Theorem sraval 14577

Description: Lemma for srabaseg 14579 through sravscag 14583. (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 2824	. . . 4 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
2	1	adantr 276	. . 3 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) → 𝑊 ∈ V)
3		df-sra 14575	. . . 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 5669	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
5	4	pweqd 3673	. . . . 5 ⊢ (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 (Base‘𝑊))
6		id 19	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊)
7		oveq1 6056	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝑤 ↾s 𝑠) = (𝑊 ↾s 𝑠))
8	7	opeq2d 3889	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ⟨(Scalar‘ndx), (𝑤 ↾s 𝑠)⟩ = ⟨(Scalar‘ndx), (𝑊 ↾s 𝑠)⟩)
9	6, 8	oveq12d 6067	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑤 sSet ⟨(Scalar‘ndx), (𝑤 ↾s 𝑠)⟩) = (𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑠)⟩))
10		fveq2 5669	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (.r‘𝑤) = (.r‘𝑊))
11	10	opeq2d 3889	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ⟨( ·𝑠 ‘ndx), (.r‘𝑤)⟩ = ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩)
12	9, 11	oveq12d 6067	. . . . . 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 3889	. . . . . 6 ⊢ (𝑤 = 𝑊 → ⟨(·𝑖‘ndx), (.r‘𝑤)⟩ = ⟨(·𝑖‘ndx), (.r‘𝑊)⟩)
14	12, 13	oveq12d 6067	. . . . 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 4191	. . . 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 2824	. . . 4 ⊢ (𝑊 ∈ V → 𝑊 ∈ V)
17		basfn 13263	. . . . . . 7 ⊢ Base Fn V
18		funfvex 5686	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝑊 ∈ dom Base) → (Base‘𝑊) ∈ V)
19	18	funfni 5457	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝑊 ∈ V) → (Base‘𝑊) ∈ V)
20	17, 19	mpan 424	. . . . . 6 ⊢ (𝑊 ∈ V → (Base‘𝑊) ∈ V)
21	20	pwexd 4293	. . . . 5 ⊢ (𝑊 ∈ V → 𝒫 (Base‘𝑊) ∈ V)
22	21	mptexd 5912	. . . 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 5770	. . 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 6065	. . . . . 6 ⊢ (((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) ∧ 𝑠 = 𝑆) → (𝑊 ↾s 𝑠) = (𝑊 ↾s 𝑆))
27	26	opeq2d 3889	. . . . 5 ⊢ (((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) ∧ 𝑠 = 𝑆) → ⟨(Scalar‘ndx), (𝑊 ↾s 𝑠)⟩ = ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩)
28	27	oveq2d 6065	. . . 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 6064	. . 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 6064	. 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 4267	. . . 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 13358	. . . . . . 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 2826	. . . . . 6 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) → 𝑆 ∈ V)
40		ressex 13270	. . . . . 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 13236	. . . . 5 ⊢ ((𝑊 ∈ 𝑉 ∧ (Scalar‘ndx) ∈ ℕ ∧ (𝑊 ↾s 𝑆) ∈ V) → (𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) ∈ V)
43	35, 38, 41, 42	syl3anc 1274	. . . 4 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) → (𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) ∈ V)
44		vscaslid 13368	. . . . . 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 13337	. . . . . 6 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
48	47	slotex 13231	. . . . 5 ⊢ (𝑊 ∈ 𝑉 → (.r‘𝑊) ∈ V)
49	48	adantr 276	. . . 4 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) → (.r‘𝑊) ∈ V)
50		setsex 13236	. . . 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 1274	. . 3 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) → ((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) ∈ V)
52		ipslid 13376	. . . . 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 13236	. . 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 1274	. 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 5757	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 1398   ∈ wcel 2203  Vcvv 2812   ⊆ wss 3210  𝒫 cpw 3668  ⟨cop 3691   ↦ cmpt 4170   Fn wfn 5346  ‘cfv 5351  (class class class)co 6049  ℕcn 9236  ndxcnx 13201   sSet csts 13202  Slot cslot 13203  Basecbs 13204   ↾_s cress 13205  ._rcmulr 13283  Scalarcsca 13285   ·_𝑠 cvsca 13286  ·_𝑖cip 13287  subringAlg csra 14573

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8217  ax-resscn 8218  ax-1re 8220  ax-addrcl 8223

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-inn 9237  df-2 9295  df-3 9296  df-4 9297  df-5 9298  df-6 9299  df-7 9300  df-8 9301  df-ndx 13207  df-slot 13208  df-base 13210  df-sets 13211  df-iress 13212  df-mulr 13296  df-sca 13298  df-vsca 13299  df-ip 13300  df-sra 14575

This theorem is referenced by:  sralemg  14578  srascag  14582  sravscag  14583  sraipg  14584  sraex  14586