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Theorem isabli 13056
Description: Properties that determine an Abelian group. (Contributed by NM, 4-Sep-2011.)
Hypotheses
Ref Expression
isabli.g 𝐺 ∈ Grp
isabli.b 𝐵 = (Base‘𝐺)
isabli.p + = (+g𝐺)
isabli.c ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
Assertion
Ref Expression
isabli 𝐺 ∈ Abel
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐺,𝑦
Allowed substitution hints:   + (𝑥,𝑦)

Proof of Theorem isabli
StepHypRef Expression
1 isabli.g . 2 𝐺 ∈ Grp
2 isabli.c . . 3 ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
32rgen2 2563 . 2 𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)
4 isabli.b . . 3 𝐵 = (Base‘𝐺)
5 isabli.p . . 3 + = (+g𝐺)
64, 5isabl2 13050 . 2 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
71, 3, 6mpbir2an 942 1 𝐺 ∈ Abel
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  wral 2455  cfv 5216  (class class class)co 5874  Basecbs 12456  +gcplusg 12530  Grpcgrp 12831  Abelcabl 13042
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-in 3135  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-iota 5178  df-fv 5224  df-ov 5877  df-grp 12834  df-cmn 13043  df-abl 13044
This theorem is referenced by: (None)
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