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Theorem isassad 20096
Description: Sufficient condition for being an associative algebra. (Contributed by Mario Carneiro, 5-Dec-2014.)
Hypotheses
Ref Expression
isassad.v (𝜑𝑉 = (Base‘𝑊))
isassad.f (𝜑𝐹 = (Scalar‘𝑊))
isassad.b (𝜑𝐵 = (Base‘𝐹))
isassad.s (𝜑· = ( ·𝑠𝑊))
isassad.t (𝜑× = (.r𝑊))
isassad.1 (𝜑𝑊 ∈ LMod)
isassad.2 (𝜑𝑊 ∈ Ring)
isassad.3 (𝜑𝐹 ∈ CRing)
isassad.4 ((𝜑 ∧ (𝑟𝐵𝑥𝑉𝑦𝑉)) → ((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)))
isassad.5 ((𝜑 ∧ (𝑟𝐵𝑥𝑉𝑦𝑉)) → (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))
Assertion
Ref Expression
isassad (𝜑𝑊 ∈ AssAlg)
Distinct variable groups:   𝑥,𝑟,𝑦,𝐵   𝜑,𝑟,𝑥,𝑦   𝑥,𝑉,𝑦   𝑊,𝑟,𝑥,𝑦
Allowed substitution hints:   · (𝑥,𝑦,𝑟)   × (𝑥,𝑦,𝑟)   𝐹(𝑥,𝑦,𝑟)   𝑉(𝑟)

Proof of Theorem isassad
StepHypRef Expression
1 isassad.1 . . 3 (𝜑𝑊 ∈ LMod)
2 isassad.2 . . 3 (𝜑𝑊 ∈ Ring)
3 isassad.f . . . 4 (𝜑𝐹 = (Scalar‘𝑊))
4 isassad.3 . . . 4 (𝜑𝐹 ∈ CRing)
53, 4eqeltrrd 2917 . . 3 (𝜑 → (Scalar‘𝑊) ∈ CRing)
61, 2, 53jca 1125 . 2 (𝜑 → (𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ (Scalar‘𝑊) ∈ CRing))
7 isassad.4 . . . . 5 ((𝜑 ∧ (𝑟𝐵𝑥𝑉𝑦𝑉)) → ((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)))
8 isassad.5 . . . . 5 ((𝜑 ∧ (𝑟𝐵𝑥𝑉𝑦𝑉)) → (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))
97, 8jca 515 . . . 4 ((𝜑 ∧ (𝑟𝐵𝑥𝑉𝑦𝑉)) → (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))))
109ralrimivvva 3187 . . 3 (𝜑 → ∀𝑟𝐵𝑥𝑉𝑦𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))))
11 isassad.b . . . . 5 (𝜑𝐵 = (Base‘𝐹))
123fveq2d 6665 . . . . 5 (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝑊)))
1311, 12eqtrd 2859 . . . 4 (𝜑𝐵 = (Base‘(Scalar‘𝑊)))
14 isassad.v . . . . 5 (𝜑𝑉 = (Base‘𝑊))
15 isassad.t . . . . . . . . 9 (𝜑× = (.r𝑊))
16 isassad.s . . . . . . . . . 10 (𝜑· = ( ·𝑠𝑊))
1716oveqd 7166 . . . . . . . . 9 (𝜑 → (𝑟 · 𝑥) = (𝑟( ·𝑠𝑊)𝑥))
18 eqidd 2825 . . . . . . . . 9 (𝜑𝑦 = 𝑦)
1915, 17, 18oveq123d 7170 . . . . . . . 8 (𝜑 → ((𝑟 · 𝑥) × 𝑦) = ((𝑟( ·𝑠𝑊)𝑥)(.r𝑊)𝑦))
20 eqidd 2825 . . . . . . . . 9 (𝜑𝑟 = 𝑟)
2115oveqd 7166 . . . . . . . . 9 (𝜑 → (𝑥 × 𝑦) = (𝑥(.r𝑊)𝑦))
2216, 20, 21oveq123d 7170 . . . . . . . 8 (𝜑 → (𝑟 · (𝑥 × 𝑦)) = (𝑟( ·𝑠𝑊)(𝑥(.r𝑊)𝑦)))
2319, 22eqeq12d 2840 . . . . . . 7 (𝜑 → (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ↔ ((𝑟( ·𝑠𝑊)𝑥)(.r𝑊)𝑦) = (𝑟( ·𝑠𝑊)(𝑥(.r𝑊)𝑦))))
24 eqidd 2825 . . . . . . . . 9 (𝜑𝑥 = 𝑥)
2516oveqd 7166 . . . . . . . . 9 (𝜑 → (𝑟 · 𝑦) = (𝑟( ·𝑠𝑊)𝑦))
