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Theorem grpoinv 27363
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 27361 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) = (𝑦𝑋 (𝑦𝐺𝐴) = 𝑈))
51, 2grpoinveu 27357 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ∃!𝑦𝑋 (𝑦𝐺𝐴) = 𝑈)
6 riotacl2 6621 . . . . . 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 2700 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ {𝑦𝑋 ∣ (𝑦𝐺𝐴) = 𝑈})
9 simpl 473 . . . . . . . . 9 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈)
109rgenw 2923 . . . . . . . 8 𝑦𝑋 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈)
1110a1i 11 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ∀𝑦𝑋 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈))
121, 2grpoidinv2 27353 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
1312simprd 479 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈))
1411, 13, 53jca 1241 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (∀𝑦𝑋 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) ∧ ∃!𝑦𝑋 (𝑦𝐺𝐴) = 𝑈))
15 reupick2 3911 . . . . . 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 488 . . . . 5 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → ((𝑦𝐺𝐴) = 𝑈 ↔ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
1716rabbidva 3186 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → {𝑦𝑋 ∣ (𝑦𝐺𝐴) = 𝑈} = {𝑦𝑋 ∣ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)})
188, 17eleqtrd 2702 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ {𝑦𝑋 ∣ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)})
19 oveq1 6654 . . . . . 6 (𝑦 = (𝑁𝐴) → (𝑦𝐺𝐴) = ((𝑁𝐴)𝐺𝐴))
2019eqeq1d 2623 . . . . 5 (𝑦 = (𝑁𝐴) → ((𝑦𝐺𝐴) = 𝑈 ↔ ((𝑁𝐴)𝐺𝐴) = 𝑈))
21 oveq2 6655 . . . . . 6 (𝑦 = (𝑁𝐴) → (𝐴𝐺𝑦) = (𝐴𝐺(𝑁𝐴)))
2221eqeq1d 2623 . . . . 5 (𝑦 = (𝑁𝐴) → ((𝐴𝐺𝑦) = 𝑈 ↔ (𝐴𝐺(𝑁𝐴)) = 𝑈))
2320, 22anbi12d 747 . . . 4 (𝑦 = (𝑁𝐴) → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) ↔ (((𝑁𝐴)𝐺𝐴) = 𝑈 ∧ (𝐴𝐺(𝑁𝐴)) = 𝑈)))
2423elrab 3361 . . 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 479 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑁𝐴)𝐺𝐴) = 𝑈 ∧ (𝐴𝐺(𝑁𝐴)) = 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1037   = wceq 1482  wcel 1989  wral 2911  wrex 2912  ∃!wreu 2913  {crab 2915  ran crn 5113  cfv 5886  crio 6607  (class class class)co 6647  GrpOpcgr 27327  GIdcgi 27328  invcgn 27329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-riota 6608  df-ov 6650  df-grpo 27331  df-gid 27332  df-ginv 27333
This theorem is referenced by:  grpolinv  27364  grporinv  27365
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