prdsvsca

Description: Scalar multiplication in a structure product. (Contributed by Stefan O'Rear, 5-Jan-2015) (Revised by Mario Carneiro, 15-Aug-2015) (Revised by Thierry Arnoux, 16-Jun-2019) (Revised by Zhi Wang, 18-Aug-2024)

	Ref	Expression
Hypotheses	prdsbas.p	$⊢ P = S ⨉_{𝑠} R$
	prdsbas.s	$⊢ φ \to S \in V$
	prdsbas.r	$⊢ φ \to R \in W$
	prdsbas.b	$⊢ B = {Base}_{P}$
	prdsbas.i	$⊢ φ \to dom R = I$
	prdsvsca.k	$⊢ K = {Base}_{S}$
	prdsvsca.m	$⊢ \underset{˙}{\cdot} = \cdot_{P}$
Assertion	prdsvsca	$⊢ φ \to \underset{˙}{\cdot} = (f \in K, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x)))$

Proof

Step	Hyp	Ref	Expression
1		prdsbas.p	$⊢ P = S ⨉_{𝑠} R$
2		prdsbas.s	$⊢ φ \to S \in V$
3		prdsbas.r	$⊢ φ \to R \in W$
4		prdsbas.b	$⊢ B = {Base}_{P}$
5		prdsbas.i	$⊢ φ \to dom R = I$
6		prdsvsca.k	$⊢ K = {Base}_{S}$
7		prdsvsca.m	$⊢ \underset{˙}{\cdot} = \cdot_{P}$
8	1 2 3 4 5	prdsbas	$⊢ φ \to B = \underset{x \in I}{⨉} {Base}_{R (x)}$
9		eqid	$⊢ +_{P} = +_{P}$
10	1 2 3 4 5 9	prdsplusg	$⊢ φ \to +_{P} = (f \in B, g \in B ⟼ (x \in I ⟼ f (x) +_{R (x)} g (x)))$
11		eqid	$⊢ \cdot_{P} = \cdot_{P}$
12	1 2 3 4 5 11	prdsmulr	$⊢ φ \to \cdot_{P} = (f \in B, g \in B ⟼ (x \in I ⟼ f (x) \cdot_{R (x)} g (x)))$
13		eqidd	$⊢ φ \to (f \in K, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x))) = (f \in K, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x)))$
14		eqidd	$⊢ φ \to (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x))) = (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x)))$
15		eqidd	$⊢ φ \to \prod_{𝑡} (TopOpen \circ R) = \prod_{𝑡} (TopOpen \circ R)$
16		eqidd	$⊢ φ \to \{⟨f, g⟩ \| (\{f, g\} \subseteq B \land \forall x \in I f (x) \leq_{R (x)} g (x))\} = \{⟨f, g⟩ \| (\{f, g\} \subseteq B \land \forall x \in I f (x) \leq_{R (x)} g (x))\}$
17		eqidd	$⊢ φ \to (f \in B, g \in B ⟼ sup (ran (x \in I ⟼ f (x) dist (R (x)) g (x)) \cup \{0\} ℝ^{} <)) = (f \in B, g \in B ⟼ sup (ran (x \in I ⟼ f (x) dist (R (x)) g (x)) \cup \{0\} ℝ^{} <))$
18		eqidd	$⊢ φ \to (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) = (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x)))$
19		eqidd	$⊢ φ \to (a \in (B \times B), c \in B ⟼ (d \in (2^{nd} (a) (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) c), e \in (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) (a) ⟼ (x \in I ⟼ d (x) (⟨1^{st} (a) (x), 2^{nd} (a) (x)⟩ comp (R (x)) c (x)) e (x)))) = (a \in (B \times B), c \in B ⟼ (d \in (2^{nd} (a) (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) c), e \in (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) (a) ⟼ (x \in I ⟼ d (x) (⟨1^{st} (a) (x), 2^{nd} (a) (x)⟩ comp (R (x)) c (x)) e (x))))$
