prdsmulr

Description: Multiplication in a structure product. (Contributed by Mario Carneiro, 11-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	$⊢ φ \to dom R = I$
	prdsmulr.t	$⊢ \underset{˙}{\cdot} = \cdot_{P}$
Assertion	prdsmulr	$⊢ φ \to \underset{˙}{\cdot} = (f \in B, g \in B ⟼ (x \in I ⟼ f (x) \cdot_{R (x)} g (x)))$

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		prdsmulr.t	$⊢ \underset{˙}{\cdot} = \cdot_{P}$
7		eqid	$⊢ {Base}_{S} = {Base}_{S}$
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		eqidd	$⊢ φ \to (f \in B, g \in B ⟼ (x \in I ⟼ f (x) \cdot_{R (x)} g (x))) = (f \in B, g \in B ⟼ (x \in I ⟼ f (x) \cdot_{R (x)} g (x)))$
12		eqidd	$⊢ φ \to (f \in {Base}_{S}, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x))) = (f \in {Base}_{S}, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x)))$
13		eqidd	$⊢ φ \to (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x))) = (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x)))$
14		eqidd	$⊢ φ \to \prod_{𝑡} (TopOpen \circ R) = \prod_{𝑡} (TopOpen \circ R)$
15		eqidd	$⊢ φ \to \{⟨f, g⟩ \| (\{f, g\} \subseteq B \land \forall x \in I f (x) \leq_{R (x)} g (x))\} = \{⟨f, g⟩ \| (\{f, g\} \subseteq B \land \forall x \in I f (x) \leq_{R (x)} g (x))\}$
16		eqidd	$⊢ φ \to (f \in B, g \in B ⟼ sup (ran (x \in I ⟼ f (x) dist (R (x)) g (x)) \cup \{0\} ℝ^{} <)) = (f \in B, g \in B ⟼ sup (ran (x \in I ⟼ f (x) dist (R (x)) g (x)) \cup \{0\} ℝ^{} <))$
17		eqidd	$⊢ φ \to (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) = (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x)))$
18		eqidd	$⊢ φ \to (a \in (B \times B), c \in B ⟼ (d \in (2^{nd} (a) (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) c), e \in (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) (a) ⟼ (x \in I ⟼ d (x) (⟨1^{st} (a) (x), 2^{nd} (a) (x)⟩ comp (R (x)) c (x)) e (x)))) = (a \in (B \times B), c \in B ⟼ (d \in (2^{nd} (a) (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) c), e \in (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) (a) ⟼ (x \in I ⟼ d (x) (⟨1^{st} (a) (x), 2^{nd} (a) (x)⟩ comp (R (x)) c (x)) e (x))))$
19	1 7 5 8 10 11 12 13 14 15 16 17 18 2 3	prdsval	$⊢ φ \to P = (\{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, +_{P}⟩, ⟨\cdot_{ndx}, (f \in B, g \in B ⟼ (x \in I ⟼ f (x) \cdot_{R (x)} g (x)))⟩\} \cup \{⟨Scalar (ndx), S⟩, ⟨\cdot_{ndx}, (f \in {Base}_{S}, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x)))⟩, ⟨\cdot_{𝑖} (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x)))⟩\}) \cup (\{⟨TopSet (ndx), \prod_{𝑡} (TopOpen \circ R)⟩, ⟨\leq_{ndx}, \{⟨f, g⟩ \| (\{f, g\} \subseteq B \land \forall x \in I f (x) \leq_{R (x)} g (x))\}⟩, ⟨dist (ndx), (f \in B, g \in B ⟼ sup (ran (x \in I ⟼ f (x) dist (R (x)) g (x)) \cup \{0\} ℝ^{*} <))⟩\} \cup \{⟨Hom (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x)))⟩, ⟨comp (ndx), (a \in (B \times B), c \in B ⟼ (d \in (2^{nd} (a) (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) c), e \in (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) (a) ⟼ (x \in I ⟼ d (x) (⟨1^{st} (a) (x), 2^{nd} (a) (x)⟩ comp (R (x)) c (x)) e (x))))⟩\})$
20		mulridx	$⊢ \cdot_{𝑟} = Slot \cdot_{ndx}$
21		ovssunirn	$⊢ f (x) \cdot_{R (x)} g (x) \subseteq ⋃ ran \cdot_{R (x)}$
22	20	strfvss	$⊢ \cdot_{R (x)} \subseteq ⋃ ran R (x)$
23		fvssunirn	$⊢ R (x) \subseteq ⋃ ran R$
24		rnss	$⊢ R (x) \subseteq ⋃ ran R \to ran R (x) \subseteq ran ⋃ ran R$
25		uniss	$⊢ ran R (x) \subseteq ran ⋃ ran R \to ⋃ ran R (x) \subseteq ⋃ ran ⋃ ran R$
26	23 24 25	mp2b	$⊢ ⋃ ran R (x) \subseteq ⋃ ran ⋃ ran R$
27	22 26	sstri	$⊢ \cdot_{R (x)} \subseteq ⋃ ran ⋃ ran R$
28		rnss	$⊢ \cdot_{R (x)} \subseteq ⋃ ran ⋃ ran R \to ran \cdot_{R (x)} \subseteq ran ⋃ ran ⋃ ran R$
29		uniss	$⊢ ran \cdot_{R (x)} \subseteq ran ⋃ ran ⋃ ran R \to ⋃ ran \cdot_{R (x)} \subseteq ⋃ ran ⋃ ran ⋃ ran R$
30	27 28 29	mp2b	$⊢ ⋃ ran \cdot_{R (x)} \subseteq ⋃ ran ⋃ ran ⋃ ran R$
31	21 30	sstri	$⊢ f (x) \cdot_{R (x)} g (x) \subseteq ⋃ ran ⋃ ran ⋃ ran R$
32		ovex	$⊢ f (x) \cdot_{R (x)} g (x) \in V$
33	32	elpw	$⊢ f (x) \cdot_{R (x)} g (x) \in 𝒫 ⋃ ran ⋃ ran ⋃ ran R \leftrightarrow f (x) \cdot_{R (x)} g (x) \subseteq ⋃ ran ⋃ ran ⋃ ran R$
34	31 33	mpbir	$⊢ f (x) \cdot_{R (x)} g (x) \in 𝒫 ⋃ ran ⋃ ran ⋃ ran R$
35	34	a1i	$⊢ (φ \land x \in I) \to f (x) \cdot_{R (x)} g (x) \in 𝒫 ⋃ ran ⋃ ran ⋃ ran R$
36	35	fmpttd	$⊢ φ \to (x \in I ⟼ f (x) \cdot_{R (x)} g (x)) : I ⟶ 𝒫 ⋃ ran ⋃ ran ⋃ ran R$
37		rnexg	$⊢ R \in W \to ran R \in V$
38		uniexg	$⊢ ran R \in V \to ⋃ ran R \in V$
39	3 37 38	3syl	$⊢ φ \to ⋃ ran R \in V$
40		rnexg	$⊢ ⋃ ran R \in V \to ran ⋃ ran R \in V$
41		uniexg	$⊢ ran ⋃ ran R \in V \to ⋃ ran ⋃ ran R \in V$
42	39 40 41	3syl	$⊢ φ \to ⋃ ran ⋃ ran R \in V$
43		rnexg	$⊢ ⋃ ran ⋃ ran R \in V \to ran ⋃ ran ⋃ ran R \in V$
44		uniexg	$⊢ ran ⋃ ran ⋃ ran R \in V \to ⋃ ran ⋃ ran ⋃ ran R \in V$
45	42 43 44	3syl	$⊢ φ \to ⋃ ran ⋃ ran ⋃ ran R \in V$
46	45	pwexd	$⊢ φ \to 𝒫 ⋃ ran ⋃ ran ⋃ ran R \in V$
47	3	dmexd	$⊢ φ \to dom R \in V$
48	5 47	eqeltrrd	$⊢ φ \to I \in V$
49	46 48	elmapd	$⊢ φ \to ((x \in I ⟼ f (x) \cdot_{R (x)} g (x)) \in ({𝒫 ⋃ ran ⋃ ran ⋃ ran R}^{I}) \leftrightarrow (x \in I ⟼ f (x) \cdot_{R (x)} g (x)) : I ⟶ 𝒫 ⋃ ran ⋃ ran ⋃ ran R)$
