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Theorem grpoinveu 27764
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
grpinveu.1 𝑋 = ran 𝐺
grpinveu.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 grpinveu.1 . . . . 5 𝑋 = ran 𝐺
2 grpinveu.2 . . . . 5 𝑈 = (GId‘𝐺)
31, 2grpoidinv2 27760 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
4 simpl 474 . . . . . 6 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈)
54reximi 3156 . . . . 5 (∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → ∃𝑦𝑋 (𝑦𝐺𝐴) = 𝑈)
65adantl 473 . . . 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 2785 . . . . . . . . . . . 12 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → (𝑦𝐺𝐴) = (𝑧𝐺𝐴))
91grporcan 27763 . . . . . . . . . . . 12 ((𝐺 ∈ GrpOp ∧ (𝑦𝑋𝑧𝑋𝐴𝑋)) → ((𝑦𝐺𝐴) = (𝑧𝐺𝐴) ↔ 𝑦 = 𝑧))
108, 9syl5ib 235 . . . . . . . . . . 11 ((𝐺 ∈ GrpOp ∧ (𝑦𝑋𝑧𝑋𝐴𝑋)) → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧))
11103exp2 1463 . . . . . . . . . 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 416 . . . . . . . 8 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑧𝑋) ∧ 𝑦𝑋) → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧))
1413an32s 642 . . . . . . 7 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) ∧ 𝑧𝑋) → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧))
1514expd 404 . . . . . 6 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) ∧ 𝑧𝑋) → ((𝑦𝐺𝐴) = 𝑈 → ((𝑧𝐺𝐴) = 𝑈𝑦 = 𝑧)))
1615ralrimdva 3115 . . . . 5 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → ((𝑦𝐺𝐴) = 𝑈 → ∀𝑧𝑋 ((𝑧𝐺𝐴) = 𝑈𝑦 = 𝑧)))
1716ancld 546 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → ((𝑦𝐺𝐴) = 𝑈 → ((𝑦𝐺𝐴) = 𝑈 ∧ ∀𝑧𝑋 ((𝑧𝐺𝐴) = 𝑈𝑦 = 𝑧))))
1817reximdva 3162 . . 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 6848 . . . 4 (𝑦 = 𝑧 → (𝑦𝐺𝐴) = (𝑧𝐺𝐴))
2120eqeq1d 2766 . . 3 (𝑦 = 𝑧 → ((𝑦𝐺𝐴) = 𝑈 ↔ (𝑧𝐺𝐴) = 𝑈))
2221reu8 3560 . 2 (∃!𝑦𝑋 (𝑦𝐺𝐴) = 𝑈 ↔ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ ∀𝑧𝑋 ((𝑧𝐺𝐴) = 𝑈𝑦 = 𝑧)))
2319, 22sylibr 225 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ∃!𝑦𝑋 (𝑦𝐺𝐴) = 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3054  wrex 3055  ∃!wreu 3056  ran crn 5277  cfv 6067  (class class class)co 6841  GrpOpcgr 27734  GIdcgi 27735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-sep 4940  ax-nul 4948  ax-pr 5061  ax-un 7146
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-nul 4079  df-if 4243  df-sn 4334  df-pr 4336  df-op 4340  df-uni 4594  df-iun 4677  df-br 4809  df-opab 4871  df-mpt 4888  df-id 5184  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-fo 6073  df-fv 6075  df-riota 6802  df-ov 6844  df-grpo 27738  df-gid 27739
This theorem is referenced by:  grpoinvcl  27769  grpoinv  27770
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