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Theorem assalem 21895
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
assalem ((𝑊 ∈ AssAlg ∧ (𝐴𝐵𝑋𝑉𝑌𝑉)) → (((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)) ∧ (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌))))

Proof of Theorem assalem
Dummy variables 𝑥 𝑟 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isassa.v . . . 4 𝑉 = (Base‘𝑊)
2 isassa.f . . . 4 𝐹 = (Scalar‘𝑊)
3 isassa.b . . . 4 𝐵 = (Base‘𝐹)
4 isassa.s . . . 4 · = ( ·𝑠𝑊)
5 isassa.t . . . 4 × = (.r𝑊)
61, 2, 3, 4, 5isassa 21894 . . 3 (𝑊 ∈ AssAlg ↔ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring) ∧ ∀𝑟𝐵𝑥𝑉𝑦𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))
76simprbi 496 . 2 (𝑊 ∈ AssAlg → ∀𝑟𝐵𝑥𝑉𝑦𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))))
8 oveq1 7438 . . . . . 6 (𝑟 = 𝐴 → (𝑟 · 𝑥) = (𝐴 · 𝑥))
98oveq1d 7446 . . . . 5 (𝑟 = 𝐴 → ((𝑟 · 𝑥) × 𝑦) = ((𝐴 · 𝑥) × 𝑦))
10 oveq1 7438 . . . . 5 (𝑟 = 𝐴 → (𝑟 · (𝑥 × 𝑦)) = (𝐴 · (𝑥 × 𝑦)))
119, 10eqeq12d 2751 . . . 4 (𝑟 = 𝐴 → (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ↔ ((𝐴 · 𝑥) × 𝑦) = (𝐴 · (𝑥 × 𝑦))))
12 oveq1 7438 . . . . . 6 (𝑟 = 𝐴 → (𝑟 · 𝑦) = (𝐴 · 𝑦))
1312oveq2d 7447 . . . . 5 (𝑟 = 𝐴 → (𝑥 × (𝑟 · 𝑦)) = (𝑥 × (𝐴 · 𝑦)))
1413, 10eqeq12d 2751 . . . 4 (𝑟 = 𝐴 → ((𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)) ↔ (𝑥 × (𝐴 · 𝑦)) = (𝐴 · (𝑥 × 𝑦))))
1511, 14anbi12d 632 . . 3 (𝑟 = 𝐴 → ((((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))) ↔ (((𝐴 · 𝑥) × 𝑦) = (𝐴 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝐴 · 𝑦)) = (𝐴 · (𝑥 × 𝑦)))))
16 oveq2 7439 . . . . . 6 (𝑥 = 𝑋 → (𝐴 · 𝑥) = (𝐴 · 𝑋))
1716oveq1d 7446 . . . . 5 (𝑥 = 𝑋 → ((𝐴 · 𝑥) × 𝑦) = ((𝐴 · 𝑋) × 𝑦))
18 oveq1 7438 . . . . . 6 (𝑥 = 𝑋 → (𝑥 × 𝑦) = (𝑋 × 𝑦))
1918oveq2d 7447 . . . . 5 (𝑥 = 𝑋 → (𝐴 · (𝑥 × 𝑦)) = (𝐴 · (𝑋 × 𝑦)))
2017, 19eqeq12d 2751 . . . 4 (𝑥 = 𝑋 → (((𝐴 · 𝑥) × 𝑦) = (𝐴 · (𝑥 × 𝑦)) ↔ ((𝐴 · 𝑋) × 𝑦) = (𝐴 · (𝑋 × 𝑦))))
21 oveq1 7438 . . . . 5 (𝑥 = 𝑋 → (𝑥 × (𝐴 · 𝑦)) = (𝑋 × (𝐴 · 𝑦)))
2221, 19eqeq12d 2751 . . . 4 (𝑥 = 𝑋 → ((𝑥 × (𝐴 · 𝑦)) = (𝐴 · (𝑥 × 𝑦)) ↔ (𝑋 × (𝐴 · 𝑦)) = (𝐴 · (𝑋 × 𝑦))))
2320, 22anbi12d 632 . . 3 (𝑥 = 𝑋 → ((((𝐴 · 𝑥) × 𝑦) = (𝐴 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝐴 · 𝑦)) = (𝐴 · (𝑥 × 𝑦))) ↔ (((𝐴 · 𝑋) × 𝑦) = (𝐴 · (𝑋 × 𝑦)) ∧ (𝑋 × (𝐴 · 𝑦)) = (𝐴 · (𝑋 × 𝑦)))))
24 oveq2 7439 . . . . 5 (𝑦 = 𝑌 → ((𝐴 · 𝑋) × 𝑦) = ((𝐴 · 𝑋) × 𝑌))
25 oveq2 7439 . . . . . 6 (𝑦 = 𝑌 → (𝑋 × 𝑦) = (𝑋 × 𝑌))
2625oveq2d 7447 . . . . 5 (𝑦 = 𝑌 → (𝐴 · (𝑋 × 𝑦)) = (𝐴 · (𝑋 × 𝑌)))
2724, 26eqeq12d 2751 . . . 4 (𝑦 = 𝑌 → (((𝐴 · 𝑋) × 𝑦) = (𝐴 · (𝑋 × 𝑦)) ↔ ((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌))))
28 oveq2 7439 . . . . . 6 (𝑦 = 𝑌 → (𝐴 · 𝑦) = (𝐴 · 𝑌))
2928oveq2d 7447 . . . . 5 (𝑦 = 𝑌 → (𝑋 × (𝐴 · 𝑦)) = (𝑋 × (𝐴 · 𝑌)))
3029, 26eqeq12d 2751 . . . 4 (𝑦 = 𝑌 → ((𝑋 × (𝐴 · 𝑦)) = (𝐴 · (𝑋 × 𝑦)) ↔ (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌))))
3127, 30anbi12d 632 . . 3 (𝑦 = 𝑌 → ((((𝐴 · 𝑋) × 𝑦) = (𝐴 · (𝑋 × 𝑦)) ∧ (𝑋 × (𝐴 · 𝑦)) = (𝐴 · (𝑋 × 𝑦))) ↔ (((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)) ∧ (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌)))))
3215, 23, 31rspc3v 3638 . 2 ((𝐴𝐵𝑋𝑉𝑌𝑉) → (∀𝑟𝐵𝑥𝑉𝑦𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))) → (((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)) ∧ (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌)))))
337, 32mpan9 506 1 ((𝑊 ∈ AssAlg ∧ (𝐴𝐵𝑋𝑉𝑌𝑉)) → (((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)) ∧ (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  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:  assaass  21896  assaassr  21897
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