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Theorem grpinveu 17678
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 17654 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ∃𝑦𝐵 (𝑦 + 𝑋) = 0 )
5 eqtr3 2782 . . . . . . . . . . . 12 (((𝑦 + 𝑋) = 0 ∧ (𝑧 + 𝑋) = 0 ) → (𝑦 + 𝑋) = (𝑧 + 𝑋))
61, 2grprcan 17677 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ (𝑦𝐵𝑧𝐵𝑋𝐵)) → ((𝑦 + 𝑋) = (𝑧 + 𝑋) ↔ 𝑦 = 𝑧))
75, 6syl5ib 234 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (𝑦𝐵𝑧𝐵𝑋𝐵)) → (((𝑦 + 𝑋) = 0 ∧ (𝑧 + 𝑋) = 0 ) → 𝑦 = 𝑧))
873exp2 1448 . . . . . . . . . 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 620 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑧𝐵) ∧ 𝑦𝐵) → (((𝑦 + 𝑋) = 0 ∧ (𝑧 + 𝑋) = 0 ) → 𝑦 = 𝑧))
1110an32s 881 . . . . . . 7 ((((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑦𝐵) ∧ 𝑧𝐵) → (((𝑦 + 𝑋) = 0 ∧ (𝑧 + 𝑋) = 0 ) → 𝑦 = 𝑧))
1211expd 451 . . . . . 6 ((((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑦𝐵) ∧ 𝑧𝐵) → ((𝑦 + 𝑋) = 0 → ((𝑧 + 𝑋) = 0𝑦 = 𝑧)))
1312ralrimdva 3108 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑦𝐵) → ((𝑦 + 𝑋) = 0 → ∀𝑧𝐵 ((𝑧 + 𝑋) = 0𝑦 = 𝑧)))
1413ancld 577 . . . 4 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑦𝐵) → ((𝑦 + 𝑋) = 0 → ((𝑦 + 𝑋) = 0 ∧ ∀𝑧𝐵 ((𝑧 + 𝑋) = 0𝑦 = 𝑧))))
1514reximdva 3156 . . 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 6822 . . . 4 (𝑦 = 𝑧 → (𝑦 + 𝑋) = (𝑧 + 𝑋))
1817eqeq1d 2763 . . 3 (𝑦 = 𝑧 → ((𝑦 + 𝑋) = 0 ↔ (𝑧 + 𝑋) = 0 ))
1918reu8 3544 . 2 (∃!𝑦𝐵 (𝑦 + 𝑋) = 0 ↔ ∃𝑦𝐵 ((𝑦 + 𝑋) = 0 ∧ ∀𝑧𝐵 ((𝑧 + 𝑋) = 0𝑦 = 𝑧)))
2016, 19sylibr 224 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ∃!𝑦𝐵 (𝑦 + 𝑋) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2140  wral 3051  wrex 3052  ∃!wreu 3053  cfv 6050  (class class class)co 6815  Basecbs 16080  +gcplusg 16164  0gc0g 16323  Grpcgrp 17644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-iota 6013  df-fun 6052  df-fv 6058  df-riota 6776  df-ov 6818  df-0g 16325  df-mgm 17464  df-sgrp 17506  df-mnd 17517  df-grp 17647
This theorem is referenced by:  grpinvf  17688  grplinv  17690  isgrpinv  17694
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