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Theorem ismgm 17224
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 6190 . . 3 (𝑚 = 𝑀 → (Base‘𝑚) ∈ V)
2 fveq2 6178 . . . 4 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
3 ismgm.b . . . 4 𝐵 = (Base‘𝑀)
42, 3syl6eqr 2672 . . 3 (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
5 fvexd 6190 . . . 4 ((𝑚 = 𝑀𝑏 = 𝐵) → (+g𝑚) ∈ V)
6 fveq2 6178 . . . . . 6 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
76adantr 481 . . . . 5 ((𝑚 = 𝑀𝑏 = 𝐵) → (+g𝑚) = (+g𝑀))
8 ismgm.o . . . . 5 = (+g𝑀)
97, 8syl6eqr 2672 . . . 4 ((𝑚 = 𝑀𝑏 = 𝐵) → (+g𝑚) = )
10 simplr 791 . . . . 5 (((𝑚 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → 𝑏 = 𝐵)
11 oveq 6641 . . . . . . . 8 (𝑜 = → (𝑥𝑜𝑦) = (𝑥 𝑦))
1211adantl 482 . . . . . . 7 (((𝑚 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (𝑥𝑜𝑦) = (𝑥 𝑦))
1312, 10eleq12d 2693 . . . . . 6 (((𝑚 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → ((𝑥𝑜𝑦) ∈ 𝑏 ↔ (𝑥 𝑦) ∈ 𝐵))
1410, 13raleqbidv 3147 . . . . 5 (((𝑚 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (∀𝑦𝑏 (𝑥𝑜𝑦) ∈ 𝑏 ↔ ∀𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
1510, 14raleqbidv 3147 . . . 4 (((𝑚 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (∀𝑥𝑏𝑦𝑏 (𝑥𝑜𝑦) ∈ 𝑏 ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
165, 9, 15sbcied2 3467 . . 3 ((𝑚 = 𝑀𝑏 = 𝐵) → ([(+g𝑚) / 𝑜]𝑥𝑏𝑦𝑏 (𝑥𝑜𝑦) ∈ 𝑏 ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
171, 4, 16sbcied2 3467 . 2 (𝑚 = 𝑀 → ([(Base‘𝑚) / 𝑏][(+g𝑚) / 𝑜]𝑥𝑏𝑦𝑏 (𝑥𝑜𝑦) ∈ 𝑏 ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
18 df-mgm 17223 . 2 Mgm = {𝑚[(Base‘𝑚) / 𝑏][(+g𝑚) / 𝑜]𝑥𝑏𝑦𝑏 (𝑥𝑜𝑦) ∈ 𝑏}
1917, 18elab2g 3347 1 (𝑀𝑉 → (𝑀 ∈ Mgm ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wral 2909  Vcvv 3195  [wsbc 3429  cfv 5876  (class class class)co 6635  Basecbs 15838  +gcplusg 15922  Mgmcmgm 17221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-nul 4780
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-iota 5839  df-fv 5884  df-ov 6638  df-mgm 17223
This theorem is referenced by:  ismgmn0  17225  mgmcl  17226  issstrmgm  17233  mgm0  17236  issgrpv  17267  0mgm  41539  ismgmd  41541  mgm2mgm  41628  lidlmmgm  41690
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