txpconn - Mathbox for Mario Carneiro

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Theorem txpconn 36066

Description: The topological product of two path-connected spaces is path-connected. (Contributed by Mario Carneiro, 12-Feb-2015.)

**Assertion**
Ref	Expression
txpconn	⊢ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (𝑅 ×_t 𝑆) ∈ PConn)

Proof of Theorem txpconn Dummy variables 𝑓 𝑥 𝑦 𝑔 ℎ 𝑡 𝑢 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pconntop 36059	. . 3 ⊢ (𝑅 ∈ PConn → 𝑅 ∈ Top)
2		pconntop 36059	. . 3 ⊢ (𝑆 ∈ PConn → 𝑆 ∈ Top)
3		txtop 23889	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×_t 𝑆) ∈ Top)
4	1, 2, 3	syl2an 608	. 2 ⊢ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (𝑅 ×_t 𝑆) ∈ Top)
5		an6 1474	. . . . . . . . . 10 ⊢ (((𝑅 ∈ PConn ∧ 𝑥 ∈ ∪ 𝑅 ∧ 𝑧 ∈ ∪ 𝑅) ∧ (𝑆 ∈ PConn ∧ 𝑦 ∈ ∪ 𝑆 ∧ 𝑤 ∈ ∪ 𝑆)) ↔ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)))
6		eqid 2760	. . . . . . . . . . . 12 ⊢ ∪ 𝑅 = ∪ 𝑅
7	6	pconncn 36058	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ PConn ∧ 𝑥 ∈ ∪ 𝑅 ∧ 𝑧 ∈ ∪ 𝑅) → ∃𝑔 ∈ (II Cn 𝑅)((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧))
8		eqid 2760	. . . . . . . . . . . 12 ⊢ ∪ 𝑆 = ∪ 𝑆
9	8	pconncn 36058	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ PConn ∧ 𝑦 ∈ ∪ 𝑆 ∧ 𝑤 ∈ ∪ 𝑆) → ∃ℎ ∈ (II Cn 𝑆)((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤))
10	7, 9	anim12i 625	. . . . . . . . . 10 ⊢ (((𝑅 ∈ PConn ∧ 𝑥 ∈ ∪ 𝑅 ∧ 𝑧 ∈ ∪ 𝑅) ∧ (𝑆 ∈ PConn ∧ 𝑦 ∈ ∪ 𝑆 ∧ 𝑤 ∈ ∪ 𝑆)) → (∃𝑔 ∈ (II Cn 𝑅)((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ∃ℎ ∈ (II Cn 𝑆)((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)))
11	5, 10	sylbir 238	. . . . . . . . 9 ⊢ (((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) → (∃𝑔 ∈ (II Cn 𝑅)((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ∃ℎ ∈ (II Cn 𝑆)((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)))
12		reeanv 3234	. . . . . . . . 9 ⊢ (∃𝑔 ∈ (II Cn 𝑅)∃ℎ ∈ (II Cn 𝑆)(((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)) ↔ (∃𝑔 ∈ (II Cn 𝑅)((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ∃ℎ ∈ (II Cn 𝑆)((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)))
13	11, 12	sylibr 237	. . . . . . . 8 ⊢ (((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) → ∃𝑔 ∈ (II Cn 𝑅)∃ℎ ∈ (II Cn 𝑆)(((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)))
14		iiuni 25203	. . . . . . . . . . . . 13 ⊢ (0[,]1) = ∪ II
15		eqid 2760	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩) = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)
16	14, 15	txcnmpt 23944	. . . . . . . . . . . 12 ⊢ ((𝑔 ∈ (II Cn 𝑅) ∧ ℎ ∈ (II Cn 𝑆)) → (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩) ∈ (II Cn (𝑅 ×_t 𝑆)))
17	16	ad2antrl 741	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ℎ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)))) → (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩) ∈ (II Cn (𝑅 ×_t 𝑆)))
18		0elunit 13601	. . . . . . . . . . . . 13 ⊢ 0 ∈ (0[,]1)
19		fveq2 6885	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 0 → (𝑔‘𝑡) = (𝑔‘0))
20		fveq2 6885	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 0 → (ℎ‘𝑡) = (ℎ‘0))
21	19, 20	opeq12d 4840	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 0 → ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩ = ⟨(𝑔‘0), (ℎ‘0)⟩)
22		opex 5431	. . . . . . . . . . . . . 14 ⊢ ⟨(𝑔‘0), (ℎ‘0)⟩ ∈ V
23	21, 15, 22	fvmpt 6993	. . . . . . . . . . . . 13 ⊢ (0 ∈ (0[,]1) → ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)‘0) = ⟨(𝑔‘0), (ℎ‘0)⟩)
24	18, 23	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)‘0) = ⟨(𝑔‘0), (ℎ‘0)⟩
25		simprrl 793	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ℎ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)))) → ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧))
26	25	simpld 500	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ℎ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)))) → (𝑔‘0) = 𝑥)
27		simprrr 794	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ℎ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)))) → ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤))
28	27	simpld 500	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ℎ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)))) → (ℎ‘0) = 𝑦)
29	26, 28	opeq12d 4840	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ℎ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)))) → ⟨(𝑔‘0), (ℎ‘0)⟩ = ⟨𝑥, 𝑦⟩)
30	24, 29	eqtrid 2807	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ℎ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)))) → ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)‘0) = ⟨𝑥, 𝑦⟩)
31		1elunit 13602	. . . . . . . . . . . . 13 ⊢ 1 ∈ (0[,]1)
32		fveq2 6885	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 1 → (𝑔‘𝑡) = (𝑔‘1))
33		fveq2 6885	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 1 → (ℎ‘𝑡) = (ℎ‘1))
34	32, 33	opeq12d 4840	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 1 → ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩ = ⟨(𝑔‘1), (ℎ‘1)⟩)
35		opex 5431	. . . . . . . . . . . . . 14 ⊢ ⟨(𝑔‘1), (ℎ‘1)⟩ ∈ V
36	34, 15, 35	fvmpt 6993	. . . . . . . . . . . . 13 ⊢ (1 ∈ (0[,]1) → ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)‘1) = ⟨(𝑔‘1), (ℎ‘1)⟩)
37	31, 36	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)‘1) = ⟨(𝑔‘1), (ℎ‘1)⟩
38	25	simprd 501	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ℎ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)))) → (𝑔‘1) = 𝑧)
39	27	simprd 501	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ℎ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)))) → (ℎ‘1) = 𝑤)
40	38, 39	opeq12d 4840	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ℎ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)))) → ⟨(𝑔‘1), (ℎ‘1)⟩ = ⟨𝑧, 𝑤⟩)
41	37, 40	eqtrid 2807	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ℎ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)))) → ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)‘1) = ⟨𝑧, 𝑤⟩)
