Theorem sraval 21082

Description: Lemma for srabase 21084 through sravsca 21088. (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 3468	. . . 4 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
2	1	adantr 480	. . 3 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) → 𝑊 ∈ V)
3		fveq2 6858	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4	3	pweqd 4580	. . . . 5 ⊢ (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 (Base‘𝑊))
5		id 22	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊)
6		oveq1 7394	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝑤 ↾s 𝑠) = (𝑊 ↾s 𝑠))
7	6	opeq2d 4844	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ⟨(Scalar‘ndx), (𝑤 ↾s 𝑠)⟩ = ⟨(Scalar‘ndx), (𝑊 ↾s 𝑠)⟩)
8	5, 7	oveq12d 7405	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑤 sSet ⟨(Scalar‘ndx), (𝑤 ↾s 𝑠)⟩) = (𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑠)⟩))
9		fveq2 6858	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (.r‘𝑤) = (.r‘𝑊))
10	9	opeq2d 4844	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ⟨( ·𝑠 ‘ndx), (.r‘𝑤)⟩ = ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩)
11	8, 10	oveq12d 7405	. . . . . 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 4844	. . . . . 6 ⊢ (𝑤 = 𝑊 → ⟨(·𝑖‘ndx), (.r‘𝑤)⟩ = ⟨(·𝑖‘ndx), (.r‘𝑊)⟩)
13	11, 12	oveq12d 7405	. . . . 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 5194	. . . 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 21080	. . . 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 6871	. . . . . 6 ⊢ (Base‘𝑊) ∈ V
17	16	pwex 5335	. . . . 5 ⊢ 𝒫 (Base‘𝑊) ∈ V
18	17	mptex 7197	. . . 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 6968	. . 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 17	. 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 484	. . . . . . 7 ⊢ (((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
22	21	oveq2d 7403	. . . . . 6 ⊢ (((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) ∧ 𝑠 = 𝑆) → (𝑊 ↾s 𝑠) = (𝑊 ↾s 𝑆))
23	22	opeq2d 4844	. . . . 5 ⊢ (((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) ∧ 𝑠 = 𝑆) → ⟨(Scalar‘ndx), (𝑊 ↾s 𝑠)⟩ = ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩)
24	23	oveq2d 7403	. . . 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 7402	. . 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 7402	. 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		simpr 484	. . 3 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) → 𝑆 ⊆ (Base‘𝑊))
28	16	elpw2 5289	. . 3 ⊢ (𝑆 ∈ 𝒫 (Base‘𝑊) ↔ 𝑆 ⊆ (Base‘𝑊))
29	27, 28	sylibr 234	. 2 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) → 𝑆 ∈ 𝒫 (Base‘𝑊))
30		ovexd 7422	. 2 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) → (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩) ∈ V)
31	20, 26, 29, 30	fvmptd 6975	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)⟩))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3447   ⊆ wss 3914  𝒫 cpw 4563  ⟨cop 4595   ↦ cmpt 5188  ‘cfv 6511  (class class class)co 7387   sSet csts 17133  ndxcnx 17163  Basecbs 17179   ↾_s cress 17200  ._rcmulr 17221  Scalarcsca 17223   ·_𝑠 cvsca 17224  ·_𝑖cip 17225  subringAlg csra 21078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-sra 21080

This theorem is referenced by:  sralem  21083  srasca  21087  sravsca  21088  sraip  21089  rlmval2  21099  resssra  33583