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Theorem isassad 21815
Description: Sufficient condition for being an associative algebra. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by SN, 2-Mar-2025.)
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.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)
31, 2jca 510 . 2 (𝜑 → (𝑊 ∈ LMod ∧ 𝑊 ∈ Ring))
4 isassad.4 . . . . 5 ((𝜑 ∧ (𝑟𝐵𝑥𝑉𝑦𝑉)) → ((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)))
5 isassad.5 . . . . 5 ((𝜑 ∧ (𝑟𝐵𝑥𝑉𝑦𝑉)) → (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))
64, 5jca 510 . . . 4 ((𝜑 ∧ (𝑟𝐵𝑥𝑉𝑦𝑉)) → (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))))
76ralrimivvva 3193 . . 3 (𝜑 → ∀𝑟𝐵𝑥𝑉𝑦𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))))
8 isassad.b . . . . 5 (𝜑𝐵 = (Base‘𝐹))
9 isassad.f . . . . . 6 (𝜑𝐹 = (Scalar‘𝑊))
109fveq2d 6900 . . . . 5 (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝑊)))
118, 10eqtrd 2765 . . . 4 (𝜑𝐵 = (Base‘(Scalar‘𝑊)))
12 isassad.v . . . . 5 (𝜑𝑉 = (Base‘𝑊))
13 isassad.t . . . . . . . . 9 (𝜑× = (.r𝑊))
14 isassad.s . . . . . . . . . 10 (𝜑· = ( ·𝑠𝑊))
1514oveqd 7436 . . . . . . . . 9 (𝜑 → (𝑟 · 𝑥) = (𝑟( ·𝑠𝑊)𝑥))
16 eqidd 2726 . . . . . . . . 9 (𝜑𝑦 = 𝑦)
1713, 15, 16oveq123d 7440 . . . . . . . 8 (𝜑 → ((𝑟 · 𝑥) × 𝑦) = ((𝑟( ·𝑠𝑊)𝑥)(.r𝑊)𝑦))
18 eqidd 2726 . . . . . . . . 9 (𝜑𝑟 = 𝑟)
1913oveqd 7436 . . . . . . . . 9 (𝜑 → (𝑥 × 𝑦) = (𝑥(.r𝑊)𝑦))
2014, 18, 19oveq123d 7440 . . . . . . . 8 (𝜑 → (𝑟 · (𝑥 × 𝑦)) = (𝑟( ·𝑠𝑊)(𝑥(.r𝑊)𝑦)))
2117, 20eqeq12d 2741 . . . . . . 7 (𝜑 → (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ↔ ((𝑟( ·𝑠𝑊)𝑥)(.r𝑊)𝑦) = (𝑟( ·𝑠𝑊)(𝑥(.r𝑊)𝑦))))
22 eqidd 2726 . . . . . . . . 9 (𝜑𝑥 = 𝑥)
2314oveqd 7436 . . . . . . . . 9 (𝜑 → (𝑟 · 𝑦) = (𝑟( ·𝑠𝑊)𝑦))
2413, 22, 23oveq123d 7440 . . . . . . . 8 (𝜑 → (𝑥 × (𝑟 · 𝑦)) = (𝑥(.r𝑊)(𝑟( ·𝑠𝑊)𝑦)))
2524, 20eqeq12d 2741 . . . . . . 7 (𝜑 → ((𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)) ↔ (𝑥(.r𝑊)(𝑟( ·𝑠𝑊)𝑦)) = (𝑟( ·𝑠𝑊)(𝑥(.r𝑊)𝑦))))
2621, 25anbi12d 630 . . . . . 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)))))
2712, 26raleqbidv 3329 . . . . 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)))))
2812, 27raleqbidv 3329 . . . 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)))))
2911, 28raleqbidv 3329 . . 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)))))
307, 29mpbid 231 . 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))))
31 eqid 2725 . . 3 (Base‘𝑊) = (Base‘𝑊)
32 eqid 2725 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
33 eqid 2725 . . 3 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
34 eqid 2725 . . 3 ( ·𝑠𝑊) = ( ·𝑠𝑊)
35 eqid 2725 . . 3 (.r𝑊) = (.r𝑊)
3631, 32, 33, 34, 35isassa 21807 . 2 (𝑊 ∈ AssAlg ↔ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring) ∧ ∀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)))))
373, 30, 36sylanbrc 581 1 (𝜑𝑊 ∈ AssAlg)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3050  cfv 6549  (class class class)co 7419  Basecbs 17183  .rcmulr 17237  Scalarcsca 17239   ·𝑠 cvsca 17240  Ringcrg 20185  LModclmod 20755  AssAlgcasa 21801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-iota 6501  df-fv 6557  df-ov 7422  df-assa 21804
This theorem is referenced by:  issubassa3  21816  sraassab  21818  sraassaOLD  21820  zlmassa  21853  psrassa  21935  matassa  22390  mendassa  42760
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