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Theorem assalem 21411
Description: The properties of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
Hypotheses
Ref Expression
isassa.v 𝑉 = (Baseβ€˜π‘Š)
isassa.f 𝐹 = (Scalarβ€˜π‘Š)
isassa.b 𝐡 = (Baseβ€˜πΉ)
isassa.s Β· = ( ·𝑠 β€˜π‘Š)
isassa.t Γ— = (.rβ€˜π‘Š)
Assertion
Ref Expression
assalem ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ 𝐡 ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉)) β†’ (((𝐴 Β· 𝑋) Γ— π‘Œ) = (𝐴 Β· (𝑋 Γ— π‘Œ)) ∧ (𝑋 Γ— (𝐴 Β· π‘Œ)) = (𝐴 Β· (𝑋 Γ— π‘Œ))))
Proof of Theorem assalem
Dummy variables π‘₯ π‘Ÿ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isassa.v . . . 4 𝑉 = (Baseβ€˜π‘Š)
2 isassa.f . . . 4 𝐹 = (Scalarβ€˜π‘Š)
3 isassa.b . . . 4 𝐡 = (Baseβ€˜πΉ)
4 isassa.s . . . 4 Β· = ( ·𝑠 β€˜π‘Š)
5 isassa.t . . . 4 Γ— = (.rβ€˜π‘Š)
61, 2, 3, 4, 5isassa 21410 . . 3 (π‘Š ∈ AssAlg ↔ ((π‘Š ∈ LMod ∧ π‘Š ∈ Ring) ∧ βˆ€π‘Ÿ ∈ 𝐡 βˆ€π‘₯ ∈ 𝑉 βˆ€π‘¦ ∈ 𝑉 (((π‘Ÿ Β· π‘₯) Γ— 𝑦) = (π‘Ÿ Β· (π‘₯ Γ— 𝑦)) ∧ (π‘₯ Γ— (π‘Ÿ Β· 𝑦)) = (π‘Ÿ Β· (π‘₯ Γ— 𝑦)))))
76simprbi 497 . 2 (π‘Š ∈ AssAlg β†’ βˆ€π‘Ÿ ∈ 𝐡 βˆ€π‘₯ ∈ 𝑉 βˆ€π‘¦ ∈ 𝑉 (((π‘Ÿ Β· π‘₯) Γ— 𝑦) = (π‘Ÿ Β· (π‘₯ Γ— 𝑦)) ∧ (π‘₯ Γ— (π‘Ÿ Β· 𝑦)) = (π‘Ÿ Β· (π‘₯ Γ— 𝑦))))
8 oveq1 7415 . . . . . 6 (π‘Ÿ = 𝐴 β†’ (π‘Ÿ Β· π‘₯) = (𝐴 Β· π‘₯))
98oveq1d 7423 . . . . 5 (π‘Ÿ = 𝐴 β†’ ((π‘Ÿ Β· π‘₯) Γ— 𝑦) = ((𝐴 Β· π‘₯) Γ— 𝑦))
10 oveq1 7415 . . . . 5 (π‘Ÿ = 𝐴 β†’ (π‘Ÿ Β· (π‘₯ Γ— 𝑦)) = (𝐴 Β· (π‘₯ Γ— 𝑦)))
119, 10eqeq12d 2748 . . . 4 (π‘Ÿ = 𝐴 β†’ (((π‘Ÿ Β· π‘₯) Γ— 𝑦) = (π‘Ÿ Β· (π‘₯ Γ— 𝑦)) ↔ ((𝐴 Β· π‘₯) Γ— 𝑦) = (𝐴 Β· (π‘₯ Γ— 𝑦))))
