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Theorem isassad 21419
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 (πœ‘ β†’ 𝑉 = (Baseβ€˜π‘Š))
isassad.f (πœ‘ β†’ 𝐹 = (Scalarβ€˜π‘Š))
isassad.b (πœ‘ β†’ 𝐡 = (Baseβ€˜πΉ))
isassad.s (πœ‘ β†’ Β· = ( ·𝑠 β€˜π‘Š))
isassad.t (πœ‘ β†’ Γ— = (.rβ€˜π‘Š))
isassad.1 (πœ‘ β†’ π‘Š ∈ LMod)
isassad.2 (πœ‘ β†’ π‘Š ∈ Ring)
isassad.4 ((πœ‘ ∧ (π‘Ÿ ∈ 𝐡 ∧ π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) β†’ ((π‘Ÿ Β· π‘₯) Γ— 𝑦) = (π‘Ÿ Β· (π‘₯ Γ— 𝑦)))
isassad.5 ((πœ‘ ∧ (π‘Ÿ ∈ 𝐡 ∧ π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) β†’ (π‘₯ Γ— (π‘Ÿ Β· 𝑦)) = (π‘Ÿ Β· (π‘₯ Γ— 𝑦)))
Assertion
Ref Expression
isassad (πœ‘ β†’ π‘Š ∈ AssAlg)
Distinct variable groups:   π‘₯,π‘Ÿ,𝑦,𝐡   πœ‘,π‘Ÿ,π‘₯,𝑦   π‘₯,𝑉,𝑦   π‘Š,π‘Ÿ,π‘₯,𝑦
Allowed substitution hints:   Β· (π‘₯,𝑦,π‘Ÿ)   Γ— (π‘₯,𝑦,π‘Ÿ)   𝐹(π‘₯,𝑦,π‘Ÿ)   𝑉(π‘Ÿ)

Proof of Theorem isassad
StepHypRef Expression
1 isassad.1 . . 3 (πœ‘ β†’ π‘Š ∈ LMod)
2 isassad.2 . . 3 (πœ‘ β†’ π‘Š ∈ Ring)
31, 2jca 513 . 2 (πœ‘ β†’ (π‘Š ∈ LMod ∧ π‘Š ∈ Ring))
4 isassad.4 . . . . 5 ((πœ‘ ∧ (π‘Ÿ ∈ 𝐡 ∧ π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) β†’ ((π‘Ÿ Β· π‘₯) Γ— 𝑦) = (π‘Ÿ Β· (π‘₯ Γ— 𝑦)))
5 isassad.5 . . . . 5 ((πœ‘ ∧ (π‘Ÿ ∈ 𝐡 ∧ π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) β†’ (π‘₯ Γ— (π‘Ÿ Β· 𝑦)) = (π‘Ÿ Β· (π‘₯ Γ— 𝑦)))
64, 5jca 513 . . . 4 ((πœ‘ ∧ (π‘Ÿ ∈ 𝐡 ∧ π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) 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))))
76ralrimivvva 3204 . . 3 (πœ‘ 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 isassad.b . . . . 5 (πœ‘ β†’ 𝐡 = (Baseβ€˜πΉ))
9 isassad.f . . . . . 6 (πœ‘ β†’ 𝐹 = (Scalarβ€˜π‘Š))
109fveq2d 6896 . . . . 5 (πœ‘ β†’ (Baseβ€˜πΉ) = (Baseβ€˜(Scalarβ€˜π‘Š)))
118, 10eqtrd 2773 . . . 4 (πœ‘ β†’ 𝐡 = (Baseβ€˜(Scalarβ€˜π‘Š)))
12 isassad.v . . . . 5 (πœ‘ β†’ 𝑉 = (Baseβ€˜π‘Š))
13 isassad.t . . . . . . . . 9 (πœ‘ β†’ Γ— = (.rβ€˜π‘Š))
14 isassad.s . . . . . . . . . 10 (πœ‘ β†’ Β· = ( ·𝑠 β€˜π‘Š))
