Theorem sraval 21430

Description: Lemma for srabase 21432 through sravsca 21436. (Contributed by Mario Carneiro, 27-Nov-2014.) (Revised by Thierry Arnoux, 16-Jun-2019.)

Assertion

Ref	Expression
sraval	⊢ ((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) → ((subringAlg ‘𝑊)‘𝑆) = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩))

Proof of Theorem sraval

Dummy variables 𝑠 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3472	. . . 4 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
2	1	adantr 486	. . 3 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) → 𝑊 ∈ V)
3		fveq2 6877	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4	3	pweqd 4574	. . . . 5 ⊢ (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 (Base‘𝑊))
5		id 23	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊)
6		oveq1 7419	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝑤 ↾s 𝑠) = (𝑊 ↾s 𝑠))
7	6	opeq2d 4840	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ⟨(Scalar‘ndx), (𝑤 ↾s 𝑠)⟩ = ⟨(Scalar‘ndx), (𝑊 ↾s 𝑠)⟩)
8	5, 7	oveq12d 7430	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑤 sSet ⟨(Scalar‘ndx), (𝑤 ↾s 𝑠)⟩) = (𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑠)⟩))
9		fveq2 6877	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (.r‘𝑤) = (.r‘𝑊))
10	9	opeq2d 4840	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ⟨( ·𝑠 ‘ndx), (.r‘𝑤)⟩ = ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩)
11	8, 10	oveq12d 7430	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((𝑤 sSet ⟨(Scalar‘ndx), (𝑤 ↾s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑤)⟩) = ((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩))
12	9	opeq2d 4840	. . . . . 6 ⊢ (𝑤 = 𝑊 → ⟨(·𝑖‘ndx), (.r‘𝑤)⟩ = ⟨(·𝑖‘ndx), (.r‘𝑊)⟩)
13	11, 12	oveq12d 7430	. . . . 5 ⊢ (𝑤 = 𝑊 → (((𝑤 sSet ⟨(Scalar‘ndx), (𝑤 ↾s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑤)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑤)⟩) = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩))
14	4, 13	mpteq12dv 5192	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ (((𝑤 sSet ⟨(Scalar‘ndx), (𝑤 ↾s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑤)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑤)⟩)) = (𝑠 ∈ 𝒫 (Base‘𝑊) ↦ (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩)))
15		df-sra 21428	. . . 4 ⊢ subringAlg = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ (((𝑤 sSet ⟨(Scalar‘ndx), (𝑤 ↾s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑤)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑤)⟩)))
16		fvex 6890	. . . . . 6 ⊢ (Base‘𝑊) ∈ V
17	16	pwex 5342	. . . . 5 ⊢ 𝒫 (Base‘𝑊) ∈ V
18	17	mptex 7221	. . . 4 ⊢ (𝑠 ∈ 𝒫 (Base‘𝑊) ↦ (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩)) ∈ V
19	14, 15, 18	fvmpt 6985	. . 3 ⊢ (𝑊 ∈ V → (subringAlg ‘𝑊) = (𝑠 ∈ 𝒫 (Base‘𝑊) ↦ (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩)))
20	2, 19	syl 18	. 2 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) → (subringAlg ‘𝑊) = (𝑠 ∈ 𝒫 (Base‘𝑊) ↦ (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩)))
21		simpr 490	. . . . . . 7 ⊢ (((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
22	21	oveq2d 7428	. . . . . 6 ⊢ (((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) ∧ 𝑠 = 𝑆) → (𝑊 ↾s 𝑠) = (𝑊 ↾s 𝑆))
23	22	opeq2d 4840	. . . . 5 ⊢ (((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) ∧ 𝑠 = 𝑆) → ⟨(Scalar‘ndx), (𝑊 ↾s 𝑠)⟩ = ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩)
24	23	oveq2d 7428	. . . 4 ⊢ (((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) ∧ 𝑠 = 𝑆) → (𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑠)⟩) = (𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩))
25	24	oveq1d 7427	. . 3 ⊢ (((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) ∧ 𝑠 = 𝑆) → ((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) = ((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩))
26	25	oveq1d 7427	. 2 ⊢ (((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) ∧ 𝑠 = 𝑆) → (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩) = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩))
27	16	elpw2 5296	. . 3 ⊢ (𝑆 ∈ 𝒫 (Base‘𝑊) ↔ 𝑆 ⊆ (Base‘𝑊))
28	27	bilanri 512	. 2 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) → 𝑆 ∈ 𝒫 (Base‘𝑊))
29		ovexd 7447	. 2 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) → (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩) ∈ V)
30	20, 26, 28, 29	fvmptd 6993	1 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) → ((subringAlg ‘𝑊)‘𝑆) = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451   ⊆ wss 3899  𝒫 cpw 4557  ⟨cop 4590   ↦ cmpt 5186  ‘cfv 6531  (class class class)co 7412   sSet csts 17321  ndxcnx 17351  Basecbs 17367   ↾_s cress 17388  ._rcmulr 17409  Scalarcsca 17411   ·_𝑠 cvsca 17412  ·_𝑖cip 17413  subringAlg csra 21426

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-sra 21428

This theorem is used by:  sralem  21431  srasca  21435  sravsca  21436  sraip  21437  rlmval2  21447  resssra  34201