12 oveq1 7415 . . . . . 6 (π‘Ÿ = 𝐴 β†’ (π‘Ÿ Β· 𝑦) = (𝐴 Β· 𝑦))
1312oveq2d 7424 . . . . 5 (π‘Ÿ = 𝐴 β†’ (π‘₯ Γ— (π‘Ÿ Β· 𝑦)) = (π‘₯ Γ— (𝐴 Β· 𝑦)))
1413, 10eqeq12d 2748 . . . 4 (π‘Ÿ = 𝐴 β†’ ((π‘₯ Γ— (π‘Ÿ Β· 𝑦)) = (π‘Ÿ Β· (π‘₯ Γ— 𝑦)) ↔ (π‘₯ Γ— (𝐴 Β· 𝑦)) = (𝐴 Β· (π‘₯ Γ— 𝑦))))
1511, 14anbi12d 631 . . 3 (π‘Ÿ = 𝐴 β†’ ((((π‘Ÿ Β· π‘₯) Γ— 𝑦) = (π‘Ÿ Β· (π‘₯ Γ— 𝑦)) ∧ (π‘₯ Γ— (π‘Ÿ Β· 𝑦)) = (π‘Ÿ Β· (π‘₯ Γ— 𝑦))) ↔ (((𝐴 Β· π‘₯) Γ— 𝑦) = (𝐴 Β· (π‘₯ Γ— 𝑦)) ∧ (π‘₯ Γ— (𝐴 Β· 𝑦)) = (𝐴 Β· (π‘₯ Γ— 𝑦)))))
16 oveq2 7416 . . . . . 6 (π‘₯ = 𝑋 β†’ (𝐴 Β· π‘₯) = (𝐴 Β· 𝑋))
1716oveq1d 7423 . . . . 5 (π‘₯ = 𝑋 β†’ ((𝐴 Β· π‘₯) Γ— 𝑦) = ((𝐴 Β· 𝑋) Γ— 𝑦))
18 oveq1 7415 . . . . . 6 (π‘₯ = 𝑋 β†’ (π‘₯ Γ— 𝑦) = (𝑋 Γ— 𝑦))
1918oveq2d 7424 . . . . 5 (π‘₯ = 𝑋 β†’ (𝐴 Β· (π‘₯ Γ— 𝑦)) = (𝐴 Β· (𝑋 Γ— 𝑦)))
2017, 19eqeq12d 2748 . . . 4 (π‘₯ = 𝑋 β†’ (((𝐴 Β· π‘₯) Γ— 𝑦) = (𝐴 Β· (π‘₯ Γ— 𝑦)) ↔ ((𝐴 Β· 𝑋) Γ— 𝑦) = (𝐴 Β· (𝑋 Γ— 𝑦))))
21 oveq1 7415 . . . . 5 (π‘₯ = 𝑋 β†’ (π‘₯ Γ— (𝐴 Β· 𝑦)) = (𝑋 Γ— (𝐴 Β· 𝑦)))
2221, 19eqeq12d 2748 . . . 4 (π‘₯ = 𝑋 β†’ ((π‘₯ Γ— (𝐴 Β· 𝑦)) = (𝐴 Β· (π‘₯ Γ— 𝑦)) ↔ (𝑋 Γ— (𝐴 Β· 𝑦)) = (𝐴 Β· (𝑋 Γ— 𝑦))))
2320, 22anbi12d 631 . . 3 (π‘₯ = 𝑋 β†’ ((((𝐴 Β· π‘₯) Γ— 𝑦) = (𝐴 Β· (π‘₯ Γ— 𝑦)) ∧ (π‘₯ Γ— (𝐴 Β· 𝑦)) = (𝐴 Β· (π‘₯ Γ— 𝑦))) ↔ (((𝐴 Β· 𝑋) Γ— 𝑦) = (𝐴 Β· (𝑋 Γ— 𝑦)) ∧ (𝑋 Γ— (𝐴 Β· 𝑦)) = (𝐴 Β· (𝑋 Γ— 𝑦)))))
24 oveq2 7416 . . . . 5 (𝑦 = π‘Œ β†’ ((𝐴 Β· 𝑋) Γ— 𝑦) = ((𝐴 Β· 𝑋) Γ— π‘Œ))
25 oveq2 7416 . . . . . 6 (𝑦 = π‘Œ β†’ (𝑋 Γ— 𝑦) = (𝑋 Γ— π‘Œ))
2625oveq2d 7424 . . . . 5 (𝑦 = π‘Œ β†’ (𝐴 Β· (𝑋 Γ— 𝑦)) = (𝐴 Β· (𝑋 Γ— π‘Œ)))
