Theorem isassad 21847

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	⊢ (𝜑 → 𝑉 = (Base‘𝑊))
isassad.f	⊢ (𝜑 → 𝐹 = (Scalar‘𝑊))
isassad.b	⊢ (𝜑 → 𝐵 = (Base‘𝐹))
isassad.s	⊢ (𝜑 → · = ( ·𝑠 ‘𝑊))
isassad.t	⊢ (𝜑 → × = (.r‘𝑊))
isassad.1	⊢ (𝜑 → 𝑊 ∈ LMod)
isassad.2	⊢ (𝜑 → 𝑊 ∈ Ring)
isassad.4	⊢ ((𝜑 ∧ (𝑟 ∈ 𝐵 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → ((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)))
isassad.5	⊢ ((𝜑 ∧ (𝑟 ∈ 𝐵 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))

Assertion

Ref	Expression
isassad	⊢ (𝜑 → 𝑊 ∈ AssAlg)

Distinct variable groups:   𝑥,𝑟,𝑦,𝐵   𝜑,𝑟,𝑥,𝑦   𝑥,𝑉,𝑦   𝑊,𝑟,𝑥,𝑦

Allowed substitution hints:   · (𝑥,𝑦,𝑟)   × (𝑥,𝑦,𝑟)   𝐹(𝑥,𝑦,𝑟)   𝑉(𝑟)

