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Theorem grpoinveu 28210
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 28206 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
4 simpl 483 . . . . . 6 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈)
54reximi 3248 . . . . 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 2848 . . . . . . . . . . . 12 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → (𝑦𝐺𝐴) = (𝑧𝐺𝐴))
91grporcan 28209 . . . . . . . . . . . 12 ((𝐺 ∈ GrpOp ∧ (𝑦𝑋𝑧𝑋𝐴𝑋)) → ((𝑦𝐺𝐴) = (𝑧𝐺𝐴) ↔ 𝑦 = 𝑧))
108, 9syl5ib 245 . . . . . . . . . . 11 ((𝐺 ∈ GrpOp ∧ (𝑦𝑋𝑧𝑋𝐴𝑋)) → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧))
11103exp2 1348 . . . . . . . . . 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 426 . . . . . . . 8 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑧𝑋) ∧ 𝑦𝑋) → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧))
1413an32s 648 . . . . . . 7 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) ∧ 𝑧𝑋) → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧))
1514expd 416 . . . . . 6 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) ∧ 𝑧𝑋) → ((𝑦𝐺𝐴) = 𝑈 → ((𝑧𝐺𝐴) = 𝑈𝑦 = 𝑧)))
1615ralrimdva 3194 . . . . 5 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → ((𝑦𝐺𝐴) = 𝑈 → ∀𝑧𝑋 ((𝑧𝐺𝐴) = 𝑈𝑦 = 𝑧)))
1716ancld 551 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → ((𝑦𝐺𝐴) = 𝑈 → ((𝑦𝐺𝐴) = 𝑈 ∧ ∀𝑧𝑋 ((𝑧𝐺𝐴) = 𝑈𝑦 = 𝑧))))
1817reximdva 3279 . . 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 7155 . . . 4 (𝑦 = 𝑧 → (𝑦𝐺𝐴) = (𝑧𝐺𝐴))
2120eqeq1d 2828 . . 3 (𝑦 = 𝑧 → ((𝑦𝐺𝐴) = 𝑈 ↔ (𝑧𝐺𝐴) = 𝑈))
2221reu8 3728 . 2 (∃!𝑦𝑋 (𝑦𝐺𝐴) = 𝑈 ↔ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ ∀𝑧𝑋 ((𝑧𝐺𝐴) = 𝑈𝑦 = 𝑧)))
2319, 22sylibr 235 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ∃!𝑦𝑋 (𝑦𝐺𝐴) = 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1081   = wceq 1530  wcel 2107  wral 3143  wrex 3144  ∃!wreu 3145  ran crn 5555  cfv 6352  (class class class)co 7148  GrpOpcgr 28180  GIdcgi 28181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-fo 6358  df-fv 6360  df-riota 7106  df-ov 7151  df-grpo 28184  df-gid 28185
This theorem is referenced by:  grpoinvcl  28215  grpoinv  28216
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