐴𝐺(𝑁𝐴)) = 𝑈))

Proof of Theorem grpoinv
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 grpinv.1 . . . . . 6 𝑋 = ran 𝐺
2 grpinv.2 . . . . . 6 𝑈 = (GId‘𝐺)
3 grpinv.3 . . . . . 6 𝑁 = (inv‘𝐺)
41, 2, 3grpoinvval 27895 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) = (𝑦𝑋 (𝑦𝐺𝐴) = 𝑈))
51, 2grpoinveu 27891 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ∃!𝑦𝑋 (𝑦𝐺𝐴) = 𝑈)
6 riotacl2 6850 . . . . . 6 (∃!𝑦𝑋 (𝑦𝐺𝐴) = 𝑈 → (𝑦𝑋 (𝑦𝐺𝐴) = 𝑈) ∈ {𝑦𝑋 ∣ (𝑦𝐺𝐴) = 𝑈})
75, 6syl 17 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑦𝑋 (𝑦𝐺𝐴) = 𝑈) ∈ {𝑦𝑋 ∣ (𝑦𝐺𝐴) = 𝑈})
84, 7eqeltrd 2876 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ {𝑦𝑋 ∣ (𝑦𝐺𝐴) = 𝑈})
9 simpl 475 . . . . . . . . 9 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈)
109rgenw 3103 . . . . . . . 8 𝑦𝑋 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈)
1110a1i 11 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ∀𝑦𝑋 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈))
121, 2grpoidinv2 27887 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
1312simprd 490 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈))
1411, 13, 53jca 1159 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (∀𝑦𝑋 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) ∧ ∃!𝑦𝑋 (𝑦𝐺𝐴) = 𝑈))
15 reupick2 4111 . . . . . 6 (((∀𝑦𝑋 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) ∧ ∃!𝑦𝑋 (𝑦𝐺𝐴) = 𝑈) ∧ 𝑦𝑋) → ((𝑦𝐺𝐴) = 𝑈 ↔ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
1614, 15sylan 576 . . . . 5 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → ((𝑦𝐺𝐴) = 𝑈 ↔ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
1716rabbidva 3370 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → {𝑦𝑋 ∣ (𝑦𝐺𝐴) = 𝑈} = {𝑦𝑋 ∣ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)})
188, 17eleqtrd 2878 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ {𝑦𝑋 ∣ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)})
19 oveq1 6883 . . . . . 6 (𝑦 = (𝑁𝐴) → (𝑦𝐺𝐴) = ((𝑁𝐴)𝐺𝐴))
2019eqeq1d 2799 . . . . 5 (𝑦 = (𝑁𝐴) → ((𝑦𝐺𝐴) = 𝑈 ↔ ((𝑁𝐴)𝐺𝐴) = 𝑈))
21 oveq2 6884 . . . . . 6 (𝑦 = (𝑁𝐴) → (𝐴𝐺𝑦) = (𝐴𝐺(𝑁𝐴)))
2221eqeq1d 2799 . . . . 5 (𝑦 = (𝑁𝐴) → ((𝐴𝐺𝑦) = 𝑈 ↔ (𝐴𝐺(𝑁𝐴)) = 𝑈))
2320, 22anbi12d 625 . . . 4 (𝑦 = (𝑁𝐴) → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) ↔ (((𝑁𝐴)𝐺𝐴) = 𝑈 ∧ (𝐴𝐺(𝑁𝐴)) = 𝑈)))
2423elrab 3554 . . 3 ((𝑁𝐴) ∈ {𝑦𝑋 ∣ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)} ↔ ((𝑁𝐴) ∈ 𝑋 ∧ (((𝑁𝐴)𝐺𝐴) = 𝑈 ∧ (𝐴𝐺(𝑁𝐴)) = 𝑈)))
2518, 24sylib 210 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁𝐴) ∈ 𝑋 ∧ (((𝑁𝐴)𝐺𝐴) = 𝑈 ∧ (𝐴𝐺(𝑁𝐴)) = 𝑈)))
2625simprd 490 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑁𝐴)𝐺𝐴) = 𝑈 ∧ (𝐴𝐺(𝑁𝐴)) = 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  wral 3087  wrex 3088  ∃!wreu 3089  {crab 3091  ran crn 5311  cfv 6099  crio 6836  (class class class)co 6876  GrpOpcgr 27861  GIdcgi 27862  invcgn 27863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-riota 6837  df-ov 6879  df-grpo 27865  df-gid 27866  df-ginv 27867
This theorem is referenced by:  grpolinv  27898  grporinv  27899
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