2724, 26eqeq12d 2748 . . . 4 (𝑦 = π‘Œ β†’ (((𝐴 Β· 𝑋) Γ— 𝑦) = (𝐴 Β· (𝑋 Γ— 𝑦)) ↔ ((𝐴 Β· 𝑋) Γ— π‘Œ) = (𝐴 Β· (𝑋 Γ— π‘Œ))))
28 oveq2 7416 . . . . . 6 (𝑦 = π‘Œ β†’ (𝐴 Β· 𝑦) = (𝐴 Β· π‘Œ))
2928oveq2d 7424 . . . . 5 (𝑦 = π‘Œ β†’ (𝑋 Γ— (𝐴 Β· 𝑦)) = (𝑋 Γ— (𝐴 Β· π‘Œ)))
3029, 26eqeq12d 2748 . . . 4 (𝑦 = π‘Œ β†’ ((𝑋 Γ— (𝐴 Β· 𝑦)) = (𝐴 Β· (𝑋 Γ— 𝑦)) ↔ (𝑋 Γ— (𝐴 Β· π‘Œ)) = (𝐴 Β· (𝑋 Γ— π‘Œ))))
3127, 30anbi12d 631 . . 3 (𝑦 = π‘Œ β†’ ((((𝐴 Β· 𝑋) Γ— 𝑦) = (𝐴 Β· (𝑋 Γ— 𝑦)) ∧ (𝑋 Γ— (𝐴 Β· 𝑦)) = (𝐴 Β· (𝑋 Γ— 𝑦))) ↔ (((𝐴 Β· 𝑋) Γ— π‘Œ) = (𝐴 Β· (𝑋 Γ— π‘Œ)) ∧ (𝑋 Γ— (𝐴 Β· π‘Œ)) = (𝐴 Β· (𝑋 Γ— π‘Œ)))))
3215, 23, 31rspc3v 3627 . 2 ((𝐴 ∈ 𝐡 ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (βˆ€π‘Ÿ ∈ 𝐡 βˆ€π‘₯ ∈ 𝑉 βˆ€π‘¦ ∈ 𝑉 (((π‘Ÿ Β· π‘₯) Γ— 𝑦) = (π‘Ÿ Β· (π‘₯ Γ— 𝑦)) ∧ (π‘₯ Γ— (π‘Ÿ Β· 𝑦)) = (π‘Ÿ Β· (π‘₯ Γ— 𝑦))) β†’ (((𝐴 Β· 𝑋) Γ— π‘Œ) = (𝐴 Β· (𝑋 Γ— π‘Œ)) ∧ (𝑋 Γ— (𝐴 Β· π‘Œ)) = (𝐴 Β· (𝑋 Γ— π‘Œ)))))
337, 32mpan9 507 1 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ 𝐡 ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉)) β†’ (((𝐴 Β· 𝑋) Γ— π‘Œ) = (𝐴 Β· (𝑋 Γ— π‘Œ)) ∧ (𝑋 Γ— (𝐴 Β· π‘Œ)) = (𝐴 Β· (𝑋 Γ— π‘Œ))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  β€˜cfv 6543  (class class class)co 7408  Basecbs 17143  .rcmulr 17197  Scalarcsca 17199   ·𝑠 cvsca 17200  Ringcrg 20055  LModclmod 20470  AssAlgcasa 21404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7411  df-assa 21407
This theorem is referenced by:  assaass  21412  assaassr  21413
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