Theorem sraval 14422

Description: Lemma for srabaseg 14424 through sravscag 14428. (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 2811	. . . 4 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
2	1	adantr 276	. . 3 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) → 𝑊 ∈ V)
3		df-sra 14420	. . . 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)⟩)))
4		fveq2 5632	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
5	4	pweqd 3654	. . . . 5 ⊢ (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 (Base‘𝑊))
6		id 19	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊)
7		oveq1 6017	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝑤 ↾s 𝑠) = (𝑊 ↾s 𝑠))
8	7	opeq2d 3864	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ⟨(Scalar‘ndx), (𝑤 ↾s 𝑠)⟩ = ⟨(Scalar‘ndx), (𝑊 ↾s 𝑠)⟩)
9	6, 8	oveq12d 6028	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑤 sSet ⟨(Scalar‘ndx), (𝑤 ↾s 𝑠)⟩) = (𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑠)⟩))
10		fveq2 5632	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (.r‘𝑤) = (.r‘𝑊))
11	10	opeq2d 3864	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ⟨( ·𝑠 ‘ndx), (.r‘𝑤)⟩ = ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩)
12	9, 11	oveq12d 6028	. . . . . 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)⟩))
13	10	opeq2d 3864	. . . . . 6 ⊢ (𝑤 = 𝑊 → ⟨(·𝑖‘ndx), (.r‘𝑤)⟩ = ⟨(·𝑖‘ndx), (.r‘𝑊)⟩)
14	12, 13	oveq12d 6028	. . . . 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)⟩))
15	5, 14	mpteq12dv 4166	. . . 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)⟩)))
16		elex 2811	. . . 4 ⊢ (𝑊 ∈ V → 𝑊 ∈ V)
17		basfn 13112	. . . . . . 7 ⊢ Base Fn V
18		funfvex 5649	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝑊 ∈ dom Base) → (Base‘𝑊) ∈ V)
19	18	funfni 5426	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝑊 ∈ V) → (Base‘𝑊) ∈ V)
20	17, 19	mpan 424	. . . . . 6 ⊢ (𝑊 ∈ V → (Base‘𝑊) ∈ V)
21	20	pwexd 4266	. . . . 5 ⊢ (𝑊 ∈ V → 𝒫 (Base‘𝑊) ∈ V)
22	21	mptexd 5873	. . . 4 ⊢ (𝑊 ∈ V → (𝑠 ∈ 𝒫 (Base‘𝑊) ↦ (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩)) ∈ V)
23	3, 15, 16, 22	fvmptd3 5733	. . 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)⟩)))
24	2, 23	syl 14	. 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)⟩)))
25		simpr 110	. . . . . . 7 ⊢ (((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
26	25	oveq2d 6026	. . . . . 6 ⊢ (((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) ∧ 𝑠 = 𝑆) → (𝑊 ↾s 𝑠) = (𝑊 ↾s 𝑆))
27	26	opeq2d 3864	. . . . 5 ⊢ (((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) ∧ 𝑠 = 𝑆) → ⟨(Scalar‘ndx), (𝑊 ↾s 𝑠)⟩ = ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩)
28	27	oveq2d 6026	. . . 4 ⊢ (((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) ∧ 𝑠 = 𝑆) → (𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑠)⟩) = (𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩))
29	28	oveq1d 6025	. . 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)⟩))
30	29	oveq1d 6025	. 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)⟩))
31		simpr 110	. . 3 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) → 𝑆 ⊆ (Base‘𝑊))
32		elpw2g 4241	. . . 4 ⊢ ((Base‘𝑊) ∈ V → (𝑆 ∈ 𝒫 (Base‘𝑊) ↔ 𝑆 ⊆ (Base‘𝑊)))
33	2, 20, 32	3syl 17	. . 3 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) → (𝑆 ∈ 𝒫 (Base‘𝑊) ↔ 𝑆 ⊆ (Base‘𝑊)))
