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Theorem ismgmn0 17846
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 2881 . . . 4 (𝐴𝐵𝐴 ∈ (Base‘𝑀))
32biimpi 219 . . 3 (𝐴𝐵𝐴 ∈ (Base‘𝑀))
43elfvexd 6679 . 2 (𝐴𝐵𝑀 ∈ V)
5 ismgmn0.o . . 3 = (+g𝑀)
61, 5ismgm 17845 . 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 209   = wceq 1538  wcel 2111  wral 3106  Vcvv 3441  cfv 6324  (class class class)co 7135  Basecbs 16475  +gcplusg 16557  Mgmcmgm 17842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-nul 5174  ax-pow 5231
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-dm 5529  df-iota 6283  df-fv 6332  df-ov 7138  df-mgm 17844
This theorem is referenced by:  mgm1  17860  opifismgm  17861  issgrpn0  17896  xrsmgm  20126  mgmpropd  44393  opmpoismgm  44425  nnsgrpmgm  44434  2zrngamgm  44561  2zrngmmgm  44568
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