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Theorem grpinvex 12819
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 12815 . . 3 (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑥𝐵𝑦𝐵 (𝑦 + 𝑥) = 0 ))
54simprbi 275 . 2 (𝐺 ∈ Grp → ∀𝑥𝐵𝑦𝐵 (𝑦 + 𝑥) = 0 )
6 oveq2 5880 . . . . 5 (𝑥 = 𝑋 → (𝑦 + 𝑥) = (𝑦 + 𝑋))
76eqeq1d 2186 . . . 4 (𝑥 = 𝑋 → ((𝑦 + 𝑥) = 0 ↔ (𝑦 + 𝑋) = 0 ))
87rexbidv 2478 . . 3 (𝑥 = 𝑋 → (∃𝑦𝐵 (𝑦 + 𝑥) = 0 ↔ ∃𝑦𝐵 (𝑦 + 𝑋) = 0 ))
98rspccva 2840 . 2 ((∀𝑥𝐵𝑦𝐵 (𝑦 + 𝑥) = 0𝑋𝐵) → ∃𝑦𝐵 (𝑦 + 𝑋) = 0 )
105, 9sylan 283 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ∃𝑦𝐵 (𝑦 + 𝑋) = 0 )
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  wral 2455  wrex 2456  cfv 5215  (class class class)co 5872  Basecbs 12454  +gcplusg 12528  0gc0g 12693  Mndcmnd 12749  Grpcgrp 12809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-iota 5177  df-fv 5223  df-ov 5875  df-grp 12812
This theorem is referenced by:  dfgrp2  12834  grprcan  12842  grpinveu  12843  grprinv  12855
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