1514oveqd 7426 . . . . . . . . 9 (πœ‘ β†’ (π‘Ÿ Β· π‘₯) = (π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯))
16 eqidd 2734 . . . . . . . . 9 (πœ‘ β†’ 𝑦 = 𝑦)
1713, 15, 16oveq123d 7430 . . . . . . . 8 (πœ‘ β†’ ((π‘Ÿ Β· π‘₯) Γ— 𝑦) = ((π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯)(.rβ€˜π‘Š)𝑦))
18 eqidd 2734 . . . . . . . . 9 (πœ‘ β†’ π‘Ÿ = π‘Ÿ)
1913oveqd 7426 . . . . . . . . 9 (πœ‘ β†’ (π‘₯ Γ— 𝑦) = (π‘₯(.rβ€˜π‘Š)𝑦))
2014, 18, 19oveq123d 7430 . . . . . . . 8 (πœ‘ β†’ (π‘Ÿ Β· (π‘₯ Γ— 𝑦)) = (π‘Ÿ( ·𝑠 β€˜π‘Š)(π‘₯(.rβ€˜π‘Š)𝑦)))
2117, 20eqeq12d 2749 . . . . . . 7 (πœ‘ β†’ (((π‘Ÿ Β· π‘₯) Γ— 𝑦) = (π‘Ÿ Β· (π‘₯ Γ— 𝑦)) ↔ ((π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯)(.rβ€˜π‘Š)𝑦) = (π‘Ÿ( ·𝑠 β€˜π‘Š)(π‘₯(.rβ€˜π‘Š)𝑦))))
22 eqidd 2734 . . . . . . . . 9 (πœ‘ β†’ π‘₯ = π‘₯)
2314oveqd 7426 . . . . . . . . 9 (πœ‘ β†’ (π‘Ÿ Β· 𝑦) = (π‘Ÿ( ·𝑠 β€˜π‘Š)𝑦))
2413, 22, 23oveq123d 7430 . . . . . . . 8 (πœ‘ β†’ (π‘₯ Γ— (π‘Ÿ Β· 𝑦)) = (π‘₯(.rβ€˜π‘Š)(π‘Ÿ( ·𝑠 β€˜π‘Š)𝑦)))
2524, 20eqeq12d 2749 . . . . . . 7 (πœ‘ β†’ ((π‘₯ Γ— (π‘Ÿ Β· 𝑦)) = (π‘Ÿ Β· (π‘₯ Γ— 𝑦)) ↔ (π‘₯(.rβ€˜π‘Š)(π‘Ÿ( ·𝑠 β€˜π‘Š)𝑦)) = (π‘Ÿ( ·𝑠 β€˜π‘Š)(π‘₯(.rβ€˜π‘Š)𝑦))))
2621, 25anbi12d 632 . . . . . 6 (πœ‘ 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))) ↔ (((π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯)(.rβ€˜π‘Š)𝑦) = (π‘Ÿ( ·𝑠 β€˜π‘Š)(π‘₯(.rβ€˜π‘Š)𝑦)) ∧ (π‘₯(.rβ€˜π‘Š)(π‘Ÿ( ·𝑠 β€˜π‘Š)𝑦)) = (π‘Ÿ( ·𝑠 β€˜π‘Š)(π‘₯(.rβ€˜π‘Š)𝑦)))))
2712, 26raleqbidv 3343 . . . . 5 (πœ‘ β†’ (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))) ↔ βˆ€π‘¦ ∈ (Baseβ€˜π‘Š)(((π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯)(.rβ€˜π‘Š)𝑦) = (π‘Ÿ( ·𝑠 β€˜π‘Š)(π‘₯(.rβ€˜π‘Š)𝑦)) ∧ (π‘₯(.rβ€˜π‘Š)(π‘Ÿ( ·𝑠 β€˜π‘Š)𝑦)) = (π‘Ÿ( ·𝑠 β€˜π‘Š)(π‘₯(.rβ€˜π‘Š)𝑦)))))
2812, 27raleqbidv 3343 . . . 4 (πœ‘ β†’ (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))) ↔ βˆ€π‘₯ ∈ (Baseβ€˜π‘Š)βˆ€π‘¦ ∈ (Baseβ€˜π‘Š)(((π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯)(.rβ€˜π‘Š)𝑦) = (π‘Ÿ( ·𝑠 β€˜π‘Š)(π‘₯(.rβ€˜π‘Š)𝑦)) ∧ (π‘₯(.rβ€˜π‘Š)(π‘Ÿ( ·𝑠 β€˜π‘Š)𝑦)) = (π‘Ÿ( ·𝑠 β€˜π‘Š)(π‘₯(.rβ€˜π‘Š)𝑦)))))