42		fveq1 6884	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩) → (𝑓‘0) = ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)‘0))
43	42	eqeq1d 2762	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩) → ((𝑓‘0) = ⟨𝑥, 𝑦⟩ ↔ ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)‘0) = ⟨𝑥, 𝑦⟩))
44		fveq1 6884	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩) → (𝑓‘1) = ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)‘1))
45	44	eqeq1d 2762	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩) → ((𝑓‘1) = ⟨𝑧, 𝑤⟩ ↔ ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)‘1) = ⟨𝑧, 𝑤⟩))
46	43, 45	anbi12d 644	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩) → (((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩) ↔ (((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)‘0) = ⟨𝑥, 𝑦⟩ ∧ ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)‘1) = ⟨𝑧, 𝑤⟩)))
47	46	rspcev 3576	. . . . . . . . . . 11 ⊢ (((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩) ∈ (II Cn (𝑅 ×_t 𝑆)) ∧ (((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)‘0) = ⟨𝑥, 𝑦⟩ ∧ ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔‘𝑡), (ℎ‘𝑡)⟩)‘1) = ⟨𝑧, 𝑤⟩)) → ∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
48	17, 30, 41, 47	syl12anc 850	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ℎ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)))) → ∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
49	48	expr 462	. . . . . . . . 9 ⊢ ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) ∧ (𝑔 ∈ (II Cn 𝑅) ∧ ℎ ∈ (II Cn 𝑆))) → ((((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)) → ∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
50	49	rexlimdvva 3219	. . . . . . . 8 ⊢ (((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) → (∃𝑔 ∈ (II Cn 𝑅)∃ℎ ∈ (II Cn 𝑆)(((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((ℎ‘0) = 𝑦 ∧ (ℎ‘1) = 𝑤)) → ∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
51	13, 50	mpd 16	. . . . . . 7 ⊢ (((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) → ∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
52	51	3expa 1136	. . . . . 6 ⊢ ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆)) ∧ (𝑧 ∈ ∪ 𝑅 ∧ 𝑤 ∈ ∪ 𝑆)) → ∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
53	52	ralrimivva 3205	. . . . 5 ⊢ (((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 ∈ ∪ 𝑅 ∧ 𝑦 ∈ ∪ 𝑆)) → ∀𝑧 ∈ ∪ 𝑅∀𝑤 ∈ ∪ 𝑆∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
54	53	ralrimivva 3205	. . . 4 ⊢ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → ∀𝑥 ∈ ∪ 𝑅∀𝑦 ∈ ∪ 𝑆∀𝑧 ∈ ∪ 𝑅∀𝑤 ∈ ∪ 𝑆∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
55		eqeq2 2772	. . . . . . . . 9 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → ((𝑓‘1) = 𝑣 ↔ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
56	55	anbi2d 642	. . . . . . . 8 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣) ↔ ((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
57	56	rexbidv 3186	. . . . . . 7 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣) ↔ ∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
58	57	ralxp 5817	. . . . . 6 ⊢ (∀𝑣 ∈ (∪ 𝑅 × ∪ 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣) ↔ ∀𝑧 ∈ ∪ 𝑅∀𝑤 ∈ ∪ 𝑆∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
59		eqeq2 2772	. . . . . . . . 9 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → ((𝑓‘0) = 𝑢 ↔ (𝑓‘0) = ⟨𝑥, 𝑦⟩))
60	59	anbi1d 643	. . . . . . . 8 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩) ↔ ((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
61	60	rexbidv 3186	. . . . . . 7 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩) ↔ ∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
62	61	2ralbidv 3226	. . . . . 6 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (∀𝑧 ∈ ∪ 𝑅∀𝑤 ∈ ∪ 𝑆∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩) ↔ ∀𝑧 ∈ ∪ 𝑅∀𝑤 ∈ ∪ 𝑆∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
63	58, 62	bitrid 286	. . . . 5 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (∀𝑣 ∈ (∪ 𝑅 × ∪ 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣) ↔ ∀𝑧 ∈ ∪ 𝑅∀𝑤 ∈ ∪ 𝑆∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
64	63	ralxp 5817	. . . 4 ⊢ (∀𝑢 ∈ (∪ 𝑅 × ∪ 𝑆)∀𝑣 ∈ (∪ 𝑅 × ∪ 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣) ↔ ∀𝑥 ∈ ∪ 𝑅∀𝑦 ∈ ∪ 𝑆∀𝑧 ∈ ∪ 𝑅∀𝑤 ∈ ∪ 𝑆∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
65	54, 64	sylibr 237	. . 3 ⊢ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → ∀𝑢 ∈ (∪ 𝑅 × ∪ 𝑆)∀𝑣 ∈ (∪ 𝑅 × ∪ 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣))
66	6, 8	txuni 23912	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×_t 𝑆))
67	1, 2, 66	syl2an 608	. . . 4 ⊢ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×_t 𝑆))
68	67	raleqdv 3319	. . . 4 ⊢ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (∀𝑣 ∈ (∪ 𝑅 × ∪ 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣) ↔ ∀𝑣 ∈ ∪ (𝑅 ×_t 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣)))
69	67, 68	raleqbidv 3334	. . 3 ⊢ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (∀𝑢 ∈ (∪ 𝑅 × ∪ 𝑆)∀𝑣 ∈ (∪ 𝑅 × ∪ 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣) ↔ ∀𝑢 ∈ ∪ (𝑅 ×_t 𝑆)∀𝑣 ∈ ∪ (𝑅 ×_t 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣)))
70	65, 69	mpbid 235	. 2 ⊢ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → ∀𝑢 ∈ ∪ (𝑅 ×_t 𝑆)∀𝑣 ∈ ∪ (𝑅 ×_t 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣))
71		eqid 2760	. . 3 ⊢ ∪ (𝑅 ×_t 𝑆) = ∪ (𝑅 ×_t 𝑆)
72	71	ispconn 36057	. 2 ⊢ ((𝑅 ×_t 𝑆) ∈ PConn ↔ ((𝑅 ×_t 𝑆) ∈ Top ∧ ∀𝑢 ∈ ∪ (𝑅 ×_t 𝑆)∀𝑣 ∈ ∪ (𝑅 ×_t 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×_t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣)))
73	4, 70, 72	sylanbrc 595	1 ⊢ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (𝑅 ×_t 𝑆) ∈ PConn)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ∀wral 3076 ∃wrex 3086 ⟨cop 4589 ∪ cuni 4866 ↦ cmpt 5185 × cxp 5648 ‘cfv 6538 (class class class)co 7420 0cc0 11201 1c1 11202 [,]cicc 13480 Topctop 23212 Cn ccn 23543 ×_t ctx 23880 IIcii 25197 PConncpconn 36053

