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Theorem ismgmn0 17445
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 2831 . . . 4 (𝐴𝐵𝐴 ∈ (Base‘𝑀))
32biimpi 206 . . 3 (𝐴𝐵𝐴 ∈ (Base‘𝑀))
43elfvexd 6383 . 2 (𝐴𝐵𝑀 ∈ V)
5 ismgmn0.o . . 3 = (+g𝑀)
61, 5ismgm 17444 . 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 196   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340  cfv 6049  (class class class)co 6813  Basecbs 16059  +gcplusg 16143  Mgmcmgm 17441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-nul 4941  ax-pow 4992
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-dm 5276  df-iota 6012  df-fv 6057  df-ov 6816  df-mgm 17443
This theorem is referenced by:  mgm1  17458  opifismgm  17459  issgrpn0  17488  xrsmgm  19983  mgmpropd  42285  opmpt2ismgm  42317  nnsgrpmgm  42326  2zrngamgm  42449  2zrngmmgm  42456
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