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Theorem ismgm 17558
Description: The predicate "is a magma". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)
Hypotheses
Ref Expression
ismgm.b 𝐵 = (Base‘𝑀)
ismgm.o = (+g𝑀)
Assertion
Ref Expression
ismgm (𝑀𝑉 → (𝑀 ∈ Mgm ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝑀,𝑦   𝑥, ,𝑦
Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem ismgm
Dummy variables 𝑏 𝑚 𝑜 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvexd 6426 . . 3 (𝑚 = 𝑀 → (Base‘𝑚) ∈ V)
2 fveq2 6411 . . . 4 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
3 ismgm.b . . . 4 𝐵 = (Base‘𝑀)
42, 3syl6eqr 2851 . . 3 (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
5 fvexd 6426 . . . 4 ((𝑚 = 𝑀𝑏 = 𝐵) → (+g𝑚) ∈ V)
6 fveq2 6411 . . . . . 6 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
76adantr 473 . . . . 5 ((𝑚 = 𝑀𝑏 = 𝐵) → (+g𝑚) = (+g𝑀))
8 ismgm.o . . . . 5 = (+g𝑀)
97, 8syl6eqr 2851 . . . 4 ((𝑚 = 𝑀𝑏 = 𝐵) → (+g𝑚) = )
10 simplr 786 . . . . 5 (((𝑚 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → 𝑏 = 𝐵)
11 oveq 6884 . . . . . . . 8 (𝑜 = → (𝑥𝑜𝑦) = (𝑥 𝑦))
1211adantl 474 . . . . . . 7 (((𝑚 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (𝑥𝑜𝑦) = (𝑥 𝑦))
1312, 10eleq12d 2872 . . . . . 6 (((𝑚 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → ((𝑥𝑜𝑦) ∈ 𝑏 ↔ (𝑥 𝑦) ∈ 𝐵))
1410, 13raleqbidv 3335 . . . . 5 (((𝑚 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (∀𝑦𝑏 (𝑥𝑜𝑦) ∈ 𝑏 ↔ ∀𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
1510, 14raleqbidv 3335 . . . 4 (((𝑚 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (∀𝑥𝑏𝑦𝑏 (𝑥𝑜𝑦) ∈ 𝑏 ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
165, 9, 15sbcied2 3671 . . 3 ((𝑚 = 𝑀𝑏 = 𝐵) → ([(+g𝑚) / 𝑜]𝑥𝑏𝑦𝑏 (𝑥𝑜𝑦) ∈ 𝑏 ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
171, 4, 16sbcied2 3671 . 2 (𝑚 = 𝑀 → ([(Base‘𝑚) / 𝑏][(+g𝑚) / 𝑜]𝑥𝑏𝑦𝑏 (𝑥𝑜𝑦) ∈ 𝑏 ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
18 df-mgm 17557 . 2 Mgm = {𝑚[(Base‘𝑚) / 𝑏][(+g𝑚) / 𝑜]𝑥𝑏𝑦𝑏 (𝑥𝑜𝑦) ∈ 𝑏}
1917, 18elab2g 3545 1 (𝑀𝑉 → (𝑀 ∈ Mgm ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wral 3089  Vcvv 3385  [wsbc 3633  cfv 6101  (class class class)co 6878  Basecbs 16184  +gcplusg 16267  Mgmcmgm 17555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-nul 4983
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-iota 6064  df-fv 6109  df-ov 6881  df-mgm 17557
This theorem is referenced by:  ismgmn0  17559  mgmcl  17560  issstrmgm  17567  mgm0  17570  issgrpv  17601  0mgm  42573  ismgmd  42575  mgm2mgm  42662  lidlmmgm  42724
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