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Theorem isassa 19363
Description: The properties of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
Hypotheses
Ref Expression
isassa.v 𝑉 = (Base‘𝑊)
isassa.f 𝐹 = (Scalar‘𝑊)
isassa.b 𝐵 = (Base‘𝐹)
isassa.s · = ( ·𝑠𝑊)
isassa.t × = (.r𝑊)
Assertion
Ref Expression
isassa (𝑊 ∈ AssAlg ↔ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ 𝐹 ∈ CRing) ∧ ∀𝑟𝐵𝑥𝑉𝑦𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
Distinct variable groups:   𝑥,𝑟,𝑦   𝐵,𝑟   𝐹,𝑟   𝑉,𝑟,𝑥,𝑦   · ,𝑟,𝑥,𝑦   × ,𝑟,𝑥,𝑦   𝑊,𝑟,𝑥,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem isassa
Dummy variables 𝑓 𝑤 𝑠 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvexd 6241 . . . 4 (𝑤 = 𝑊 → (Scalar‘𝑤) ∈ V)
2 fveq2 6229 . . . . 5 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
3 isassa.f . . . . 5 𝐹 = (Scalar‘𝑊)
42, 3syl6eqr 2703 . . . 4 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
5 simpr 476 . . . . . 6 ((𝑤 = 𝑊𝑓 = 𝐹) → 𝑓 = 𝐹)
65eleq1d 2715 . . . . 5 ((𝑤 = 𝑊𝑓 = 𝐹) → (𝑓 ∈ CRing ↔ 𝐹 ∈ CRing))
75fveq2d 6233 . . . . . . 7 ((𝑤 = 𝑊𝑓 = 𝐹) → (Base‘𝑓) = (Base‘𝐹))
8 isassa.b . . . . . . 7 𝐵 = (Base‘𝐹)
97, 8syl6eqr 2703 . . . . . 6 ((𝑤 = 𝑊𝑓 = 𝐹) → (Base‘𝑓) = 𝐵)
10 fveq2 6229 . . . . . . . . 9 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
11 isassa.v . . . . . . . . 9 𝑉 = (Base‘𝑊)
1210, 11syl6eqr 2703 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
13 fvexd 6241 . . . . . . . . . 10 (𝑤 = 𝑊 → ( ·𝑠𝑤) ∈ V)
14 fvexd 6241 . . . . . . . . . . 11 ((𝑤 = 𝑊𝑠 = ( ·𝑠𝑤)) → (.r𝑤) ∈ V)
15 simpr 476 . . . . . . . . . . . . . . 15 (((𝑤 = 𝑊𝑠 = ( ·𝑠𝑤)) ∧ 𝑡 = (.r𝑤)) → 𝑡 = (.r𝑤))
16 fveq2 6229 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑊 → (.r𝑤) = (.r𝑊))
1716ad2antrr 762 . . . . . . . . . . . . . . . 16 (((𝑤 = 𝑊𝑠 = ( ·𝑠𝑤)) ∧ 𝑡 = (.r𝑤)) → (.r𝑤) = (.r𝑊))
18 isassa.t . . . . . . . . . . . . . . . 16 × = (.r𝑊)
1917, 18syl6eqr 2703 . . . . . . . . . . . . . . 15 (((𝑤 = 𝑊𝑠 = ( ·𝑠𝑤)) ∧ 𝑡 = (.r𝑤)) → (.r𝑤) = × )
2015, 19eqtrd 2685 . . . . . . . . . . . . . 14 (((𝑤 = 𝑊𝑠 = ( ·𝑠𝑤)) ∧ 𝑡 = (.r𝑤)) → 𝑡 = × )
21 simplr 807 . . . . . . . . . . . . . . . 16 (((𝑤 = 𝑊𝑠 = ( ·𝑠𝑤)) ∧ 𝑡 = (.r𝑤)) → 𝑠 = ( ·𝑠𝑤))
22 fveq2 6229 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑊 → ( ·𝑠𝑤) = ( ·𝑠𝑊))
2322ad2antrr 762 . . . . . . . . . . . . . . . . 17 (((𝑤 = 𝑊𝑠 = ( ·𝑠𝑤)) ∧ 𝑡 = (.r𝑤)) → ( ·𝑠𝑤) = ( ·𝑠𝑊))
24 isassa.s . . . . . . . . . . . . . . . . 17 · = ( ·𝑠𝑊)
2523, 24syl6eqr 2703 . . . . . . . . . . . . . . . 16 (((𝑤 = 𝑊𝑠 = ( ·𝑠𝑤)) ∧ 𝑡 = (.r𝑤)) → ( ·𝑠𝑤) = · )
2621, 25eqtrd 2685 . . . . . . . . . . . . . . 15 (((𝑤 = 𝑊𝑠 = ( ·𝑠𝑤)) ∧ 𝑡 = (.r𝑤)) → 𝑠 = · )
