prdsip

Description: Inner product in a structure product. (Contributed 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	$⊢ φ \to dom R = I$
	prdsip.m	$⊢ \underset{˙}{,} = \cdot_{𝑖} (P)$
Assertion	prdsip	$⊢ φ \to \underset{˙}{,} = (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x)))$

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		prdsip.m	$⊢ \underset{˙}{,} = \cdot_{𝑖} (P)$
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		ipid	$⊢ \cdot_{𝑖} = Slot \cdot_{𝑖} (ndx)$
21	4	fvexi	$⊢ B \in V$
22	21	a1i	$⊢ φ \to B \in V$
23		mpoexga	$⊢ (B \in V \land B \in V) \to (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x))) \in V$
24	22 21 23	sylancl	$⊢ φ \to (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x))) \in V$
25		snsstp3	$⊢ \{⟨\cdot_{𝑖} (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x)))⟩\} \subseteq \{⟨Scalar (ndx), S⟩, ⟨\cdot_{ndx}, (f \in {Base}_{S}, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x)))⟩, ⟨\cdot_{𝑖} (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x)))⟩\}$
26		ssun2	$⊢ \{⟨Scalar (ndx), S⟩, ⟨\cdot_{ndx}, (f \in {Base}_{S}, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x)))⟩, ⟨\cdot_{𝑖} (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x)))⟩\} \subseteq \{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, +_{P}⟩, ⟨\cdot_{ndx}, (f \in B, g \in B ⟼ (x \in I ⟼ f (x) \cdot_{R (x)} g (x)))⟩\} \cup \{⟨Scalar (ndx), S⟩, ⟨\cdot_{ndx}, (f \in {Base}_{S}, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x)))⟩, ⟨\cdot_{𝑖} (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x)))⟩\}$
27	25 26	sstri	$⊢ \{⟨\cdot_{𝑖} (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x)))⟩\} \subseteq \{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, +_{P}⟩, ⟨\cdot_{ndx}, (f \in B, g \in B ⟼ (x \in I ⟼ f (x) \cdot_{R (x)} g (x)))⟩\} \cup \{⟨Scalar (ndx), S⟩, ⟨\cdot_{ndx}, (f \in {Base}_{S}, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x)))⟩, ⟨\cdot_{𝑖} (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x)))⟩\}$
28		ssun1	$⊢ \{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, +_{P}⟩, ⟨\cdot_{ndx}, (f \in B, g \in B ⟼ (x \in I ⟼ f (x) \cdot_{R (x)} g (x)))⟩\} \cup \{⟨Scalar (ndx), S⟩, ⟨\cdot_{ndx}, (f \in {Base}_{S}, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x)))⟩, ⟨\cdot_{𝑖} (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x)))⟩\} \subseteq (\{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, +_{P}⟩, ⟨\cdot_{ndx}, (f \in B, g \in B ⟼ (x \in I ⟼ f (x) \cdot_{R (x)} g (x)))⟩\} \cup \{⟨Scalar (ndx), S⟩, ⟨\cdot_{ndx}, (f \in {Base}_{S}, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x)))⟩, ⟨\cdot_{𝑖} (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x)))⟩\}) \cup (\{⟨TopSet (ndx), \prod_{𝑡} (TopOpen \circ R)⟩, ⟨\leq_{ndx}, \{⟨f, g⟩ \| (\{f, g\} \subseteq B \land \forall x \in I f (x) \leq_{R (x)} g (x))\}⟩, ⟨dist (ndx), (f \in B, g \in B ⟼ sup (ran (x \in I ⟼ f (x) dist (R (x)) g (x)) \cup \{0\} ℝ^{*} <))⟩\} \cup \{⟨Hom (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x)))⟩, ⟨comp (ndx), (a \in (B \times B), c \in B ⟼ (d \in (2^{nd} (a) (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) c), e \in (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) (a) ⟼ (x \in I ⟼ d (x) (⟨1^{st} (a) (x), 2^{nd} (a) (x)⟩ comp (R (x)) c (x)) e (x))))⟩\})$
29	27 28	sstri	$⊢ \{⟨\cdot_{𝑖} (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x)))⟩\} \subseteq (\{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, +_{P}⟩, ⟨\cdot_{ndx}, (f \in B, g \in B ⟼ (x \in I ⟼ f (x) \cdot_{R (x)} g (x)))⟩\} \cup \{⟨Scalar (ndx), S⟩, ⟨\cdot_{ndx}, (f \in {Base}_{S}, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x)))⟩, ⟨\cdot_{𝑖} (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x)))⟩\}) \cup (\{⟨TopSet (ndx), \prod_{𝑡} (TopOpen \circ R)⟩, ⟨\leq_{ndx}, \{⟨f, g⟩ \| (\{f, g\} \subseteq B \land \forall x \in I f (x) \leq_{R (x)} g (x))\}⟩, ⟨dist (ndx), (f \in B, g \in B ⟼ sup (ran (x \in I ⟼ f (x) dist (R (x)) g (x)) \cup \{0\} ℝ^{*} <))⟩\} \cup \{⟨Hom (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x)))⟩, ⟨comp (ndx), (a \in (B \times B), c \in B ⟼ (d \in (2^{nd} (a) (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) c), e \in (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) (a) ⟼ (x \in I ⟼ d (x) (⟨1^{st} (a) (x), 2^{nd} (a) (x)⟩ comp (R (x)) c (x)) e (x))))⟩\})$
30	19 6 20 24 29	prdsvallem	$⊢ φ \to \underset{˙}{,} = (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x)))$

Metamath Proof Explorer

Theorem prdsip

Proof