20	1 6 5 8 10 12 13 14 15 16 17 18 19 2 3	prdsval	$⊢ φ \to P = (\{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, +_{P}⟩, ⟨\cdot_{ndx}, \cdot_{P}⟩\} \cup \{⟨Scalar (ndx), S⟩, ⟨\cdot_{ndx}, (f \in K, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x)))⟩, ⟨\cdot_{𝑖} (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x)))⟩\}) \cup (\{⟨TopSet (ndx), \prod_{𝑡} (TopOpen \circ R)⟩, ⟨\leq_{ndx}, \{⟨f, g⟩ \| (\{f, g\} \subseteq B \land \forall x \in I f (x) \leq_{R (x)} g (x))\}⟩, ⟨dist (ndx), (f \in B, g \in B ⟼ sup (ran (x \in I ⟼ f (x) dist (R (x)) g (x)) \cup \{0\} ℝ^{*} <))⟩\} \cup \{⟨Hom (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x)))⟩, ⟨comp (ndx), (a \in (B \times B), c \in B ⟼ (d \in (2^{nd} (a) (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) c), e \in (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) (a) ⟼ (x \in I ⟼ d (x) (⟨1^{st} (a) (x), 2^{nd} (a) (x)⟩ comp (R (x)) c (x)) e (x))))⟩\})$
21		vscaid	$⊢ \cdot_{𝑠} = Slot \cdot_{ndx}$
22		ovssunirn	$⊢ f \cdot_{R (x)} g (x) \subseteq ⋃ ran \cdot_{R (x)}$
23	21	strfvss	$⊢ \cdot_{R (x)} \subseteq ⋃ ran R (x)$
24		fvssunirn	$⊢ R (x) \subseteq ⋃ ran R$
25		rnss	$⊢ R (x) \subseteq ⋃ ran R \to ran R (x) \subseteq ran ⋃ ran R$
26		uniss	$⊢ ran R (x) \subseteq ran ⋃ ran R \to ⋃ ran R (x) \subseteq ⋃ ran ⋃ ran R$
27	24 25 26	mp2b	$⊢ ⋃ ran R (x) \subseteq ⋃ ran ⋃ ran R$
28	23 27	sstri	$⊢ \cdot_{R (x)} \subseteq ⋃ ran ⋃ ran R$
29		rnss	$⊢ \cdot_{R (x)} \subseteq ⋃ ran ⋃ ran R \to ran \cdot_{R (x)} \subseteq ran ⋃ ran ⋃ ran R$
30		uniss	$⊢ ran \cdot_{R (x)} \subseteq ran ⋃ ran ⋃ ran R \to ⋃ ran \cdot_{R (x)} \subseteq ⋃ ran ⋃ ran ⋃ ran R$
31	28 29 30	mp2b	$⊢ ⋃ ran \cdot_{R (x)} \subseteq ⋃ ran ⋃ ran ⋃ ran R$
32	22 31	sstri	$⊢ f \cdot_{R (x)} g (x) \subseteq ⋃ ran ⋃ ran ⋃ ran R$
33		ovex	$⊢ f \cdot_{R (x)} g (x) \in V$
34	33	elpw	$⊢ f \cdot_{R (x)} g (x) \in 𝒫 ⋃ ran ⋃ ran ⋃ ran R \leftrightarrow f \cdot_{R (x)} g (x) \subseteq ⋃ ran ⋃ ran ⋃ ran R$
35	32 34	mpbir	$⊢ f \cdot_{R (x)} g (x) \in 𝒫 ⋃ ran ⋃ ran ⋃ ran R$
36	35	a1i	$⊢ (φ \land x \in I) \to f \cdot_{R (x)} g (x) \in 𝒫 ⋃ ran ⋃ ran ⋃ ran R$
37	36	fmpttd	$⊢ φ \to (x \in I ⟼ f \cdot_{R (x)} g (x)) : I ⟶ 𝒫 ⋃ ran ⋃ ran ⋃ ran R$
38		rnexg	$⊢ R \in W \to ran R \in V$
39		uniexg	$⊢ ran R \in V \to ⋃ ran R \in V$
40	3 38 39	3syl	$⊢ φ \to ⋃ ran R \in V$
41		rnexg	$⊢ ⋃ ran R \in V \to ran ⋃ ran R \in V$
42		uniexg	$⊢ ran ⋃ ran R \in V \to ⋃ ran ⋃ ran R \in V$
43	40 41 42	3syl	$⊢ φ \to ⋃ ran ⋃ ran R \in V$
44		rnexg	$⊢ ⋃ ran ⋃ ran R \in V \to ran ⋃ ran ⋃ ran R \in V$
45		uniexg	$⊢ ran ⋃ ran ⋃ ran R \in V \to ⋃ ran ⋃ ran ⋃ ran R \in V$
46		pwexg	$⊢ ⋃ ran ⋃ ran ⋃ ran R \in V \to 𝒫 ⋃ ran ⋃ ran ⋃ ran R \in V$
