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 Description: Sufficient condition for being an associative algebra. (Contributed by Mario Carneiro, 5-Dec-2014.)
Hypotheses
Ref Expression
isassad.4 ((𝜑 ∧ (𝑟𝐵𝑥𝑉𝑦𝑉)) → ((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)))
isassad.5 ((𝜑 ∧ (𝑟𝐵𝑥𝑉𝑦𝑉)) → (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))
Assertion
Ref Expression
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)

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 2860 . . 3 (𝜑 → (Scalar‘𝑊) ∈ CRing)
61, 2, 53jca 1119 . 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 507 . . . 4 ((𝜑 ∧ (𝑟𝐵𝑥𝑉𝑦𝑉)) → (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))))
109ralrimivvva 3154 . . 3 (𝜑 → ∀𝑟𝐵𝑥𝑉𝑦𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))))
11 isassad.b . . . . 5 (𝜑𝐵 = (Base‘𝐹))
123fveq2d 6452 . . . . 5 (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝑊)))
1311, 12eqtrd 2814 . . . 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 6941 . . . . . . . . 9 (𝜑 → (𝑟 · 𝑥) = (𝑟( ·𝑠𝑊)𝑥))
18 eqidd 2779 . . . . . . . . 9 (𝜑𝑦 = 𝑦)
1915, 17, 18oveq123d 6945 . . . . . . . 8 (𝜑 → ((𝑟 · 𝑥) × 𝑦) = ((𝑟( ·𝑠𝑊)𝑥)(.r𝑊)𝑦))
20 eqidd 2779 . . . . . . . . 9 (𝜑𝑟 = 𝑟)
2115oveqd 6941 . . . . . . . . 9 (𝜑 → (𝑥 × 𝑦) = (𝑥(.r𝑊)𝑦))
2216, 20, 21oveq123d 6945 . . . . . . . 8 (𝜑 → (𝑟 · (𝑥 × 𝑦)) = (𝑟( ·𝑠𝑊)(𝑥(.r𝑊)𝑦)))
2319, 22eqeq12d 2793 . . . . . . 7 (𝜑 → (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ↔ ((𝑟( ·𝑠𝑊)𝑥)(.r𝑊)𝑦) = (𝑟( ·𝑠𝑊)(𝑥(.r𝑊)𝑦))))
24 eqidd 2779 . . . . . . . . 9 (𝜑𝑥 = 𝑥)
2516oveqd 6941 . . . . . . . . 9 (𝜑 → (𝑟 · 𝑦) = (𝑟( ·𝑠𝑊)𝑦))
2615, 24, 25oveq123d 6945 . . . . . . . 8 (𝜑 → (𝑥 × (𝑟 · 𝑦)) = (𝑥(.r𝑊)(𝑟( ·𝑠𝑊)𝑦)))
2726, 22eqeq12d 2793 . . . . . . 7 (𝜑 → ((𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)) ↔ (𝑥(.r𝑊)(𝑟( ·𝑠𝑊)𝑦)) = (𝑟( ·𝑠𝑊)(𝑥(.r𝑊)𝑦))))
2823, 27anbi12d 624 . . . . . 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 3326 . . . . 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 3326 . . . 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 3326 . . 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 224 . 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 2778 . . 3 (Base‘𝑊) = (Base‘𝑊)
34 eqid 2778 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
35 eqid 2778 . . 3 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
36 eqid 2778 . . 3 ( ·𝑠𝑊) = ( ·𝑠𝑊)
37 eqid 2778 . . 3 (.r𝑊) = (.r𝑊)
3833, 34, 35, 36, 37isassa 19723 . 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 578 1 (𝜑𝑊 ∈ AssAlg)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107  ∀wral 3090  ‘cfv 6137  (class class class)co 6924  Basecbs 16266  .rcmulr 16350  Scalarcsca 16352   ·𝑠 cvsca 16353  Ringcrg 18945  CRingccrg 18946  LModclmod 19266  AssAlgcasa 19717 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-nul 5027 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-iota 6101  df-fv 6145  df-ov 6927  df-assa 19720 This theorem is referenced by:  issubassa  19732  sraassa  19733  psrassa  19822  zlmassa  20279  matassa  20665  mendassa  38737
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