2911, 28raleqbidv 3343 . . 3 (πœ‘ 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))) ↔ βˆ€π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))βˆ€π‘₯ ∈ (Baseβ€˜π‘Š)βˆ€π‘¦ ∈ (Baseβ€˜π‘Š)(((π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯)(.rβ€˜π‘Š)𝑦) = (π‘Ÿ( ·𝑠 β€˜π‘Š)(π‘₯(.rβ€˜π‘Š)𝑦)) ∧ (π‘₯(.rβ€˜π‘Š)(π‘Ÿ( ·𝑠 β€˜π‘Š)𝑦)) = (π‘Ÿ( ·𝑠 β€˜π‘Š)(π‘₯(.rβ€˜π‘Š)𝑦)))))
307, 29mpbid 231 . 2 (πœ‘ β†’ βˆ€π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))βˆ€π‘₯ ∈ (Baseβ€˜π‘Š)βˆ€π‘¦ ∈ (Baseβ€˜π‘Š)(((π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯)(.rβ€˜π‘Š)𝑦) = (π‘Ÿ( ·𝑠 β€˜π‘Š)(π‘₯(.rβ€˜π‘Š)𝑦)) ∧ (π‘₯(.rβ€˜π‘Š)(π‘Ÿ( ·𝑠 β€˜π‘Š)𝑦)) = (π‘Ÿ( ·𝑠 β€˜π‘Š)(π‘₯(.rβ€˜π‘Š)𝑦))))
31 eqid 2733 . . 3 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
32 eqid 2733 . . 3 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
33 eqid 2733 . . 3 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
34 eqid 2733 . . 3 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
35 eqid 2733 . . 3 (.rβ€˜π‘Š) = (.rβ€˜π‘Š)
3631, 32, 33, 34, 35isassa 21411 . 2 (π‘Š ∈ AssAlg ↔ ((π‘Š ∈ LMod ∧ π‘Š ∈ Ring) ∧ βˆ€π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))βˆ€π‘₯ ∈ (Baseβ€˜π‘Š)βˆ€π‘¦ ∈ (Baseβ€˜π‘Š)(((π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯)(.rβ€˜π‘Š)𝑦) = (π‘Ÿ( ·𝑠 β€˜π‘Š)(π‘₯(.rβ€˜π‘Š)𝑦)) ∧ (π‘₯(.rβ€˜π‘Š)(π‘Ÿ( ·𝑠 β€˜π‘Š)𝑦)) = (π‘Ÿ( ·𝑠 β€˜π‘Š)(π‘₯(.rβ€˜π‘Š)𝑦)))))
373, 30, 36sylanbrc 584 1 (πœ‘ β†’ π‘Š ∈ AssAlg)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  .rcmulr 17198  Scalarcsca 17200   ·𝑠 cvsca 17201  Ringcrg 20056  LModclmod 20471  AssAlgcasa 21405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-assa 21408
This theorem is referenced by:  issubassa3  21420  sraassab  21422  sraassaOLD  21424  zlmassa  21456  psrassa  21534  matassa  21946  mendassa  41936
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