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Theorem grpinveu 18790
Description: The left inverse element of a group is unique. Lemma 2.2.1(b) of [Herstein] p. 55. (Contributed by NM, 24-Aug-2011.)
Hypotheses
Ref Expression
grpinveu.b 𝐵 = (Base‘𝐺)
grpinveu.p + = (+g𝐺)
grpinveu.o 0 = (0g𝐺)
Assertion
Ref Expression
grpinveu ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ∃!𝑦𝐵 (𝑦 + 𝑋) = 0 )
Distinct variable groups:   𝑦,𝐵   𝑦,𝐺   𝑦, +   𝑦, 0   𝑦,𝑋

Proof of Theorem grpinveu
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 grpinveu.b . . . 4 𝐵 = (Base‘𝐺)
2 grpinveu.p . . . 4 + = (+g𝐺)
3 grpinveu.o . . . 4 0 = (0g𝐺)
41, 2, 3grpinvex 18763 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ∃𝑦𝐵 (𝑦 + 𝑋) = 0 )
5 eqtr3 2759 . . . . . . . . . . . 12 (((𝑦 + 𝑋) = 0 ∧ (𝑧 + 𝑋) = 0 ) → (𝑦 + 𝑋) = (𝑧 + 𝑋))
61, 2grprcan 18789 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ (𝑦𝐵𝑧𝐵𝑋𝐵)) → ((𝑦 + 𝑋) = (𝑧 + 𝑋) ↔ 𝑦 = 𝑧))
75, 6imbitrid 243 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (𝑦𝐵𝑧𝐵𝑋𝐵)) → (((𝑦 + 𝑋) = 0 ∧ (𝑧 + 𝑋) = 0 ) → 𝑦 = 𝑧))
873exp2 1355 . . . . . . . . . 10 (𝐺 ∈ Grp → (𝑦𝐵 → (𝑧𝐵 → (𝑋𝐵 → (((𝑦 + 𝑋) = 0 ∧ (𝑧 + 𝑋) = 0 ) → 𝑦 = 𝑧)))))
98com24 95 . . . . . . . . 9 (𝐺 ∈ Grp → (𝑋𝐵 → (𝑧𝐵 → (𝑦𝐵 → (((𝑦 + 𝑋) = 0 ∧ (𝑧 + 𝑋) = 0 ) → 𝑦 = 𝑧)))))
109imp41 427 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑧𝐵) ∧ 𝑦𝐵) → (((𝑦 + 𝑋) = 0 ∧ (𝑧 + 𝑋) = 0 ) → 𝑦 = 𝑧))
1110an32s 651 . . . . . . 7 ((((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑦𝐵) ∧ 𝑧𝐵) → (((𝑦 + 𝑋) = 0 ∧ (𝑧 + 𝑋) = 0 ) → 𝑦 = 𝑧))
1211expd 417 . . . . . 6 ((((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑦𝐵) ∧ 𝑧𝐵) → ((𝑦 + 𝑋) = 0 → ((𝑧 + 𝑋) = 0𝑦 = 𝑧)))
1312ralrimdva 3148 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑦𝐵) → ((𝑦 + 𝑋) = 0 → ∀𝑧𝐵 ((𝑧 + 𝑋) = 0𝑦 = 𝑧)))
1413ancld 552 . . . 4 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑦𝐵) → ((𝑦 + 𝑋) = 0 → ((𝑦 + 𝑋) = 0 ∧ ∀𝑧𝐵 ((𝑧 + 𝑋) = 0𝑦 = 𝑧))))
1514reximdva 3162 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (∃𝑦𝐵 (𝑦 + 𝑋) = 0 → ∃𝑦𝐵 ((𝑦 + 𝑋) = 0 ∧ ∀𝑧𝐵 ((𝑧 + 𝑋) = 0𝑦 = 𝑧))))
164, 15mpd 15 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ∃𝑦𝐵 ((𝑦 + 𝑋) = 0 ∧ ∀𝑧𝐵 ((𝑧 + 𝑋) = 0𝑦 = 𝑧)))
17 oveq1 7365 . . . 4 (𝑦 = 𝑧 → (𝑦 + 𝑋) = (𝑧 + 𝑋))
1817eqeq1d 2735 . . 3 (𝑦 = 𝑧 → ((𝑦 + 𝑋) = 0 ↔ (𝑧 + 𝑋) = 0 ))
1918reu8 3692 . 2 (∃!𝑦𝐵 (𝑦 + 𝑋) = 0 ↔ ∃𝑦𝐵 ((𝑦 + 𝑋) = 0 ∧ ∀𝑧𝐵 ((𝑧 + 𝑋) = 0𝑦 = 𝑧)))
2016, 19sylibr 233 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ∃!𝑦𝐵 (𝑦 + 𝑋) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3061  wrex 3070  ∃!wreu 3350  cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  0gc0g 17326  Grpcgrp 18753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-riota 7314  df-ov 7361  df-0g 17328  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-grp 18756
This theorem is referenced by:  grpinvf  18802  grplinv  18805  isgrpinv  18809
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