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Theorem isabli 18128
 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))
32rgen2a 2971 . 2 𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)
4 isabli.b . . 3 𝐵 = (Base‘𝐺)
5 isabli.p . . 3 + = (+g𝐺)
64, 5isabl2 18122 . 2 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
71, 3, 6mpbir2an 954 1 𝐺 ∈ Abel
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∀wral 2907  ‘cfv 5847  (class class class)co 6604  Basecbs 15781  +gcplusg 15862  Grpcgrp 17343  Abelcabl 18115 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 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-iota 5810  df-fv 5855  df-ov 6607  df-grp 17346  df-cmn 18116  df-abl 18117 This theorem is referenced by:  cnaddablx  18192  cnaddabl  18193  zaddablx  18196
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