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Theorem isassa 21894
Description: The properties of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by SN, 2-Mar-2025.)
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) ∧ ∀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 6922 . . . 4 (𝑤 = 𝑊 → (Scalar‘𝑤) ∈ V)
2 fveq2 6907 . . . . 5 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
3 isassa.f . . . . 5 𝐹 = (Scalar‘𝑊)
42, 3eqtr4di 2793 . . . 4 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
5 fveq2 6907 . . . . . . 7 (𝑓 = 𝐹 → (Base‘𝑓) = (Base‘𝐹))
6 isassa.b . . . . . . 7 𝐵 = (Base‘𝐹)
75, 6eqtr4di 2793 . . . . . 6 (𝑓 = 𝐹 → (Base‘𝑓) = 𝐵)
87adantl 481 . . . . 5 ((𝑤 = 𝑊𝑓 = 𝐹) → (Base‘𝑓) = 𝐵)
9 fveq2 6907 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
10 isassa.v . . . . . . . 8 𝑉 = (Base‘𝑊)
119, 10eqtr4di 2793 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
12 isassa.s . . . . . . . . 9 · = ( ·𝑠𝑊)
13 isassa.t . . . . . . . . 9 × = (.r𝑊)
14 simpr 484 . . . . . . . . . . . 12 ((𝑠 = ·𝑡 = × ) → 𝑡 = × )
15 simpl 482 . . . . . . . . . . . . 13 ((𝑠 = ·𝑡 = × ) → 𝑠 = · )
1615oveqd 7448 . . . . . . . . . . . 12 ((𝑠 = ·𝑡 = × ) → (𝑟𝑠𝑥) = (𝑟 · 𝑥))
17 eqidd 2736 . . . . . . . . . . . 12 ((𝑠 = ·𝑡 = × ) → 𝑦 = 𝑦)
1814, 16, 17oveq123d 7452 . . . . . . . . . . 11 ((𝑠 = ·𝑡 = × ) → ((𝑟𝑠𝑥)𝑡𝑦) = ((𝑟 · 𝑥) × 𝑦))
19 eqidd 2736 . . . . . . . . . . . 12 ((𝑠 = ·𝑡 = × ) → 𝑟 = 𝑟)
2014oveqd 7448 . . . . . . . . . . . 12 ((𝑠 = ·𝑡 = × ) → (𝑥𝑡𝑦) = (𝑥 × 𝑦))
2115, 19, 20oveq123d 7452 . . . . . . . . . . 11 ((𝑠 = ·𝑡 = × ) → (𝑟𝑠(𝑥𝑡𝑦)) = (𝑟 · (𝑥 × 𝑦)))
2218, 21eqeq12d 2751 . . . . . . . . . 10 ((𝑠 = ·𝑡 = × ) → (((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ↔ ((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦))))
23 eqidd 2736 . . . . . . . . . . . 12 ((𝑠 = ·𝑡 = × ) → 𝑥 = 𝑥)
2415oveqd 7448 . . . . . . . . . . . 12 ((𝑠 = ·𝑡 = × ) → (𝑟𝑠𝑦) = (𝑟 · 𝑦))
2514, 23, 24oveq123d 7452 . . . . . . . . . . 11 ((𝑠 = ·𝑡 = × ) → (𝑥𝑡(𝑟𝑠𝑦)) = (𝑥 × (𝑟 · 𝑦)))
2625, 21eqeq12d 2751 . . . . . . . . . 10 ((𝑠 = ·𝑡 = × ) → ((𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)) ↔ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))))
2722, 26anbi12d 632 . . . . . . . . 9 ((𝑠 = ·𝑡 = × ) → ((((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦))) ↔ (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
2812, 13, 27sbcie2s 17195 . . . . . . . 8 (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)))))
2911, 28raleqbidv 3344 . . . . . . 7 (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)))))
3011, 29raleqbidv 3344 . . . . . 6 (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)))))
3130adantr 480 . . . . 5 ((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)))))
328, 31raleqbidv 3344 . . . 4 ((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)))))
331, 4, 32sbcied2 3839 . . 3 (𝑤 = 𝑊 → ([(Scalar‘𝑤) / 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)))))
34 df-assa 21891 . . 3 AssAlg = {𝑤 ∈ (LMod ∩ Ring) ∣ [(Scalar‘𝑤) / 𝑓]𝑟 ∈ (Base‘𝑓)∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)[( ·𝑠𝑤) / 𝑠][(.r𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)))}
3533, 34elrab2 3698 . 2 (𝑊 ∈ AssAlg ↔ (𝑊 ∈ (LMod ∩ Ring) ∧ ∀𝑟𝐵𝑥𝑉𝑦𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
36 elin 3979 . . 3 (𝑊 ∈ (LMod ∩ Ring) ↔ (𝑊 ∈ LMod ∧ 𝑊 ∈ Ring))
3736anbi1i 624 . 2 ((𝑊 ∈ (LMod ∩ Ring) ∧ ∀𝑟𝐵𝑥𝑉𝑦𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))) ↔ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring) ∧ ∀𝑟𝐵𝑥𝑉𝑦𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
3835, 37bitri 275 1 (𝑊 ∈ AssAlg ↔ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring) ∧ ∀𝑟𝐵𝑥𝑉𝑦𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  [wsbc 3791  cin 3962  cfv 6563  (class class class)co 7431  Basecbs 17245  .rcmulr 17299  Scalarcsca 17301   ·𝑠 cvsca 17302  Ringcrg 20251  LModclmod 20875  AssAlgcasa 21888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-assa 21891
This theorem is referenced by:  assalem  21895  assalmod  21898  assaring  21899  isassad  21903  assapropd  21910
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