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Theorem grpoinv 27709
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 27707 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) = (𝑦𝑋 (𝑦𝐺𝐴) = 𝑈))
51, 2grpoinveu 27703 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ∃!𝑦𝑋 (𝑦𝐺𝐴) = 𝑈)
6 riotacl2 6788 . . . . . 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 2839 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ {𝑦𝑋 ∣ (𝑦𝐺𝐴) = 𝑈})
9 simpl 474 . . . . . . . . 9 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈)
109rgenw 3062 . . . . . . . 8 𝑦𝑋 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈)
1110a1i 11 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ∀𝑦𝑋 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈))
121, 2grpoidinv2 27699 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
1312simprd 482 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈))
1411, 13, 53jca 1123 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (∀𝑦𝑋 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) ∧ ∃!𝑦𝑋 (𝑦𝐺𝐴) = 𝑈))
15 reupick2 4056 . . . . . 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 489 . . . . 5 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → ((𝑦𝐺𝐴) = 𝑈 ↔ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
1716rabbidva 3328 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → {𝑦𝑋 ∣ (𝑦𝐺𝐴) = 𝑈} = {𝑦𝑋 ∣ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)})
188, 17eleqtrd 2841 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ {𝑦𝑋 ∣ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)})
19 oveq1 6821 . . . . . 6 (𝑦 = (𝑁𝐴) → (𝑦𝐺𝐴) = ((𝑁𝐴)𝐺𝐴))
2019eqeq1d 2762 . . . . 5 (𝑦 = (𝑁𝐴) → ((𝑦𝐺𝐴) = 𝑈 ↔ ((𝑁𝐴)𝐺𝐴) = 𝑈))
21 oveq2 6822 . . . . . 6 (𝑦 = (𝑁𝐴) → (𝐴𝐺𝑦) = (𝐴𝐺(𝑁𝐴)))
2221eqeq1d 2762 . . . . 5 (𝑦 = (𝑁𝐴) → ((𝐴𝐺𝑦) = 𝑈 ↔ (𝐴𝐺(𝑁𝐴)) = 𝑈))
2320, 22anbi12d 749 . . . 4 (𝑦 = (𝑁𝐴) → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) ↔ (((𝑁𝐴)𝐺𝐴) = 𝑈 ∧ (𝐴𝐺(𝑁𝐴)) = 𝑈)))
2423elrab 3504 . . 3 ((𝑁𝐴) ∈ {𝑦𝑋 ∣ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)} ↔ ((𝑁𝐴) ∈ 𝑋 ∧ (((𝑁𝐴)𝐺𝐴) = 𝑈 ∧ (𝐴𝐺(𝑁𝐴)) = 𝑈)))
2518, 24sylib 208 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁𝐴) ∈ 𝑋 ∧ (((𝑁𝐴)𝐺𝐴) = 𝑈 ∧ (𝐴𝐺(𝑁𝐴)) = 𝑈)))
2625simprd 482 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑁𝐴)𝐺𝐴) = 𝑈 ∧ (𝐴𝐺(𝑁𝐴)) = 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  wrex 3051  ∃!wreu 3052  {crab 3054  ran crn 5267  cfv 6049  crio 6774  (class class class)co 6814  GrpOpcgr 27673  GIdcgi 27674  invcgn 27675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-grpo 27677  df-gid 27678  df-ginv 27679
This theorem is referenced by:  grpolinv  27710  grporinv  27711
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