34	31, 33	mpbird 167	. 2 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) → 𝑆 ∈ 𝒫 (Base‘𝑊))
35		simpl 109	. . . . 5 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) → 𝑊 ∈ 𝑉)
36		scaslid 13207	. . . . . . 7 ⊢ (Scalar = Slot (Scalar‘ndx) ∧ (Scalar‘ndx) ∈ ℕ)
37	36	simpri 113	. . . . . 6 ⊢ (Scalar‘ndx) ∈ ℕ
38	37	a1i 9	. . . . 5 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) → (Scalar‘ndx) ∈ ℕ)
39	34	elexd 2813	. . . . . 6 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) → 𝑆 ∈ V)
40		ressex 13119	. . . . . 6 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑆 ∈ V) → (𝑊 ↾s 𝑆) ∈ V)
41	39, 40	syldan 282	. . . . 5 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) → (𝑊 ↾s 𝑆) ∈ V)
42		setsex 13085	. . . . 5 ⊢ ((𝑊 ∈ 𝑉 ∧ (Scalar‘ndx) ∈ ℕ ∧ (𝑊 ↾s 𝑆) ∈ V) → (𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) ∈ V)
43	35, 38, 41, 42	syl3anc 1271	. . . 4 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) → (𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) ∈ V)
44		vscaslid 13217	. . . . . 6 ⊢ ( ·𝑠 = Slot ( ·𝑠 ‘ndx) ∧ ( ·𝑠 ‘ndx) ∈ ℕ)
45	44	simpri 113	. . . . 5 ⊢ ( ·𝑠 ‘ndx) ∈ ℕ
46	45	a1i 9	. . . 4 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) → ( ·𝑠 ‘ndx) ∈ ℕ)
47		mulrslid 13186	. . . . . 6 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
48	47	slotex 13080	. . . . 5 ⊢ (𝑊 ∈ 𝑉 → (.r‘𝑊) ∈ V)
49	48	adantr 276	. . . 4 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) → (.r‘𝑊) ∈ V)
50		setsex 13085	. . . 4 ⊢ (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) ∈ V ∧ ( ·𝑠 ‘ndx) ∈ ℕ ∧ (.r‘𝑊) ∈ V) → ((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) ∈ V)
51	43, 46, 49, 50	syl3anc 1271	. . 3 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) → ((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) ∈ V)
52		ipslid 13225	. . . . 5 ⊢ (·𝑖 = Slot (·𝑖‘ndx) ∧ (·𝑖‘ndx) ∈ ℕ)
53	52	simpri 113	. . . 4 ⊢ (·𝑖‘ndx) ∈ ℕ
54	53	a1i 9	. . 3 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) → (·𝑖‘ndx) ∈ ℕ)
55		setsex 13085	. . 3 ⊢ ((((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) ∈ V ∧ (·𝑖‘ndx) ∈ ℕ ∧ (.r‘𝑊) ∈ V) → (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩) ∈ V)
56	51, 54, 49, 55	syl3anc 1271	. 2 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) → (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩) ∈ V)
57	24, 30, 34, 56	fvmptd 5720	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 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200  Vcvv 2799   ⊆ wss 3197  𝒫 cpw 3649  ⟨cop 3669   ↦ cmpt 4145   Fn wfn 5316  ‘cfv 5321  (class class class)co 6010  ℕcn 9126  ndxcnx 13050   sSet csts 13051  Slot cslot 13052  Basecbs 13053   ↾_s cress 13054  ._rcmulr 13132  Scalarcsca 13134   ·_𝑠 cvsca 13135  ·_𝑖cip 13136  subringAlg csra 14418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-cnex 8106  ax-resscn 8107  ax-1re 8109  ax-addrcl 8112

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4385  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-ov 6013  df-oprab 6014  df-mpo 6015  df-inn 9127  df-2 9185  df-3 9186  df-4 9187  df-5 9188  df-6 9189  df-7 9190  df-8 9191  df-ndx 13056  df-slot 13057  df-base 13059  df-sets 13060  df-iress 13061  df-mulr 13145  df-sca 13147  df-vsca 13148  df-ip 13149  df-sra 14420

This theorem is referenced by:  sralemg  14423  srascag  14427  sravscag  14428  sraipg  14429  sraex  14431