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Theorem grpinvex 17640
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 17636 . . 3 (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑥𝐵𝑦𝐵 (𝑦 + 𝑥) = 0 ))
54simprbi 478 . 2 (𝐺 ∈ Grp → ∀𝑥𝐵𝑦𝐵 (𝑦 + 𝑥) = 0 )
6 oveq2 6801 . . . . 5 (𝑥 = 𝑋 → (𝑦 + 𝑥) = (𝑦 + 𝑋))
76eqeq1d 2773 . . . 4 (𝑥 = 𝑋 → ((𝑦 + 𝑥) = 0 ↔ (𝑦 + 𝑋) = 0 ))
87rexbidv 3200 . . 3 (𝑥 = 𝑋 → (∃𝑦𝐵 (𝑦 + 𝑥) = 0 ↔ ∃𝑦𝐵 (𝑦 + 𝑋) = 0 ))
98rspccva 3459 . 2 ((∀𝑥𝐵𝑦𝐵 (𝑦 + 𝑥) = 0𝑋𝐵) → ∃𝑦𝐵 (𝑦 + 𝑋) = 0 )
105, 9sylan 561 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ∃𝑦𝐵 (𝑦 + 𝑋) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wral 3061  wrex 3062  cfv 6031  (class class class)co 6793  Basecbs 16064  +gcplusg 16149  0gc0g 16308  Mndcmnd 17502  Grpcgrp 17630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-ov 6796  df-grp 17633
This theorem is referenced by:  dfgrp2  17655  grprcan  17663  grpinveu  17664  grprinv  17677
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