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Theorem ismgm 17290
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 6241 . . 3 (𝑚 = 𝑀 → (Base‘𝑚) ∈ V)
2 fveq2 6229 . . . 4 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
3 ismgm.b . . . 4 𝐵 = (Base‘𝑀)
42, 3syl6eqr 2703 . . 3 (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
5 fvexd 6241 . . . 4 ((𝑚 = 𝑀𝑏 = 𝐵) → (+g𝑚) ∈ V)
6 fveq2 6229 . . . . . 6 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
76adantr 480 . . . . 5 ((𝑚 = 𝑀𝑏 = 𝐵) → (+g𝑚) = (+g𝑀))
8 ismgm.o . . . . 5 = (+g𝑀)
97, 8syl6eqr 2703 . . . 4 ((𝑚 = 𝑀𝑏 = 𝐵) → (+g𝑚) = )
10 simplr 807 . . . . 5 (((𝑚 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → 𝑏 = 𝐵)
11 oveq 6696 . . . . . . . 8 (𝑜 = → (𝑥𝑜𝑦) = (𝑥 𝑦))
1211adantl 481 . . . . . . 7 (((𝑚 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (𝑥𝑜𝑦) = (𝑥 𝑦))
1312, 10eleq12d 2724 . . . . . 6 (((𝑚 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → ((𝑥𝑜𝑦) ∈ 𝑏 ↔ (𝑥 𝑦) ∈ 𝐵))
1410, 13raleqbidv 3182 . . . . 5 (((𝑚 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (∀𝑦𝑏 (𝑥𝑜𝑦) ∈ 𝑏 ↔ ∀𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
1510, 14raleqbidv 3182 . . . 4 (((𝑚 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (∀𝑥𝑏𝑦𝑏 (𝑥𝑜𝑦) ∈ 𝑏 ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
165, 9, 15sbcied2 3506 . . 3 ((𝑚 = 𝑀𝑏 = 𝐵) → ([(+g𝑚) / 𝑜]𝑥𝑏𝑦𝑏 (𝑥𝑜𝑦) ∈ 𝑏 ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
171, 4, 16sbcied2 3506 . 2 (𝑚 = 𝑀 → ([(Base‘𝑚) / 𝑏][(+g𝑚) / 𝑜]𝑥𝑏𝑦𝑏 (𝑥𝑜𝑦) ∈ 𝑏 ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
18 df-mgm 17289 . 2 Mgm = {𝑚[(Base‘𝑚) / 𝑏][(+g𝑚) / 𝑜]𝑥𝑏𝑦𝑏 (𝑥𝑜𝑦) ∈ 𝑏}
1917, 18elab2g 3385 1 (𝑀𝑉 → (𝑀 ∈ Mgm ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  [wsbc 3468  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  Mgmcmgm 17287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-ov 6693  df-mgm 17289
This theorem is referenced by:  ismgmn0  17291  mgmcl  17292  issstrmgm  17299  mgm0  17302  issgrpv  17333  0mgm  42099  ismgmd  42101  mgm2mgm  42188  lidlmmgm  42250
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