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Theorem ismgmn0 18655
Description: The predicate "is a magma" for a structure with a nonempty base set. (Contributed by AV, 29-Jan-2020.)
Hypotheses
Ref Expression
ismgmn0.b 𝐵 = (Base‘𝑀)
ismgmn0.o = (+g𝑀)
Assertion
Ref Expression
ismgmn0 (𝐴𝐵 → (𝑀 ∈ Mgm ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝑀,𝑦   𝑥, ,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem ismgmn0
StepHypRef Expression
1 ismgmn0.b . . . . 5 𝐵 = (Base‘𝑀)
21eleq2i 2833 . . . 4 (𝐴𝐵𝐴 ∈ (Base‘𝑀))
32biimpi 216 . . 3 (𝐴𝐵𝐴 ∈ (Base‘𝑀))
43elfvexd 6945 . 2 (𝐴𝐵𝑀 ∈ V)
5 ismgmn0.o . . 3 = (+g𝑀)
61, 5ismgm 18654 . 2 (𝑀 ∈ V → (𝑀 ∈ Mgm ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
74, 6syl 17 1 (𝐴𝐵 → (𝑀 ∈ Mgm ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  Mgmcmgm 18651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-dm 5695  df-iota 6514  df-fv 6569  df-ov 7434  df-mgm 18653
This theorem is referenced by:  mgmpropd  18664  mgm1  18671  opifismgm  18672  issgrpn0  18735  xrsmgm  21419  opmpoismgm  48083  nnsgrpmgm  48092  2zrngamgm  48161  2zrngmmgm  48168
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