prdsle

Description: Structure product weak ordering. (Contributed 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	$⊢ φ \to dom R = I$
	prdsle.l	$⊢ \underset{˙}{\leq} = \leq_{P}$
Assertion	prdsle	$⊢ φ \to \underset{˙}{\leq} = \{⟨f, g⟩ \| (\{f, g\} \subseteq B \land \forall x \in I f (x) \leq_{R (x)} g (x))\}$

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		prdsle.l	$⊢ \underset{˙}{\leq} = \leq_{P}$
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		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		eqid	$⊢ \cdot_{P} = \cdot_{P}$
14	1 2 3 4 5 7 13	prdsvsca	$⊢ φ \to \cdot_{P} = (f \in {Base}_{S}, g \in B ⟼ (x \in I ⟼ f \cdot_{R (x)} g (x)))$
15		eqidd	$⊢ φ \to (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x))) = (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x)))$
16		eqidd	$⊢ φ \to \prod_{𝑡} (TopOpen \circ R) = \prod_{𝑡} (TopOpen \circ R)$
17		eqidd	$⊢ φ \to \{⟨f, g⟩ \| (\{f, g\} \subseteq B \land \forall x \in I f (x) \leq_{R (x)} g (x))\} = \{⟨f, g⟩ \| (\{f, g\} \subseteq B \land \forall x \in I f (x) \leq_{R (x)} g (x))\}$
18		eqidd	$⊢ φ \to (f \in B, g \in B ⟼ sup (ran (x \in I ⟼ f (x) dist (R (x)) g (x)) \cup \{0\} ℝ^{} <)) = (f \in B, g \in B ⟼ sup (ran (x \in I ⟼ f (x) dist (R (x)) g (x)) \cup \{0\} ℝ^{} <))$
19		eqidd	$⊢ φ \to (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) = (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x)))$
20		eqidd	$⊢ φ \to (a \in (B \times B), c \in B ⟼ (d \in (2^{nd} (a) (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) c), e \in (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) (a) ⟼ (x \in I ⟼ d (x) (⟨1^{st} (a) (x), 2^{nd} (a) (x)⟩ comp (R (x)) c (x)) e (x)))) = (a \in (B \times B), c \in B ⟼ (d \in (2^{nd} (a) (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) c), e \in (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) (a) ⟼ (x \in I ⟼ d (x) (⟨1^{st} (a) (x), 2^{nd} (a) (x)⟩ comp (R (x)) c (x)) e (x))))$
21	1 7 5 8 10 12 14 15 16 17 18 19 20 2 3	prdsval	$⊢ φ \to P = (\{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, +_{P}⟩, ⟨\cdot_{ndx}, \cdot_{P}⟩\} \cup \{⟨Scalar (ndx), S⟩, ⟨\cdot_{ndx}, \cdot_{P}⟩, ⟨\cdot_{𝑖} (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x)))⟩\}) \cup (\{⟨TopSet (ndx), \prod_{𝑡} (TopOpen \circ R)⟩, ⟨\leq_{ndx}, \{⟨f, g⟩ \| (\{f, g\} \subseteq B \land \forall x \in I f (x) \leq_{R (x)} g (x))\}⟩, ⟨dist (ndx), (f \in B, g \in B ⟼ sup (ran (x \in I ⟼ f (x) dist (R (x)) g (x)) \cup \{0\} ℝ^{*} <))⟩\} \cup \{⟨Hom (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x)))⟩, ⟨comp (ndx), (a \in (B \times B), c \in B ⟼ (d \in (2^{nd} (a) (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) c), e \in (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) (a) ⟼ (x \in I ⟼ d (x) (⟨1^{st} (a) (x), 2^{nd} (a) (x)⟩ comp (R (x)) c (x)) e (x))))⟩\})$
