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Theorem grpoinveu 27234
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 27230 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
4 simpl 473 . . . . . 6 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈)
54reximi 3005 . . . . 5 (∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → ∃𝑦𝑋 (𝑦𝐺𝐴) = 𝑈)
65adantl 482 . . . 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 2642 . . . . . . . . . . . 12 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → (𝑦𝐺𝐴) = (𝑧𝐺𝐴))
91grporcan 27233 . . . . . . . . . . . 12 ((𝐺 ∈ GrpOp ∧ (𝑦𝑋𝑧𝑋𝐴𝑋)) → ((𝑦𝐺𝐴) = (𝑧𝐺𝐴) ↔ 𝑦 = 𝑧))
108, 9syl5ib 234 . . . . . . . . . . 11 ((𝐺 ∈ GrpOp ∧ (𝑦𝑋𝑧𝑋𝐴𝑋)) → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧))
11103exp2 1282 . . . . . . . . . 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 618 . . . . . . . 8 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑧𝑋) ∧ 𝑦𝑋) → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧))
1413an32s 845 . . . . . . 7 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) ∧ 𝑧𝑋) → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧))
1514expd 452 . . . . . 6 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) ∧ 𝑧𝑋) → ((𝑦𝐺𝐴) = 𝑈 → ((𝑧𝐺𝐴) = 𝑈𝑦 = 𝑧)))
1615ralrimdva 2963 . . . . 5 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → ((𝑦𝐺𝐴) = 𝑈 → ∀𝑧𝑋 ((𝑧𝐺𝐴) = 𝑈𝑦 = 𝑧)))
1716ancld 575 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → ((𝑦𝐺𝐴) = 𝑈 → ((𝑦𝐺𝐴) = 𝑈 ∧ ∀𝑧𝑋 ((𝑧𝐺𝐴) = 𝑈𝑦 = 𝑧))))
1817reximdva 3011 . . 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 6614 . . . 4 (𝑦 = 𝑧 → (𝑦𝐺𝐴) = (𝑧𝐺𝐴))
2120eqeq1d 2623 . . 3 (𝑦 = 𝑧 → ((𝑦𝐺𝐴) = 𝑈 ↔ (𝑧𝐺𝐴) = 𝑈))
2221reu8 3385 . 2 (∃!𝑦𝑋 (𝑦𝐺𝐴) = 𝑈 ↔ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ ∀𝑧𝑋 ((𝑧𝐺𝐴) = 𝑈𝑦 = 𝑧)))
2319, 22sylibr 224 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ∃!𝑦𝑋 (𝑦𝐺𝐴) = 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  ∃!wreu 2909  ran crn 5077  cfv 5849  (class class class)co 6607  GrpOpcgr 27204  GIdcgi 27205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-fo 5855  df-fv 5857  df-riota 6568  df-ov 6610  df-grpo 27208  df-gid 27209
This theorem is referenced by:  grpoinvcl  27239  grpoinv  27240
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