47	43 44 45 46	4syl	$⊢ φ \to 𝒫 ⋃ ran ⋃ ran ⋃ ran R \in V$
48	3	dmexd	$⊢ φ \to dom R \in V$
49	5 48	eqeltrrd	$⊢ φ \to I \in V$
50	47 49	elmapd	$⊢ φ \to ((x \in I ⟼ f \cdot_{R (x)} g (x)) \in ({𝒫 ⋃ ran ⋃ ran ⋃ ran R}^{I}) \leftrightarrow (x \in I ⟼ f \cdot_{R (x)} g (x)) : I ⟶ 𝒫 ⋃ ran ⋃ ran ⋃ ran R)$
51	37 50	mpbird	$⊢ φ \to (x \in I ⟼ f \cdot_{R (x)} g (x)) \in ({𝒫 ⋃ ran ⋃ ran ⋃ ran R}^{I})$
52	51	ralrimivw	$⊢ φ \to \forall g \in B (x \in I ⟼ f \cdot_{R (x)} g (x)) \in ({𝒫 ⋃ ran ⋃ ran ⋃ ran R}^{I})$
53	52	ralrimivw	$⊢ φ \to \forall f \in K \forall g \in B (x \in I ⟼ f \cdot_{R (x)} g (x)) \in ({𝒫 ⋃ ran ⋃ ran ⋃ ran R}^{I})$
54		eqid	$⊢ (f \in K, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x))) = (f \in K, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x)))$
55	54	fmpo	$⊢ \forall f \in K \forall g \in B (x \in I ⟼ f \cdot_{R (x)} g (x)) \in ({𝒫 ⋃ ran ⋃ ran ⋃ ran R}^{I}) \leftrightarrow (f \in K, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x))) : K \times B ⟶ {𝒫 ⋃ ran ⋃ ran ⋃ ran R}^{I}$
56	53 55	sylib	$⊢ φ \to (f \in K, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x))) : K \times B ⟶ {𝒫 ⋃ ran ⋃ ran ⋃ ran R}^{I}$
57	6	fvexi	$⊢ K \in V$
58	4	fvexi	$⊢ B \in V$
59	57 58	xpex	$⊢ K \times B \in V$
60		ovex	$⊢ {𝒫 ⋃ ran ⋃ ran ⋃ ran R}^{I} \in V$
61		fex2	$⊢ ((f \in K, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x))) : K \times B ⟶ {𝒫 ⋃ ran ⋃ ran ⋃ ran R}^{I} \land K \times B \in V \land {𝒫 ⋃ ran ⋃ ran ⋃ ran R}^{I} \in V) \to (f \in K, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x))) \in V$
62	59 60 61	mp3an23	$⊢ (f \in K, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x))) : K \times B ⟶ {𝒫 ⋃ ran ⋃ ran ⋃ ran R}^{I} \to (f \in K, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x))) \in V$
63	56 62	syl	$⊢ φ \to (f \in K, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x))) \in V$
64		snsstp2	$⊢ \{⟨\cdot_{ndx}, (f \in K, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x)))⟩\} \subseteq \{⟨Scalar (ndx), S⟩, ⟨\cdot_{ndx}, (f \in K, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x)))⟩, ⟨\cdot_{𝑖} (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x)))⟩\}$
65		ssun2	$⊢ \{⟨Scalar (ndx), S⟩, ⟨\cdot_{ndx}, (f \in K, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x)))⟩, ⟨\cdot_{𝑖} (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x)))⟩\} \subseteq \{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, +_{P}⟩, ⟨\cdot_{ndx}, \cdot_{P}⟩\} \cup \{⟨Scalar (ndx), S⟩, ⟨\cdot_{ndx}, (f \in K, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x)))⟩, ⟨\cdot_{𝑖} (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x)))⟩\}$
