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Theorem isabli 18635
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 3191 . 2 𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)
4 isabli.b . . 3 𝐵 = (Base‘𝐺)
5 isabli.p . . 3 + = (+g𝐺)
64, 5isabl2 18629 . 2 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
71, 3, 6mpbir2an 707 1 𝐺 ∈ Abel
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1520  wcel 2079  wral 3103  cfv 6217  (class class class)co 7007  Basecbs 16300  +gcplusg 16382  Grpcgrp 17849  Abelcabl 18622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-br 4957  df-iota 6181  df-fv 6225  df-ov 7010  df-grp 17852  df-cmn 18623  df-abl 18624
This theorem is referenced by:  cnaddablx  18699  cnaddabl  18700  zaddablx  18703
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