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Theorem grpinvex 18759
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 18755 . . 3 (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑥𝐵𝑦𝐵 (𝑦 + 𝑥) = 0 ))
54simprbi 498 . 2 (𝐺 ∈ Grp → ∀𝑥𝐵𝑦𝐵 (𝑦 + 𝑥) = 0 )
6 oveq2 7366 . . . . 5 (𝑥 = 𝑋 → (𝑦 + 𝑥) = (𝑦 + 𝑋))
76eqeq1d 2739 . . . 4 (𝑥 = 𝑋 → ((𝑦 + 𝑥) = 0 ↔ (𝑦 + 𝑋) = 0 ))
87rexbidv 3176 . . 3 (𝑥 = 𝑋 → (∃𝑦𝐵 (𝑦 + 𝑥) = 0 ↔ ∃𝑦𝐵 (𝑦 + 𝑋) = 0 ))
98rspccva 3581 . 2 ((∀𝑥𝐵𝑦𝐵 (𝑦 + 𝑥) = 0𝑋𝐵) → ∃𝑦𝐵 (𝑦 + 𝑋) = 0 )
105, 9sylan 581 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ∃𝑦𝐵 (𝑦 + 𝑋) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3065  wrex 3074  cfv 6497  (class class class)co 7358  Basecbs 17084  +gcplusg 17134  0gc0g 17322  Mndcmnd 18557  Grpcgrp 18749
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-ext 2708
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-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  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-iota 6449  df-fv 6505  df-ov 7361  df-grp 18752
This theorem is referenced by:  dfgrp2  18776  grprcan  18785  grpinveu  18786  grprinv  18802
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