22		pleid	$⊢ le = Slot \leq_{ndx}$
23	4	fvexi	$⊢ B \in V$
24	23 23	xpex	$⊢ B \times B \in V$
25		vex	$⊢ f \in V$
26		vex	$⊢ g \in V$
27	25 26	prss	$⊢ (f \in B \land g \in B) \leftrightarrow \{f, g\} \subseteq B$
28	27	anbi1i	$⊢ ((f \in B \land g \in B) \land \forall x \in I f (x) \leq_{R (x)} g (x)) \leftrightarrow (\{f, g\} \subseteq B \land \forall x \in I f (x) \leq_{R (x)} g (x))$
29	28	opabbii	$⊢ \{⟨f, g⟩ \| ((f \in B \land g \in B) \land \forall x \in I f (x) \leq_{R (x)} g (x))\} = \{⟨f, g⟩ \| (\{f, g\} \subseteq B \land \forall x \in I f (x) \leq_{R (x)} g (x))\}$
30		opabssxp	$⊢ \{⟨f, g⟩ \| ((f \in B \land g \in B) \land \forall x \in I f (x) \leq_{R (x)} g (x))\} \subseteq B \times B$
31	29 30	eqsstrri	$⊢ \{⟨f, g⟩ \| (\{f, g\} \subseteq B \land \forall x \in I f (x) \leq_{R (x)} g (x))\} \subseteq B \times B$
32	24 31	ssexi	$⊢ \{⟨f, g⟩ \| (\{f, g\} \subseteq B \land \forall x \in I f (x) \leq_{R (x)} g (x))\} \in V$
33	32	a1i	$⊢ φ \to \{⟨f, g⟩ \| (\{f, g\} \subseteq B \land \forall x \in I f (x) \leq_{R (x)} g (x))\} \in V$
34		snsstp2	$⊢ \{⟨\leq_{ndx}, \{⟨f, g⟩ \| (\{f, g\} \subseteq B \land \forall x \in I f (x) \leq_{R (x)} g (x))\}⟩\} \subseteq \{⟨TopSet (ndx), \prod_{𝑡} (TopOpen \circ R)⟩, ⟨\leq_{ndx}, \{⟨f, g⟩ \| (\{f, g\} \subseteq B \land \forall x \in I f (x) \leq_{R (x)} g (x))\}⟩, ⟨dist (ndx), (f \in B, g \in B ⟼ sup (ran (x \in I ⟼ f (x) dist (R (x)) g (x)) \cup \{0\} ℝ^{*} <))⟩\}$
35		ssun1	$⊢ \{⟨TopSet (ndx), \prod_{𝑡} (TopOpen \circ R)⟩, ⟨\leq_{ndx}, \{⟨f, g⟩ \| (\{f, g\} \subseteq B \land \forall x \in I f (x) \leq_{R (x)} g (x))\}⟩, ⟨dist (ndx), (f \in B, g \in B ⟼ sup (ran (x \in I ⟼ f (x) dist (R (x)) g (x)) \cup \{0\} ℝ^{} <))⟩\} \subseteq \{⟨TopSet (ndx), \prod_{𝑡} (TopOpen \circ R)⟩, ⟨\leq_{ndx}, \{⟨f, g⟩ \| (\{f, g\} \subseteq B \land \forall x \in I f (x) \leq_{R (x)} g (x))\}⟩, ⟨dist (ndx), (f \in B, g \in B ⟼ sup (ran (x \in I ⟼ f (x) dist (R (x)) g (x)) \cup \{0\} ℝ^{} <))⟩\} \cup \{⟨Hom (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x)))⟩, ⟨comp (ndx), (a \in (B \times B), c \in B ⟼ (d \in (2^{nd} (a) (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) c), e \in (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) (a) ⟼ (x \in I ⟼ d (x) (⟨1^{st} (a) (x), 2^{nd} (a) (x)⟩ comp (R (x)) c (x)) e (x))))⟩\}$
36	34 35	sstri	$⊢ \{⟨\leq_{ndx}, \{⟨f, g⟩ \| (\{f, g\} \subseteq B \land \forall x \in I f (x) \leq_{R (x)} g (x))\}⟩\} \subseteq \{⟨TopSet (ndx), \prod_{𝑡} (TopOpen \circ R)⟩, ⟨\leq_{ndx}, \{⟨f, g⟩ \| (\{f, g\} \subseteq B \land \forall x \in I f (x) \leq_{R (x)} g (x))\}⟩, ⟨dist (ndx), (f \in B, g \in B ⟼ sup (ran (x \in I ⟼ f (x) dist (R (x)) g (x)) \cup \{0\} ℝ^{*} <))⟩\} \cup \{⟨Hom (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x)))⟩, ⟨comp (ndx), (a \in (B \times B), c \in B ⟼ (d \in (2^{nd} (a) (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) c), e \in (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) (a) ⟼ (x \in I ⟼ d (x) (⟨1^{st} (a) (x), 2^{nd} (a) (x)⟩ comp (R (x)) c (x)) e (x))))⟩\}$
