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Theorem grpoinv 28294
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 28292 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) = (𝑦𝑋 (𝑦𝐺𝐴) = 𝑈))
51, 2grpoinveu 28288 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ∃!𝑦𝑋 (𝑦𝐺𝐴) = 𝑈)
6 riotacl2 7122 . . . . . 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 2911 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ {𝑦𝑋 ∣ (𝑦𝐺𝐴) = 𝑈})
9 simpl 485 . . . . . . . . 9 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈)
109rgenw 3148 . . . . . . . 8 𝑦𝑋 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈)
1110a1i 11 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ∀𝑦𝑋 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈))
121, 2grpoidinv2 28284 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
1312simprd 498 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈))
1411, 13, 53jca 1122 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (∀𝑦𝑋 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) ∧ ∃!𝑦𝑋 (𝑦𝐺𝐴) = 𝑈))
15 reupick2 4287 . . . . . 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 582 . . . . 5 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → ((𝑦𝐺𝐴) = 𝑈 ↔ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
1716rabbidva 3477 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → {𝑦𝑋 ∣ (𝑦𝐺𝐴) = 𝑈} = {𝑦𝑋 ∣ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)})
188, 17eleqtrd 2913 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ {𝑦𝑋 ∣ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)})
19 oveq1 7155 . . . . . 6 (𝑦 = (𝑁𝐴) → (𝑦𝐺𝐴) = ((𝑁𝐴)𝐺𝐴))
2019eqeq1d 2821 . . . . 5 (𝑦 = (𝑁𝐴) → ((𝑦𝐺𝐴) = 𝑈 ↔ ((𝑁𝐴)𝐺𝐴) = 𝑈))
21 oveq2 7156 . . . . . 6 (𝑦 = (𝑁𝐴) → (𝐴𝐺𝑦) = (𝐴𝐺(𝑁𝐴)))
2221eqeq1d 2821 . . . . 5 (𝑦 = (𝑁𝐴) → ((𝐴𝐺𝑦) = 𝑈 ↔ (𝐴𝐺(𝑁𝐴)) = 𝑈))
2320, 22anbi12d 632 . . . 4 (𝑦 = (𝑁𝐴) → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) ↔ (((𝑁𝐴)𝐺𝐴) = 𝑈 ∧ (𝐴𝐺(𝑁𝐴)) = 𝑈)))
2423elrab 3678 . . 3 ((𝑁𝐴) ∈ {𝑦𝑋 ∣ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)} ↔ ((𝑁𝐴) ∈ 𝑋 ∧ (((𝑁𝐴)𝐺𝐴) = 𝑈 ∧ (𝐴𝐺(𝑁𝐴)) = 𝑈)))
2518, 24sylib 220 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁𝐴) ∈ 𝑋 ∧ (((𝑁𝐴)𝐺𝐴) = 𝑈 ∧ (𝐴𝐺(𝑁𝐴)) = 𝑈)))
2625simprd 498 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑁𝐴)𝐺𝐴) = 𝑈 ∧ (𝐴𝐺(𝑁𝐴)) = 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1081   = wceq 1530  wcel 2107  wral 3136  wrex 3137  ∃!wreu 3138  {crab 3140  ran crn 5549  cfv 6348  crio 7105  (class class class)co 7148  GrpOpcgr 28258  GIdcgi 28259  invcgn 28260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-grpo 28262  df-gid 28263  df-ginv 28264
This theorem is referenced by:  grpolinv  28295  grporinv  28296
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