50	36 49	mpbird	$⊢ φ \to (x \in I ⟼ f (x) \cdot_{R (x)} g (x)) \in ({𝒫 ⋃ ran ⋃ ran ⋃ ran R}^{I})$
51	50	ralrimivw	$⊢ φ \to \forall g \in B (x \in I ⟼ f (x) \cdot_{R (x)} g (x)) \in ({𝒫 ⋃ ran ⋃ ran ⋃ ran R}^{I})$
52	51	ralrimivw	$⊢ φ \to \forall f \in B \forall g \in B (x \in I ⟼ f (x) \cdot_{R (x)} g (x)) \in ({𝒫 ⋃ ran ⋃ ran ⋃ ran R}^{I})$
53		eqid	$⊢ (f \in B, g \in B ⟼ (x \in I ⟼ f (x) \cdot_{R (x)} g (x))) = (f \in B, g \in B ⟼ (x \in I ⟼ f (x) \cdot_{R (x)} g (x)))$
54	53	fmpo	$⊢ \forall f \in B \forall g \in B (x \in I ⟼ f (x) \cdot_{R (x)} g (x)) \in ({𝒫 ⋃ ran ⋃ ran ⋃ ran R}^{I}) \leftrightarrow (f \in B, g \in B ⟼ (x \in I ⟼ f (x) \cdot_{R (x)} g (x))) : B \times B ⟶ {𝒫 ⋃ ran ⋃ ran ⋃ ran R}^{I}$
55	52 54	sylib	$⊢ φ \to (f \in B, g \in B ⟼ (x \in I ⟼ f (x) \cdot_{R (x)} g (x))) : B \times B ⟶ {𝒫 ⋃ ran ⋃ ran ⋃ ran R}^{I}$
56	4	fvexi	$⊢ B \in V$
57	56 56	xpex	$⊢ B \times B \in V$
58		ovex	$⊢ {𝒫 ⋃ ran ⋃ ran ⋃ ran R}^{I} \in V$
59		fex2	$⊢ ((f \in B, g \in B ⟼ (x \in I ⟼ f (x) \cdot_{R (x)} g (x))) : B \times B ⟶ {𝒫 ⋃ ran ⋃ ran ⋃ ran R}^{I} \land B \times B \in V \land {𝒫 ⋃ ran ⋃ ran ⋃ ran R}^{I} \in V) \to (f \in B, g \in B ⟼ (x \in I ⟼ f (x) \cdot_{R (x)} g (x))) \in V$
60	57 58 59	mp3an23	$⊢ (f \in B, g \in B ⟼ (x \in I ⟼ f (x) \cdot_{R (x)} g (x))) : B \times B ⟶ {𝒫 ⋃ ran ⋃ ran ⋃ ran R}^{I} \to (f \in B, g \in B ⟼ (x \in I ⟼ f (x) \cdot_{R (x)} g (x))) \in V$
61	55 60	syl	$⊢ φ \to (f \in B, g \in B ⟼ (x \in I ⟼ f (x) \cdot_{R (x)} g (x))) \in V$
62		snsstp3	$⊢ \{⟨\cdot_{ndx}, (f \in B, g \in B ⟼ (x \in I ⟼ f (x) \cdot_{R (x)} g (x)))⟩\} \subseteq \{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, +_{P}⟩, ⟨\cdot_{ndx}, (f \in B, g \in B ⟼ (x \in I ⟼ f (x) \cdot_{R (x)} g (x)))⟩\}$
63		ssun1	$⊢ \{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, +_{P}⟩, ⟨\cdot_{ndx}, (f \in B, g \in B ⟼ (x \in I ⟼ f (x) \cdot_{R (x)} g (x)))⟩\} \subseteq \{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, +_{P}⟩, ⟨\cdot_{ndx}, (f \in B, g \in B ⟼ (x \in I ⟼ f (x) \cdot_{R (x)} g (x)))⟩\} \cup \{⟨Scalar (ndx), S⟩, ⟨\cdot_{ndx}, (f \in {Base}_{S}, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x)))⟩, ⟨\cdot_{𝑖} (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x)))⟩\}$
64	62 63	sstri	$⊢ \{⟨\cdot_{ndx}, (f \in B, g \in B ⟼ (x \in I ⟼ f (x) \cdot_{R (x)} g (x)))⟩\} \subseteq \{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, +_{P}⟩, ⟨\cdot_{ndx}, (f \in B, g \in B ⟼ (x \in I ⟼ f (x) \cdot_{R (x)} g (x)))⟩\} \cup \{⟨Scalar (ndx), S⟩, ⟨\cdot_{ndx}, (f \in {Base}_{S}, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x)))⟩, ⟨\cdot_{𝑖} (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x)))⟩\}$
