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Theorem isabli 18404
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 3165 . 2 𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)
4 isabli.b . . 3 𝐵 = (Base‘𝐺)
5 isabli.p . . 3 + = (+g𝐺)
64, 5isabl2 18398 . 2 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
71, 3, 6mpbir2an 693 1 𝐺 ∈ Abel
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wcel 2156  wral 3096  cfv 6097  (class class class)co 6870  Basecbs 16064  +gcplusg 16149  Grpcgrp 17623  Abelcabl 18391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-iota 6060  df-fv 6105  df-ov 6873  df-grp 17626  df-cmn 18392  df-abl 18393
This theorem is referenced by:  cnaddablx  18468  cnaddabl  18469  zaddablx  18472
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