2615, 24, 25oveq123d 7170 . . . . . . . 8 (𝜑 → (𝑥 × (𝑟 · 𝑦)) = (𝑥(.r𝑊)(𝑟( ·𝑠𝑊)𝑦)))
2726, 22eqeq12d 2840 . . . . . . 7 (𝜑 → ((𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)) ↔ (𝑥(.r𝑊)(𝑟( ·𝑠𝑊)𝑦)) = (𝑟( ·𝑠𝑊)(𝑥(.r𝑊)𝑦))))
2823, 27anbi12d 633 . . . . . 6 (𝜑 → ((((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))) ↔ (((𝑟( ·𝑠𝑊)𝑥)(.r𝑊)𝑦) = (𝑟( ·𝑠𝑊)(𝑥(.r𝑊)𝑦)) ∧ (𝑥(.r𝑊)(𝑟( ·𝑠𝑊)𝑦)) = (𝑟( ·𝑠𝑊)(𝑥(.r𝑊)𝑦)))))
2914, 28raleqbidv 3392 . . . . 5 (𝜑 → (∀𝑦𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))) ↔ ∀𝑦 ∈ (Base‘𝑊)(((𝑟( ·𝑠𝑊)𝑥)(.r𝑊)𝑦) = (𝑟( ·𝑠𝑊)(𝑥(.r𝑊)𝑦)) ∧ (𝑥(.r𝑊)(𝑟( ·𝑠𝑊)𝑦)) = (𝑟( ·𝑠𝑊)(𝑥(.r𝑊)𝑦)))))
3014, 29raleqbidv 3392 . . . 4 (𝜑 → (∀𝑥𝑉𝑦𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))) ↔ ∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)(((𝑟( ·𝑠𝑊)𝑥)(.r𝑊)𝑦) = (𝑟( ·𝑠𝑊)(𝑥(.r𝑊)𝑦)) ∧ (𝑥(.r𝑊)(𝑟( ·𝑠𝑊)𝑦)) = (𝑟( ·𝑠𝑊)(𝑥(.r𝑊)𝑦)))))
3113, 30raleqbidv 3392 . . 3 (𝜑 → (∀𝑟𝐵𝑥𝑉𝑦𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))) ↔ ∀𝑟 ∈ (Base‘(Scalar‘𝑊))∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)(((𝑟( ·𝑠𝑊)𝑥)(.r𝑊)𝑦) = (𝑟( ·𝑠𝑊)(𝑥(.r𝑊)𝑦)) ∧ (𝑥(.r𝑊)(𝑟( ·𝑠𝑊)𝑦)) = (𝑟( ·𝑠𝑊)(𝑥(.r𝑊)𝑦)))))
3210, 31mpbid 235 . 2 (𝜑 → ∀𝑟 ∈ (Base‘(Scalar‘𝑊))∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)(((𝑟( ·𝑠𝑊)𝑥)(.r𝑊)𝑦) = (𝑟( ·𝑠𝑊)(𝑥(.r𝑊)𝑦)) ∧ (𝑥(.r𝑊)(𝑟( ·𝑠𝑊)𝑦)) = (𝑟( ·𝑠𝑊)(𝑥(.r𝑊)𝑦))))
33 eqid 2824 . . 3 (Base‘𝑊) = (Base‘𝑊)
34 eqid 2824 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
35 eqid 2824 . . 3 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
36 eqid 2824 . . 3 ( ·𝑠𝑊) = ( ·𝑠𝑊)
37 eqid 2824 . . 3 (.r𝑊) = (.r𝑊)
3833, 34, 35, 36, 37isassa 20088 . 2 (𝑊 ∈ AssAlg ↔ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ (Scalar‘𝑊) ∈ CRing) ∧ ∀𝑟 ∈ (Base‘(Scalar‘𝑊))∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)(((𝑟( ·𝑠𝑊)𝑥)(.r𝑊)𝑦) = (𝑟( ·𝑠𝑊)(𝑥(.r𝑊)𝑦)) ∧ (𝑥(.r𝑊)(𝑟( ·𝑠𝑊)𝑦)) = (𝑟( ·𝑠𝑊)(𝑥(.r𝑊)𝑦)))))
396, 32, 38sylanbrc 586 1 (𝜑𝑊 ∈ AssAlg)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3133  cfv 6343  (class class class)co 7149  Basecbs 16483  .rcmulr 16566  Scalarcsca 16568   ·𝑠 cvsca 16569  Ringcrg 19297  CRingccrg 19298  LModclmod 19634  AssAlgcasa 20082
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-nul 5196
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-iota 6302  df-fv 6351  df-ov 7152  df-assa 20085
This theorem is referenced by:  issubassa3  20097  sraassa  20099  psrassa  20194  zlmassa  20671  matassa  21053  mendassa  40054
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