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Theorem grpoinveu 30303
Description: The left inverse element of a group is unique. Lemma 2.2.1(b) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpoinveu.1 𝑋 = ran 𝐺
grpoinveu.2 𝑈 = (GId‘𝐺)
Assertion
Ref Expression
grpoinveu ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ∃!𝑦𝑋 (𝑦𝐺𝐴) = 𝑈)
Distinct variable groups:   𝑦,𝐴   𝑦,𝐺   𝑦,𝑈   𝑦,𝑋

Proof of Theorem grpoinveu
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 grpoinveu.1 . . . . 5 𝑋 = ran 𝐺
2 grpoinveu.2 . . . . 5 𝑈 = (GId‘𝐺)
31, 2grpoidinv2 30299 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
4 simpl 482 . . . . . 6 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈)
54reximi 3079 . . . . 5 (∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → ∃𝑦𝑋 (𝑦𝐺𝐴) = 𝑈)
65adantl 481 . . . 4 ((((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)) → ∃𝑦𝑋 (𝑦𝐺𝐴) = 𝑈)
73, 6syl 17 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ∃𝑦𝑋 (𝑦𝐺𝐴) = 𝑈)
8 eqtr3 2753 . . . . . . . . . . . 12 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → (𝑦𝐺𝐴) = (𝑧𝐺𝐴))
91grporcan 30302 . . . . . . . . . . . 12 ((𝐺 ∈ GrpOp ∧ (𝑦𝑋𝑧𝑋𝐴𝑋)) → ((𝑦𝐺𝐴) = (𝑧𝐺𝐴) ↔ 𝑦 = 𝑧))
108, 9imbitrid 243 . . . . . . . . . . 11 ((𝐺 ∈ GrpOp ∧ (𝑦𝑋𝑧𝑋𝐴𝑋)) → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧))
11103exp2 1352 . . . . . . . . . 10 (𝐺 ∈ GrpOp → (𝑦𝑋 → (𝑧𝑋 → (𝐴𝑋 → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧)))))
1211com24 95 . . . . . . . . 9 (𝐺 ∈ GrpOp → (𝐴𝑋 → (𝑧𝑋 → (𝑦𝑋 → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧)))))
1312imp41 425 . . . . . . . 8 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑧𝑋) ∧ 𝑦𝑋) → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧))
1413an32s 651 . . . . . . 7 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) ∧ 𝑧𝑋) → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧))
1514expd 415 . . . . . 6 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) ∧ 𝑧𝑋) → ((𝑦𝐺𝐴) = 𝑈 → ((𝑧𝐺𝐴) = 𝑈𝑦 = 𝑧)))
1615ralrimdva 3149 . . . . 5 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → ((𝑦𝐺𝐴) = 𝑈 → ∀𝑧𝑋 ((𝑧𝐺𝐴) = 𝑈𝑦 = 𝑧)))
1716ancld 550 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → ((𝑦𝐺𝐴) = 𝑈 → ((𝑦𝐺𝐴) = 𝑈 ∧ ∀𝑧𝑋 ((𝑧𝐺𝐴) = 𝑈𝑦 = 𝑧))))
1817reximdva 3163 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (∃𝑦𝑋 (𝑦𝐺𝐴) = 𝑈 → ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ ∀𝑧𝑋 ((𝑧𝐺𝐴) = 𝑈𝑦 = 𝑧))))
197, 18mpd 15 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ ∀𝑧𝑋 ((𝑧𝐺𝐴) = 𝑈𝑦 = 𝑧)))
20 oveq1 7421 . . . 4 (𝑦 = 𝑧 → (𝑦𝐺𝐴) = (𝑧𝐺𝐴))
2120eqeq1d 2729 . . 3 (𝑦 = 𝑧 → ((𝑦𝐺𝐴) = 𝑈 ↔ (𝑧𝐺𝐴) = 𝑈))
2221reu8 3726 . 2 (∃!𝑦𝑋 (𝑦𝐺𝐴) = 𝑈 ↔ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ ∀𝑧𝑋 ((𝑧𝐺𝐴) = 𝑈𝑦 = 𝑧)))
2319, 22sylibr 233 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ∃!𝑦𝑋 (𝑦𝐺𝐴) = 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085   = wceq 1534  wcel 2099  wral 3056  wrex 3065  ∃!wreu 3369  ran crn 5673  cfv 6542  (class class class)co 7414  GrpOpcgr 30273  GIdcgi 30274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fo 6548  df-fv 6550  df-riota 7370  df-ov 7417  df-grpo 30277  df-gid 30278
This theorem is referenced by:  grpoinvcl  30308  grpoinv  30309
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