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Theorem grpinvex 17364
Description: Every member of a group has a left inverse. (Contributed by NM, 16-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
grpcl.b 𝐵 = (Base‘𝐺)
grpcl.p + = (+g𝐺)
grpinvex.p 0 = (0g𝐺)
Assertion
Ref Expression
grpinvex ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ∃𝑦𝐵 (𝑦 + 𝑋) = 0 )
Distinct variable groups:   𝑦,𝐵   𝑦,𝐺   𝑦,𝑋
Allowed substitution hints:   + (𝑦)   0 (𝑦)

Proof of Theorem grpinvex
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 grpcl.b . . . 4 𝐵 = (Base‘𝐺)
2 grpcl.p . . . 4 + = (+g𝐺)
3 grpinvex.p . . . 4 0 = (0g𝐺)
41, 2, 3isgrp 17360 . . 3 (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑥𝐵𝑦𝐵 (𝑦 + 𝑥) = 0 ))
54simprbi 480 . 2 (𝐺 ∈ Grp → ∀𝑥𝐵𝑦𝐵 (𝑦 + 𝑥) = 0 )
6 oveq2 6618 . . . . 5 (𝑥 = 𝑋 → (𝑦 + 𝑥) = (𝑦 + 𝑋))
76eqeq1d 2623 . . . 4 (𝑥 = 𝑋 → ((𝑦 + 𝑥) = 0 ↔ (𝑦 + 𝑋) = 0 ))
87rexbidv 3046 . . 3 (𝑥 = 𝑋 → (∃𝑦𝐵 (𝑦 + 𝑥) = 0 ↔ ∃𝑦𝐵 (𝑦 + 𝑋) = 0 ))
98rspccva 3297 . 2 ((∀𝑥𝐵𝑦𝐵 (𝑦 + 𝑥) = 0𝑋𝐵) → ∃𝑦𝐵 (𝑦 + 𝑋) = 0 )
105, 9sylan 488 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ∃𝑦𝐵 (𝑦 + 𝑋) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  cfv 5852  (class class class)co 6610  Basecbs 15792  +gcplusg 15873  0gc0g 16032  Mndcmnd 17226  Grpcgrp 17354
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-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
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-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-iota 5815  df-fv 5860  df-ov 6613  df-grp 17357
This theorem is referenced by:  dfgrp2  17379  grprcan  17387  grpinveu  17388  grprinv  17401
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