Proof of Theorem isassad

Step	Hyp	Ref	Expression
1		isassad.1	. . 3 ⊢ (𝜑 → 𝑊 ∈ LMod)
2		isassad.2	. . 3 ⊢ (𝜑 → 𝑊 ∈ Ring)
3	1, 2	jca 516	. 2 ⊢ (𝜑 → (𝑊 ∈ LMod ∧ 𝑊 ∈ Ring))
4		isassad.4	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝐵 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → ((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)))
5		isassad.5	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝐵 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))
6	4, 5	jca 516	. . . 4 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝐵 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))))
7	6	ralrimivvva 3186	. . 3 ⊢ (𝜑 → ∀𝑟 ∈ 𝐵 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))))
8		isassad.b	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐹))
9		isassad.f	. . . . . 6 ⊢ (𝜑 → 𝐹 = (Scalar‘𝑊))
10	9	fveq2d 6838	. . . . 5 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝑊)))
11	8, 10	eqtrd 2775	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(Scalar‘𝑊)))
12		isassad.v	. . . . 5 ⊢ (𝜑 → 𝑉 = (Base‘𝑊))
13		isassad.t	. . . . . . . . 9 ⊢ (𝜑 → × = (.r‘𝑊))
14		isassad.s	. . . . . . . . . 10 ⊢ (𝜑 → · = ( ·𝑠 ‘𝑊))
15	14	oveqd 7380	. . . . . . . . 9 ⊢ (𝜑 → (𝑟 · 𝑥) = (𝑟( ·𝑠 ‘𝑊)𝑥))
16		eqidd 2741	. . . . . . . . 9 ⊢ (𝜑 → 𝑦 = 𝑦)
17	13, 15, 16	oveq123d 7384	. . . . . . . 8 ⊢ (𝜑 → ((𝑟 · 𝑥) × 𝑦) = ((𝑟( ·𝑠 ‘𝑊)𝑥)(.r‘𝑊)𝑦))
18		eqidd 2741	. . . . . . . . 9 ⊢ (𝜑 → 𝑟 = 𝑟)
19	13	oveqd 7380	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 × 𝑦) = (𝑥(.r‘𝑊)𝑦))
20	14, 18, 19	oveq123d 7384	. . . . . . . 8 ⊢ (𝜑 → (𝑟 · (𝑥 × 𝑦)) = (𝑟( ·𝑠 ‘𝑊)(𝑥(.r‘𝑊)𝑦)))
21	17, 20	eqeq12d 2756	. . . . . . 7 ⊢ (𝜑 → (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ↔ ((𝑟( ·𝑠 ‘𝑊)𝑥)(.r‘𝑊)𝑦) = (𝑟( ·𝑠 ‘𝑊)(𝑥(.r‘𝑊)𝑦))))
22		eqidd 2741	. . . . . . . . 9 ⊢ (𝜑 → 𝑥 = 𝑥)
23	14	oveqd 7380	. . . . . . . . 9 ⊢ (𝜑 → (𝑟 · 𝑦) = (𝑟( ·𝑠 ‘𝑊)𝑦))
24	13, 22, 23	oveq123d 7384	. . . . . . . 8 ⊢ (𝜑 → (𝑥 × (𝑟 · 𝑦)) = (𝑥(.r‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑦)))
25	24, 20	eqeq12d 2756	. . . . . . 7 ⊢ (𝜑 → ((𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)) ↔ (𝑥(.r‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑦)) = (𝑟( ·𝑠 ‘𝑊)(𝑥(.r‘𝑊)𝑦))))
26	21, 25	anbi12d 638	. . . . . 6 ⊢ (𝜑 → ((((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))) ↔ (((𝑟( ·𝑠 ‘𝑊)𝑥)(.r‘𝑊)𝑦) = (𝑟( ·𝑠 ‘𝑊)(𝑥(.r‘𝑊)𝑦)) ∧ (𝑥(.r‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑦)) = (𝑟( ·𝑠 ‘𝑊)(𝑥(.r‘𝑊)𝑦)))))
27	12, 26	raleqbidv 3314	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ 𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))) ↔ ∀𝑦 ∈ (Base‘𝑊)(((𝑟( ·𝑠 ‘𝑊)𝑥)(.r‘𝑊)𝑦) = (𝑟( ·𝑠 ‘𝑊)(𝑥(.r‘𝑊)𝑦)) ∧ (𝑥(.r‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑦)) = (𝑟( ·𝑠 ‘𝑊)(𝑥(.r‘𝑊)𝑦)))))
28	12, 27	raleqbidv 3314	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))) ↔ ∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)(((𝑟( ·𝑠 ‘𝑊)𝑥)(.r‘𝑊)𝑦) = (𝑟( ·𝑠 ‘𝑊)(𝑥(.r‘𝑊)𝑦)) ∧ (𝑥(.r‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑦)) = (𝑟( ·𝑠 ‘𝑊)(𝑥(.r‘𝑊)𝑦)))))
29	11, 28	raleqbidv 3314	. . 3 ⊢ (𝜑 → (∀𝑟 ∈ 𝐵 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))) ↔ ∀𝑟 ∈ (Base‘(Scalar‘𝑊))∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)(((𝑟( ·𝑠 ‘𝑊)𝑥)(.r‘𝑊)𝑦) = (𝑟( ·𝑠 ‘𝑊)(𝑥(.r‘𝑊)𝑦)) ∧ (𝑥(.r‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑦)) = (𝑟( ·𝑠 ‘𝑊)(𝑥(.r‘𝑊)𝑦)))))
30	7, 29	mpbid 233	. 2 ⊢ (𝜑 → ∀𝑟 ∈ (Base‘(Scalar‘𝑊))∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)(((𝑟( ·𝑠 ‘𝑊)𝑥)(.r‘𝑊)𝑦) = (𝑟( ·𝑠 ‘𝑊)(𝑥(.r‘𝑊)𝑦)) ∧ (𝑥(.r‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑦)) = (𝑟( ·𝑠 ‘𝑊)(𝑥(.r‘𝑊)𝑦))))
31		eqid 2740	. . 3 ⊢ (Base‘𝑊) = (Base‘𝑊)
32		eqid 2740	. . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
33		eqid 2740	. . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
34		eqid 2740	. . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
35		eqid 2740	. . 3 ⊢ (.r‘𝑊) = (.r‘𝑊)
36	31, 32, 33, 34, 35	isassa 21838	. 2 ⊢ (𝑊 ∈ AssAlg ↔ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring) ∧ ∀𝑟 ∈ (Base‘(Scalar‘𝑊))∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)(((𝑟( ·𝑠 ‘𝑊)𝑥)(.r‘𝑊)𝑦) = (𝑟( ·𝑠 ‘𝑊)(𝑥(.r‘𝑊)𝑦)) ∧ (𝑥(.r‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑦)) = (𝑟( ·𝑠 ‘𝑊)(𝑥(.r‘𝑊)𝑦)))))
37	3, 30, 36	sylanbrc 589	1 ⊢ (𝜑 → 𝑊 ∈ AssAlg)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3054  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  ._rcmulr 17219  Scalarcsca 17221   ·_𝑠 cvsca 17222  Ringcrg 20212  LModclmod 20857  AssAlgcasa 21832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-nul 5235

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rab 3393  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-iota 6448  df-fv 6500  df-ov 7366  df-assa 21835

This theorem is referenced by:  issubassa3  21848  sraassab  21850  zlmassa  21885  psrassa  21954  matassa  22434  mendassa  43642