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Theorem isassad 19242
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 2699 . . 3 (𝜑 → (Scalar‘𝑊) ∈ CRing)
61, 2, 53jca 1240 . 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 554 . . . 4 ((𝜑 ∧ (𝑟𝐵𝑥𝑉𝑦𝑉)) → (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))))
109ralrimivvva 2966 . . 3 (𝜑 → ∀𝑟𝐵𝑥𝑉𝑦𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))))
11 isassad.b . . . . 5 (𝜑𝐵 = (Base‘𝐹))
123fveq2d 6152 . . . . 5 (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝑊)))
1311, 12eqtrd 2655 . . . 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 6621 . . . . . . . . 9 (𝜑 → (𝑟 · 𝑥) = (𝑟( ·𝑠𝑊)𝑥))
18 eqidd 2622 . . . . . . . . 9 (𝜑𝑦 = 𝑦)
1915, 17, 18oveq123d 6625 . . . . . . . 8 (𝜑 → ((𝑟 · 𝑥) × 𝑦) = ((𝑟( ·𝑠𝑊)𝑥)(.r𝑊)𝑦))
20 eqidd 2622 . . . . . . . . 9 (𝜑𝑟 = 𝑟)
2115oveqd 6621 . . . . . . . . 9 (𝜑 → (𝑥 × 𝑦) = (𝑥(.r𝑊)𝑦))
2216, 20, 21oveq123d 6625 . . . . . . . 8 (𝜑 → (𝑟 · (𝑥 × 𝑦)) = (𝑟( ·𝑠𝑊)(𝑥(.r𝑊)𝑦)))
2319, 22eqeq12d 2636 . . . . . . 7 (𝜑 → (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ↔ ((𝑟( ·𝑠𝑊)𝑥)(.r𝑊)𝑦) = (𝑟( ·𝑠𝑊)(𝑥(.r𝑊)𝑦))))
24 eqidd 2622 . . . . . . . . 9 (𝜑𝑥 = 𝑥)
2516oveqd 6621 . . . . . . . . 9 (𝜑 → (𝑟 · 𝑦) = (𝑟( ·𝑠𝑊)𝑦))
2615, 24, 25oveq123d 6625 . . . . . . . 8 (𝜑 → (𝑥 × (𝑟 · 𝑦)) = (𝑥(.r𝑊)(𝑟( ·𝑠𝑊)𝑦)))
2726, 22eqeq12d 2636 . . . . . . 7 (𝜑 → ((𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)) ↔ (𝑥(.r𝑊)(𝑟( ·𝑠𝑊)𝑦)) = (𝑟( ·𝑠𝑊)(𝑥(.r𝑊)𝑦))))
2823, 27anbi12d 746 . . . . . 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 3141 . . . . 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 3141 . . . 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 3141 . . 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 222 . 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 2621 . . 3 (Base‘𝑊) = (Base‘𝑊)
34 eqid 2621 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
35 eqid 2621 . . 3 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
36 eqid 2621 . . 3 ( ·𝑠𝑊) = ( ·𝑠𝑊)
37 eqid 2621 . . 3 (.r𝑊) = (.r𝑊)
3833, 34, 35, 36, 37isassa 19234 . 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 697 1 (𝜑𝑊 ∈ AssAlg)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  cfv 5847  (class class class)co 6604  Basecbs 15781  .rcmulr 15863  Scalarcsca 15865   ·𝑠 cvsca 15866  Ringcrg 18468  CRingccrg 18469  LModclmod 18784  AssAlgcasa 19228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4749
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-iota 5810  df-fv 5855  df-ov 6607  df-assa 19231
This theorem is referenced by:  issubassa  19243  sraassa  19244  psrassa  19333  zlmassa  19791  matassa  20169  mendassa  37245
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