66	64 65	sstri	$⊢ \{⟨\cdot_{ndx}, (f \in K, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x)))⟩\} \subseteq \{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, +_{P}⟩, ⟨\cdot_{ndx}, \cdot_{P}⟩\} \cup \{⟨Scalar (ndx), S⟩, ⟨\cdot_{ndx}, (f \in K, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x)))⟩, ⟨\cdot_{𝑖} (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x)))⟩\}$
67		ssun1	$⊢ \{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, +_{P}⟩, ⟨\cdot_{ndx}, \cdot_{P}⟩\} \cup \{⟨Scalar (ndx), S⟩, ⟨\cdot_{ndx}, (f \in K, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x)))⟩, ⟨\cdot_{𝑖} (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x)))⟩\} \subseteq (\{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, +_{P}⟩, ⟨\cdot_{ndx}, \cdot_{P}⟩\} \cup \{⟨Scalar (ndx), S⟩, ⟨\cdot_{ndx}, (f \in K, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x)))⟩, ⟨\cdot_{𝑖} (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x)))⟩\}) \cup (\{⟨TopSet (ndx), \prod_{𝑡} (TopOpen \circ R)⟩, ⟨\leq_{ndx}, \{⟨f, g⟩ \| (\{f, g\} \subseteq B \land \forall x \in I f (x) \leq_{R (x)} g (x))\}⟩, ⟨dist (ndx), (f \in B, g \in B ⟼ sup (ran (x \in I ⟼ f (x) dist (R (x)) g (x)) \cup \{0\} ℝ^{*} <))⟩\} \cup \{⟨Hom (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x)))⟩, ⟨comp (ndx), (a \in (B \times B), c \in B ⟼ (d \in (2^{nd} (a) (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) c), e \in (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) (a) ⟼ (x \in I ⟼ d (x) (⟨1^{st} (a) (x), 2^{nd} (a) (x)⟩ comp (R (x)) c (x)) e (x))))⟩\})$
68	66 67	sstri	$⊢ \{⟨\cdot_{ndx}, (f \in K, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x)))⟩\} \subseteq (\{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, +_{P}⟩, ⟨\cdot_{ndx}, \cdot_{P}⟩\} \cup \{⟨Scalar (ndx), S⟩, ⟨\cdot_{ndx}, (f \in K, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x)))⟩, ⟨\cdot_{𝑖} (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x)))⟩\}) \cup (\{⟨TopSet (ndx), \prod_{𝑡} (TopOpen \circ R)⟩, ⟨\leq_{ndx}, \{⟨f, g⟩ \| (\{f, g\} \subseteq B \land \forall x \in I f (x) \leq_{R (x)} g (x))\}⟩, ⟨dist (ndx), (f \in B, g \in B ⟼ sup (ran (x \in I ⟼ f (x) dist (R (x)) g (x)) \cup \{0\} ℝ^{*} <))⟩\} \cup \{⟨Hom (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x)))⟩, ⟨comp (ndx), (a \in (B \times B), c \in B ⟼ (d \in (2^{nd} (a) (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) c), e \in (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) (a) ⟼ (x \in I ⟼ d (x) (⟨1^{st} (a) (x), 2^{nd} (a) (x)⟩ comp (R (x)) c (x)) e (x))))⟩\})$
69	20 7 21 63 68	prdsvallem	$⊢ φ \to \underset{˙}{\cdot} = (f \in K, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x)))$

Metamath Proof Explorer

Theorem prdsvsca

Proof