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Theorem grpoinveu 27713
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 27709 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
4 simpl 468 . . . . . 6 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈)
54reximi 3159 . . . . 5 (∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → ∃𝑦𝑋 (𝑦𝐺𝐴) = 𝑈)
65adantl 467 . . . 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 2792 . . . . . . . . . . . 12 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → (𝑦𝐺𝐴) = (𝑧𝐺𝐴))
91grporcan 27712 . . . . . . . . . . . 12 ((𝐺 ∈ GrpOp ∧ (𝑦𝑋𝑧𝑋𝐴𝑋)) → ((𝑦𝐺𝐴) = (𝑧𝐺𝐴) ↔ 𝑦 = 𝑧))
108, 9syl5ib 234 . . . . . . . . . . 11 ((𝐺 ∈ GrpOp ∧ (𝑦𝑋𝑧𝑋𝐴𝑋)) → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧))
11103exp2 1447 . . . . . . . . . 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 412 . . . . . . . 8 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑧𝑋) ∧ 𝑦𝑋) → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧))
1413an32s 631 . . . . . . 7 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) ∧ 𝑧𝑋) → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧))
1514expd 400 . . . . . 6 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) ∧ 𝑧𝑋) → ((𝑦𝐺𝐴) = 𝑈 → ((𝑧𝐺𝐴) = 𝑈𝑦 = 𝑧)))
1615ralrimdva 3118 . . . . 5 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → ((𝑦𝐺𝐴) = 𝑈 → ∀𝑧𝑋 ((𝑧𝐺𝐴) = 𝑈𝑦 = 𝑧)))
1716ancld 540 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → ((𝑦𝐺𝐴) = 𝑈 → ((𝑦𝐺𝐴) = 𝑈 ∧ ∀𝑧𝑋 ((𝑧𝐺𝐴) = 𝑈𝑦 = 𝑧))))
1817reximdva 3165 . . 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 6800 . . . 4 (𝑦 = 𝑧 → (𝑦𝐺𝐴) = (𝑧𝐺𝐴))
2120eqeq1d 2773 . . 3 (𝑦 = 𝑧 → ((𝑦𝐺𝐴) = 𝑈 ↔ (𝑧𝐺𝐴) = 𝑈))
2221reu8 3554 . 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 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  wrex 3062  ∃!wreu 3063  ran crn 5250  cfv 6031  (class class class)co 6793  GrpOpcgr 27683  GIdcgi 27684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fo 6037  df-fv 6039  df-riota 6754  df-ov 6796  df-grpo 27687  df-gid 27688
This theorem is referenced by:  grpoinvcl  27718  grpoinv  27719
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