65		ssun1	$⊢ \{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, +_{P}⟩, ⟨\cdot_{ndx}, (f \in B, g \in B ⟼ (x \in I ⟼ f (x) \cdot_{R (x)} g (x)))⟩\} \cup \{⟨Scalar (ndx), S⟩, ⟨\cdot_{ndx}, (f \in {Base}_{S}, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x)))⟩, ⟨\cdot_{𝑖} (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x)))⟩\} \subseteq (\{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, +_{P}⟩, ⟨\cdot_{ndx}, (f \in B, g \in B ⟼ (x \in I ⟼ f (x) \cdot_{R (x)} g (x)))⟩\} \cup \{⟨Scalar (ndx), S⟩, ⟨\cdot_{ndx}, (f \in {Base}_{S}, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x)))⟩, ⟨\cdot_{𝑖} (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x)))⟩\}) \cup (\{⟨TopSet (ndx), \prod_{𝑡} (TopOpen \circ R)⟩, ⟨\leq_{ndx}, \{⟨f, g⟩ \| (\{f, g\} \subseteq B \land \forall x \in I f (x) \leq_{R (x)} g (x))\}⟩, ⟨dist (ndx), (f \in B, g \in B ⟼ sup (ran (x \in I ⟼ f (x) dist (R (x)) g (x)) \cup \{0\} ℝ^{*} <))⟩\} \cup \{⟨Hom (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x)))⟩, ⟨comp (ndx), (a \in (B \times B), c \in B ⟼ (d \in (2^{nd} (a) (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) c), e \in (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) (a) ⟼ (x \in I ⟼ d (x) (⟨1^{st} (a) (x), 2^{nd} (a) (x)⟩ comp (R (x)) c (x)) e (x))))⟩\})$
66	64 65	sstri	$⊢ \{⟨\cdot_{ndx}, (f \in B, g \in B ⟼ (x \in I ⟼ f (x) \cdot_{R (x)} g (x)))⟩\} \subseteq (\{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, +_{P}⟩, ⟨\cdot_{ndx}, (f \in B, g \in B ⟼ (x \in I ⟼ f (x) \cdot_{R (x)} g (x)))⟩\} \cup \{⟨Scalar (ndx), S⟩, ⟨\cdot_{ndx}, (f \in {Base}_{S}, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x)))⟩, ⟨\cdot_{𝑖} (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x)))⟩\}) \cup (\{⟨TopSet (ndx), \prod_{𝑡} (TopOpen \circ R)⟩, ⟨\leq_{ndx}, \{⟨f, g⟩ \| (\{f, g\} \subseteq B \land \forall x \in I f (x) \leq_{R (x)} g (x))\}⟩, ⟨dist (ndx), (f \in B, g \in B ⟼ sup (ran (x \in I ⟼ f (x) dist (R (x)) g (x)) \cup \{0\} ℝ^{*} <))⟩\} \cup \{⟨Hom (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x)))⟩, ⟨comp (ndx), (a \in (B \times B), c \in B ⟼ (d \in (2^{nd} (a) (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) c), e \in (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) (a) ⟼ (x \in I ⟼ d (x) (⟨1^{st} (a) (x), 2^{nd} (a) (x)⟩ comp (R (x)) c (x)) e (x))))⟩\})$
67	19 6 20 61 66	prdsbaslem	$⊢ φ \to \underset{˙}{\cdot} = (f \in B, g \in B ⟼ (x \in I ⟼ f (x) \cdot_{R (x)} g (x)))$

Metamath Proof Explorer

Theorem prdsmulr

Proof