xpstps

Description: A binary product of topologies is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015)

		Ref	Expression
	Hypothesis	xpstps.t	$⊢ T = R \times_{𝑠} S$
	Assertion	xpstps	$⊢ (R \in TopSp \land S \in TopSp) \to T \in TopSp$

Proof

Step	Hyp	Ref	Expression
1		xpstps.t	$⊢ T = R \times_{𝑠} S$
2		eqid	$⊢ {Base}_{R} = {Base}_{R}$
3		eqid	$⊢ {Base}_{S} = {Base}_{S}$
4		simpl	$⊢ (R \in TopSp \land S \in TopSp) \to R \in TopSp$
5		simpr	$⊢ (R \in TopSp \land S \in TopSp) \to S \in TopSp$
6		eqid	$⊢ (x \in {Base}_{R}, y \in {Base}_{S} ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\}) = (x \in {Base}_{R}, y \in {Base}_{S} ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\})$
7		eqid	$⊢ Scalar (R) = Scalar (R)$
8		eqid	$⊢ Scalar (R) ⨉_{𝑠} \{⟨\emptyset, R⟩, ⟨1_{𝑜}, S⟩\} = Scalar (R) ⨉_{𝑠} \{⟨\emptyset, R⟩, ⟨1_{𝑜}, S⟩\}$
9	1 2 3 4 5 6 7 8	xpsval	$⊢ (R \in TopSp \land S \in TopSp) \to T = {(x \in {Base}_{R}, y \in {Base}_{S} ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\})}^{-1} “_{𝑠} (Scalar (R) ⨉_{𝑠} \{⟨\emptyset, R⟩, ⟨1_{𝑜}, S⟩\})$
10	1 2 3 4 5 6 7 8	xpsrnbas	$⊢ (R \in TopSp \land S \in TopSp) \to ran (x \in {Base}_{R}, y \in {Base}_{S} ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\}) = {Base}_{(Scalar (R) ⨉_{𝑠} \{⟨\emptyset, R⟩, ⟨1_{𝑜}, S⟩\})}$
11	6	xpsff1o2	$⊢ (x \in {Base}_{R}, y \in {Base}_{S} ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\}) : {Base}_{R} \times {Base}_{S} \underset{1-1 onto}{⟶} ran (x \in {Base}_{R}, y \in {Base}_{S} ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\})$
12	11	a1i	$⊢ (R \in TopSp \land S \in TopSp) \to (x \in {Base}_{R}, y \in {Base}_{S} ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\}) : {Base}_{R} \times {Base}_{S} \underset{1-1 onto}{⟶} ran (x \in {Base}_{R}, y \in {Base}_{S} ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\})$
13		f1ocnv	$⊢ (x \in {Base}_{R}, y \in {Base}_{S} ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\}) : {Base}_{R} \times {Base}_{S} \underset{1-1 onto}{⟶} ran (x \in {Base}_{R}, y \in {Base}_{S} ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\}) \to {(x \in {Base}_{R}, y \in {Base}_{S} ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\})}^{-1} : ran (x \in {Base}_{R}, y \in {Base}_{S} ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\}) \underset{1-1 onto}{⟶} {Base}_{R} \times {Base}_{S}$
14		f1ofo	$⊢ {(x \in {Base}_{R}, y \in {Base}_{S} ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\})}^{-1} : ran (x \in {Base}_{R}, y \in {Base}_{S} ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\}) \underset{1-1 onto}{⟶} {Base}_{R} \times {Base}_{S} \to {(x \in {Base}_{R}, y \in {Base}_{S} ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\})}^{-1} : ran (x \in {Base}_{R}, y \in {Base}_{S} ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\}) \underset{onto}{⟶} {Base}_{R} \times {Base}_{S}$
15	12 13 14	3syl	$⊢ (R \in TopSp \land S \in TopSp) \to {(x \in {Base}_{R}, y \in {Base}_{S} ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\})}^{-1} : ran (x \in {Base}_{R}, y \in {Base}_{S} ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\}) \underset{onto}{⟶} {Base}_{R} \times {Base}_{S}$
16		fvexd	$⊢ (R \in TopSp \land S \in TopSp) \to Scalar (R) \in V$
17		2on	$⊢ 2_{𝑜} \in On$
18	17	a1i	$⊢ (R \in TopSp \land S \in TopSp) \to 2_{𝑜} \in On$
19		xpscf	$⊢ \{⟨\emptyset, R⟩, ⟨1_{𝑜}, S⟩\} : 2_{𝑜} ⟶ TopSp \leftrightarrow (R \in TopSp \land S \in TopSp)$
20	19	biimpri	$⊢ (R \in TopSp \land S \in TopSp) \to \{⟨\emptyset, R⟩, ⟨1_{𝑜}, S⟩\} : 2_{𝑜} ⟶ TopSp$
21	8 16 18 20	prdstps	$⊢ (R \in TopSp \land S \in TopSp) \to Scalar (R) ⨉_{𝑠} \{⟨\emptyset, R⟩, ⟨1_{𝑜}, S⟩\} \in TopSp$
22	9 10 15 21	imastps	$⊢ (R \in TopSp \land S \in TopSp) \to T \in TopSp$

Metamath Proof Explorer

Theorem xpstps

Proof