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11257 ax-resscn 11258 ax-1cn 11259 ax-icn 11260 ax-addcl 11261 ax-addrcl 11262 ax-mulcl 11263 ax-mulrcl 11264 ax-mulcom 11265 ax-addass 11266 ax-mulass 11267 ax-distr 11268 ax-i2m1 11269 ax-1ne0 11270 ax-1rid 11271 ax-rnegex 11272 ax-rrecex 11273 ax-cnre 11274 ax-pre-lttri 11275 ax-pre-lttrn 11276 ax-pre-ltadd 11277 ax-pre-mulgt0 11278 ax-pre-sup 11279

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8300 df-wrecs 8331 df-recs 8380 df-rdg 8419 df-er 8718 df-map 8850 df-en 8975 df-dom 8976 df-sdom 8977 df-sup 9435 df-inf 9436 df-pnf 11346 df-mnf 11347 df-xr 11348 df-ltxr 11349 df-le 11350 df-sub 11544 df-neg 11545 df-div 11975 df-nn 12337 df-2 12406 df-3 12407 df-n0 12608 df-z 12695 df-uz 12967 df-q 13077 df-rp 13122 df-xneg 13242 df-xadd 13243 df-xmul 13244 df-icc 13484 df-seq 14146 df-exp 14206 df-cj 15267 df-re 15268 df-im 15269 df-sqrt 15403 df-abs 15404 df-topgen 17615 df-psmet 21671 df-xmet 21672 df-met 21673 df-bl 21674 df-mopn 21675 df-top 23213 df-topon 23230 df-bases 23265 df-cn 23546 df-tx 23882 df-ii 25199 df-pconn 36055

This theorem is used by: txsconn 36075

W3C validator