2726oveqd 6707 . . . . . . . . . . . . . 14 (((𝑤 = 𝑊𝑠 = ( ·𝑠𝑤)) ∧ 𝑡 = (.r𝑤)) → (𝑟𝑠𝑥) = (𝑟 · 𝑥))
28 eqidd 2652 . . . . . . . . . . . . . 14 (((𝑤 = 𝑊𝑠 = ( ·𝑠𝑤)) ∧ 𝑡 = (.r𝑤)) → 𝑦 = 𝑦)
2920, 27, 28oveq123d 6711 . . . . . . . . . . . . 13 (((𝑤 = 𝑊𝑠 = ( ·𝑠𝑤)) ∧ 𝑡 = (.r𝑤)) → ((𝑟𝑠𝑥)𝑡𝑦) = ((𝑟 · 𝑥) × 𝑦))
30 eqidd 2652 . . . . . . . . . . . . . 14 (((𝑤 = 𝑊𝑠 = ( ·𝑠𝑤)) ∧ 𝑡 = (.r𝑤)) → 𝑟 = 𝑟)
3120oveqd 6707 . . . . . . . . . . . . . 14 (((𝑤 = 𝑊𝑠 = ( ·𝑠𝑤)) ∧ 𝑡 = (.r𝑤)) → (𝑥𝑡𝑦) = (𝑥 × 𝑦))
3226, 30, 31oveq123d 6711 . . . . . . . . . . . . 13 (((𝑤 = 𝑊𝑠 = ( ·𝑠𝑤)) ∧ 𝑡 = (.r𝑤)) → (𝑟𝑠(𝑥𝑡𝑦)) = (𝑟 · (𝑥 × 𝑦)))
3329, 32eqeq12d 2666 . . . . . . . . . . . 12 (((𝑤 = 𝑊𝑠 = ( ·𝑠𝑤)) ∧ 𝑡 = (.r𝑤)) → (((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ↔ ((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦))))
34 eqidd 2652 . . . . . . . . . . . . . 14 (((𝑤 = 𝑊𝑠 = ( ·𝑠𝑤)) ∧ 𝑡 = (.r𝑤)) → 𝑥 = 𝑥)
3526oveqd 6707 . . . . . . . . . . . . . 14 (((𝑤 = 𝑊𝑠 = ( ·𝑠𝑤)) ∧ 𝑡 = (.r𝑤)) → (𝑟𝑠𝑦) = (𝑟 · 𝑦))
3620, 34, 35oveq123d 6711 . . . . . . . . . . . . 13 (((𝑤 = 𝑊𝑠 = ( ·𝑠𝑤)) ∧ 𝑡 = (.r𝑤)) → (𝑥𝑡(𝑟𝑠𝑦)) = (𝑥 × (𝑟 · 𝑦)))
3736, 32eqeq12d 2666 . . . . . . . . . . . 12 (((𝑤 = 𝑊𝑠 = ( ·𝑠𝑤)) ∧ 𝑡 = (.r𝑤)) → ((𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)) ↔ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))))
3833, 37anbi12d 747 . . . . . . . . . . 11 (((𝑤 = 𝑊𝑠 = ( ·𝑠𝑤)) ∧ 𝑡 = (.r𝑤)) → ((((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦))) ↔ (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
3914, 38sbcied 3505 . . . . . . . . . 10 ((𝑤 = 𝑊𝑠 = ( ·𝑠𝑤)) → ([(.r𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦))) ↔ (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
4013, 39sbcied 3505 . . . . . . . . 9 (𝑤 = 𝑊 → ([( ·𝑠𝑤) / 𝑠][(.r𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦))) ↔ (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
4112, 40raleqbidv 3182 . . . . . . . 8 (𝑤 = 𝑊 → (∀𝑦 ∈ (Base‘𝑤)[( ·𝑠𝑤) / 𝑠][(.r𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦))) ↔ ∀𝑦𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
4212, 41raleqbidv 3182 . . . . . . 7 (𝑤 = 𝑊 → (∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)[( ·𝑠𝑤) / 𝑠][(.r𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦))) ↔ ∀𝑥𝑉𝑦𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
4342adantr 480 . . . . . 6 ((𝑤 = 𝑊𝑓 = 𝐹) → (∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)[( ·𝑠𝑤) / 𝑠][(.r𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦))) ↔ ∀𝑥𝑉𝑦𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
449, 43raleqbidv 3182 . . . . 5 ((𝑤 = 𝑊𝑓 = 𝐹) → (∀𝑟 ∈ (Base‘𝑓)∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)[( ·𝑠𝑤) / 𝑠][(.r𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦))) ↔ ∀𝑟𝐵𝑥𝑉𝑦𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