37		ssun2	$⊢ \{⟨TopSet (ndx), \prod_{𝑡} (TopOpen \circ R)⟩, ⟨\leq_{ndx}, \{⟨f, g⟩ \| (\{f, g\} \subseteq B \land \forall x \in I f (x) \leq_{R (x)} g (x))\}⟩, ⟨dist (ndx), (f \in B, g \in B ⟼ sup (ran (x \in I ⟼ f (x) dist (R (x)) g (x)) \cup \{0\} ℝ^{} <))⟩\} \cup \{⟨Hom (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x)))⟩, ⟨comp (ndx), (a \in (B \times B), c \in B ⟼ (d \in (2^{nd} (a) (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) c), e \in (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) (a) ⟼ (x \in I ⟼ d (x) (⟨1^{st} (a) (x), 2^{nd} (a) (x)⟩ comp (R (x)) c (x)) e (x))))⟩\} \subseteq (\{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, +_{P}⟩, ⟨\cdot_{ndx}, \cdot_{P}⟩\} \cup \{⟨Scalar (ndx), S⟩, ⟨\cdot_{ndx}, \cdot_{P}⟩, ⟨\cdot_{𝑖} (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x)))⟩\}) \cup (\{⟨TopSet (ndx), \prod_{𝑡} (TopOpen \circ R)⟩, ⟨\leq_{ndx}, \{⟨f, g⟩ \| (\{f, g\} \subseteq B \land \forall x \in I f (x) \leq_{R (x)} g (x))\}⟩, ⟨dist (ndx), (f \in B, g \in B ⟼ sup (ran (x \in I ⟼ f (x) dist (R (x)) g (x)) \cup \{0\} ℝ^{} <))⟩\} \cup \{⟨Hom (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x)))⟩, ⟨comp (ndx), (a \in (B \times B), c \in B ⟼ (d \in (2^{nd} (a) (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) c), e \in (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) (a) ⟼ (x \in I ⟼ d (x) (⟨1^{st} (a) (x), 2^{nd} (a) (x)⟩ comp (R (x)) c (x)) e (x))))⟩\})$
38	36 37	sstri	$⊢ \{⟨\leq_{ndx}, \{⟨f, g⟩ \| (\{f, g\} \subseteq B \land \forall x \in I f (x) \leq_{R (x)} g (x))\}⟩\} \subseteq (\{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, +_{P}⟩, ⟨\cdot_{ndx}, \cdot_{P}⟩\} \cup \{⟨Scalar (ndx), S⟩, ⟨\cdot_{ndx}, \cdot_{P}⟩, ⟨\cdot_{𝑖} (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{\sum_{S}} (f (x) \cdot_{𝑖} (R (x)) g (x)))⟩\}) \cup (\{⟨TopSet (ndx), \prod_{𝑡} (TopOpen \circ R)⟩, ⟨\leq_{ndx}, \{⟨f, g⟩ \| (\{f, g\} \subseteq B \land \forall x \in I f (x) \leq_{R (x)} g (x))\}⟩, ⟨dist (ndx), (f \in B, g \in B ⟼ sup (ran (x \in I ⟼ f (x) dist (R (x)) g (x)) \cup \{0\} ℝ^{*} <))⟩\} \cup \{⟨Hom (ndx), (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x)))⟩, ⟨comp (ndx), (a \in (B \times B), c \in B ⟼ (d \in (2^{nd} (a) (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) c), e \in (f \in B, g \in B ⟼ \underset{x \in I}{⨉} (f (x) Hom (R (x)) g (x))) (a) ⟼ (x \in I ⟼ d (x) (⟨1^{st} (a) (x), 2^{nd} (a) (x)⟩ comp (R (x)) c (x)) e (x))))⟩\})$
39	21 6 22 33 38	prdsbaslem	$⊢ φ \to \underset{˙}{\leq} = \{⟨f, g⟩ \| (\{f, g\} \subseteq B \land \forall x \in I f (x) \leq_{R (x)} g (x))\}$

Metamath Proof Explorer

Theorem prdsle

Proof