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Theorem ismgm 17924
 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 6677 . . 3 (𝑚 = 𝑀 → (Base‘𝑚) ∈ V)
2 fveq2 6662 . . . 4 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
3 ismgm.b . . . 4 𝐵 = (Base‘𝑀)
42, 3eqtr4di 2811 . . 3 (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
5 fvexd 6677 . . . 4 ((𝑚 = 𝑀𝑏 = 𝐵) → (+g𝑚) ∈ V)
6 fveq2 6662 . . . . . 6 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
76adantr 484 . . . . 5 ((𝑚 = 𝑀𝑏 = 𝐵) → (+g𝑚) = (+g𝑀))
8 ismgm.o . . . . 5 = (+g𝑀)
97, 8eqtr4di 2811 . . . 4 ((𝑚 = 𝑀𝑏 = 𝐵) → (+g𝑚) = )
10 simplr 768 . . . . 5 (((𝑚 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → 𝑏 = 𝐵)
11 oveq 7161 . . . . . . . 8 (𝑜 = → (𝑥𝑜𝑦) = (𝑥 𝑦))
1211adantl 485 . . . . . . 7 (((𝑚 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (𝑥𝑜𝑦) = (𝑥 𝑦))
1312, 10eleq12d 2846 . . . . . 6 (((𝑚 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → ((𝑥𝑜𝑦) ∈ 𝑏 ↔ (𝑥 𝑦) ∈ 𝐵))
1410, 13raleqbidv 3319 . . . . 5 (((𝑚 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (∀𝑦𝑏 (𝑥𝑜𝑦) ∈ 𝑏 ↔ ∀𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
1510, 14raleqbidv 3319 . . . 4 (((𝑚 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (∀𝑥𝑏𝑦𝑏 (𝑥𝑜𝑦) ∈ 𝑏 ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
165, 9, 15sbcied2 3742 . . 3 ((𝑚 = 𝑀𝑏 = 𝐵) → ([(+g𝑚) / 𝑜]𝑥𝑏𝑦𝑏 (𝑥𝑜𝑦) ∈ 𝑏 ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
171, 4, 16sbcied2 3742 . 2 (𝑚 = 𝑀 → ([(Base‘𝑚) / 𝑏][(+g𝑚) / 𝑜]𝑥𝑏𝑦𝑏 (𝑥𝑜𝑦) ∈ 𝑏 ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
18 df-mgm 17923 . 2 Mgm = {𝑚[(Base‘𝑚) / 𝑏][(+g𝑚) / 𝑜]𝑥𝑏𝑦𝑏 (𝑥𝑜𝑦) ∈ 𝑏}
1917, 18elab2g 3591 1 (𝑀𝑉 → (𝑀 ∈ Mgm ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  Vcvv 3409  [wsbc 3698  ‘cfv 6339  (class class class)co 7155  Basecbs 16546  +gcplusg 16628  Mgmcmgm 17921 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 2729  ax-nul 5179 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5036  df-iota 6298  df-fv 6347  df-ov 7158  df-mgm 17923 This theorem is referenced by:  ismgmn0  17925  mgmcl  17926  issstrmgm  17934  mgm0  17937  issgrpv  17974  efmndmgm  18121  smndex1mgm  18143  0mgm  44789  ismgmd  44791  mgm2mgm  44882  lidlmmgm  44944
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