456, 44anbi12d 747 . . . 4 ((𝑤 = 𝑊𝑓 = 𝐹) → ((𝑓 ∈ CRing ∧ ∀𝑟 ∈ (Base‘𝑓)∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)[( ·𝑠𝑤) / 𝑠][(.r𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)))) ↔ (𝐹 ∈ CRing ∧ ∀𝑟𝐵𝑥𝑉𝑦𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))))))
461, 4, 45sbcied2 3506 . . 3 (𝑤 = 𝑊 → ([(Scalar‘𝑤) / 𝑓](𝑓 ∈ CRing ∧ ∀𝑟 ∈ (Base‘𝑓)∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)[( ·𝑠𝑤) / 𝑠][(.r𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)))) ↔ (𝐹 ∈ CRing ∧ ∀𝑟𝐵𝑥𝑉𝑦𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))))))
47 df-assa 19360 . . 3 AssAlg = {𝑤 ∈ (LMod ∩ Ring) ∣ [(Scalar‘𝑤) / 𝑓](𝑓 ∈ CRing ∧ ∀𝑟 ∈ (Base‘𝑓)∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)[( ·𝑠𝑤) / 𝑠][(.r𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦))))}
4846, 47elrab2 3399 . 2 (𝑊 ∈ AssAlg ↔ (𝑊 ∈ (LMod ∩ Ring) ∧ (𝐹 ∈ CRing ∧ ∀𝑟𝐵𝑥𝑉𝑦𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))))))
49 anass 682 . 2 (((𝑊 ∈ (LMod ∩ Ring) ∧ 𝐹 ∈ CRing) ∧ ∀𝑟𝐵𝑥𝑉𝑦𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))) ↔ (𝑊 ∈ (LMod ∩ Ring) ∧ (𝐹 ∈ CRing ∧ ∀𝑟𝐵𝑥𝑉𝑦𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))))))
50 elin 3829 . . . . 5 (𝑊 ∈ (LMod ∩ Ring) ↔ (𝑊 ∈ LMod ∧ 𝑊 ∈ Ring))
5150anbi1i 731 . . . 4 ((𝑊 ∈ (LMod ∩ Ring) ∧ 𝐹 ∈ CRing) ↔ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring) ∧ 𝐹 ∈ CRing))
52 df-3an 1056 . . . 4 ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ 𝐹 ∈ CRing) ↔ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring) ∧ 𝐹 ∈ CRing))
5351, 52bitr4i 267 . . 3 ((𝑊 ∈ (LMod ∩ Ring) ∧ 𝐹 ∈ CRing) ↔ (𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ 𝐹 ∈ CRing))
5453anbi1i 731 . 2 (((𝑊 ∈ (LMod ∩ Ring) ∧ 𝐹 ∈ CRing) ∧ ∀𝑟𝐵𝑥𝑉𝑦𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))) ↔ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ 𝐹 ∈ CRing) ∧ ∀𝑟𝐵𝑥𝑉𝑦𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
5548, 49, 543bitr2i 288 1 (𝑊 ∈ AssAlg ↔ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ 𝐹 ∈ CRing) ∧ ∀𝑟𝐵𝑥𝑉𝑦𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  [wsbc 3468  cin 3606  cfv 5926  (class class class)co 6690  Basecbs 15904  .rcmulr 15989  Scalarcsca 15991   ·𝑠 cvsca 15992  Ringcrg 18593  CRingccrg 18594  LModclmod 18911  AssAlgcasa 19357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-ov 6693  df-assa 19360
This theorem is referenced by:  assalem  19364  assalmod  19367  assaring  19368  assasca  19369  isassad  19371  assapropd  19375
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