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Theorem grpinveu 17396
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 17372 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ∃𝑦𝐵 (𝑦 + 𝑋) = 0 )
5 eqtr3 2642 . . . . . . . . . . . 12 (((𝑦 + 𝑋) = 0 ∧ (𝑧 + 𝑋) = 0 ) → (𝑦 + 𝑋) = (𝑧 + 𝑋))
61, 2grprcan 17395 . . . . . . . . . . . 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 1282 . . . . . . . . . 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 618 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑧𝐵) ∧ 𝑦𝐵) → (((𝑦 + 𝑋) = 0 ∧ (𝑧 + 𝑋) = 0 ) → 𝑦 = 𝑧))
1110an32s 845 . . . . . . 7 ((((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑦𝐵) ∧ 𝑧𝐵) → (((𝑦 + 𝑋) = 0 ∧ (𝑧 + 𝑋) = 0 ) → 𝑦 = 𝑧))
1211expd 452 . . . . . 6 ((((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑦𝐵) ∧ 𝑧𝐵) → ((𝑦 + 𝑋) = 0 → ((𝑧 + 𝑋) = 0𝑦 = 𝑧)))
1312ralrimdva 2965 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑦𝐵) → ((𝑦 + 𝑋) = 0 → ∀𝑧𝐵 ((𝑧 + 𝑋) = 0𝑦 = 𝑧)))
1413ancld 575 . . . 4 (((𝐺 ∈ Grp ∧ 𝑋𝐵) ∧ 𝑦𝐵) → ((𝑦 + 𝑋) = 0 → ((𝑦 + 𝑋) = 0 ∧ ∀𝑧𝐵 ((𝑧 + 𝑋) = 0𝑦 = 𝑧))))
1514reximdva 3013 . . 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 6622 . . . 4 (𝑦 = 𝑧 → (𝑦 + 𝑋) = (𝑧 + 𝑋))
1817eqeq1d 2623 . . 3 (𝑦 = 𝑧 → ((𝑦 + 𝑋) = 0 ↔ (𝑧 + 𝑋) = 0 ))
1918reu8 3389 . 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 384  w3a 1036   = wceq 1480  wcel 1987  wral 2908  wrex 2909  ∃!wreu 2910  cfv 5857  (class class class)co 6615  Basecbs 15800  +gcplusg 15881  0gc0g 16040  Grpcgrp 17362
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 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
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-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-riota 6576  df-ov 6618  df-0g 16042  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-grp 17365
This theorem is referenced by:  grpinvf  17406  grplinv  17408  isgrpinv  17412
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