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Theorem ismgmn0 18572
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 2819 . . . 4 (𝐴𝐵𝐴 ∈ (Base‘𝑀))
32biimpi 215 . . 3 (𝐴𝐵𝐴 ∈ (Base‘𝑀))
43elfvexd 6923 . 2 (𝐴𝐵𝑀 ∈ V)
5 ismgmn0.o . . 3 = (+g𝑀)
61, 5ismgm 18571 . 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 205   = wceq 1533  wcel 2098  wral 3055  Vcvv 3468  cfv 6536  (class class class)co 7404  Basecbs 17150  +gcplusg 17203  Mgmcmgm 18568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-dm 5679  df-iota 6488  df-fv 6544  df-ov 7407  df-mgm 18570
This theorem is referenced by:  mgmpropd  18581  mgm1  18588  opifismgm  18589  issgrpn0  18652  xrsmgm  21290  opmpoismgm  47099  nnsgrpmgm  47108  2zrngamgm  47177  2zrngmmgm  47184
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