Theorem txsconn 33203

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

Assertion

Ref	Expression
txsconn	⊢ ((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) → (𝑅 ×t 𝑆) ∈ SConn)

Proof of Theorem txsconn

Dummy variables 𝑓 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sconnpconn 33189	. . 3 ⊢ (𝑅 ∈ SConn → 𝑅 ∈ PConn)
2		sconnpconn 33189	. . 3 ⊢ (𝑆 ∈ SConn → 𝑆 ∈ PConn)
3		txpconn 33194	. . 3 ⊢ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (𝑅 ×t 𝑆) ∈ PConn)
4	1, 2, 3	syl2an 596	. 2 ⊢ ((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) → (𝑅 ×t 𝑆) ∈ PConn)
5		simpll 764	. . . . . . . . 9 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑅 ∈ SConn)
6		simprl 768	. . . . . . . . . 10 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓 ∈ (II Cn (𝑅 ×t 𝑆)))
7		sconntop 33190	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ SConn → 𝑅 ∈ Top)
8	7	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑅 ∈ Top)
9		eqid 2738	. . . . . . . . . . . . 13 ⊢ ∪ 𝑅 = ∪ 𝑅
10	9	toptopon 22066	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘∪ 𝑅))
11	8, 10	sylib 217	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑅 ∈ (TopOn‘∪ 𝑅))
12		sconntop 33190	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ SConn → 𝑆 ∈ Top)
13	12	ad2antlr 724	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑆 ∈ Top)
14		eqid 2738	. . . . . . . . . . . . 13 ⊢ ∪ 𝑆 = ∪ 𝑆
15	14	toptopon 22066	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘∪ 𝑆))
16	13, 15	sylib 217	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑆 ∈ (TopOn‘∪ 𝑆))
17		tx1cn 22760	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ (TopOn‘∪ 𝑅) ∧ 𝑆 ∈ (TopOn‘∪ 𝑆)) → (1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
18	11, 16, 17	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
19		cnco 22417	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅)) → ((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑅))
20	6, 18, 19	syl2anc 584	. . . . . . . . 9 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑅))
21		simprr 770	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑓‘0) = (𝑓‘1))
22	21	fveq2d 6778	. . . . . . . . . 10 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝑓‘0)) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝑓‘1)))
23		iitopon 24042	. . . . . . . . . . . . 13 ⊢ II ∈ (TopOn‘(0[,]1))
24	23	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → II ∈ (TopOn‘(0[,]1)))
25		txtopon 22742	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ (TopOn‘∪ 𝑅) ∧ 𝑆 ∈ (TopOn‘∪ 𝑆)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(∪ 𝑅 × ∪ 𝑆)))
26	11, 16, 25	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑅 ×t 𝑆) ∈ (TopOn‘(∪ 𝑅 × ∪ 𝑆)))
27		cnf2 22400	. . . . . . . . . . . 12 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ (𝑅 ×t 𝑆) ∈ (TopOn‘(∪ 𝑅 × ∪ 𝑆)) ∧ 𝑓 ∈ (II Cn (𝑅 ×t 𝑆))) → 𝑓:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆))
28	24, 26, 6, 27	syl3anc 1370	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆))
29		0elunit 13201	. . . . . . . . . . 11 ⊢ 0 ∈ (0[,]1)
30		fvco3 6867	. . . . . . . . . . 11 ⊢ ((𝑓:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 0 ∈ (0[,]1)) → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝑓‘0)))
31	28, 29, 30	sylancl 586	. . . . . . . . . 10 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝑓‘0)))
32		1elunit 13202	. . . . . . . . . . 11 ⊢ 1 ∈ (0[,]1)
33		fvco3 6867	. . . . . . . . . . 11 ⊢ ((𝑓:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 1 ∈ (0[,]1)) → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘1) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝑓‘1)))
34	28, 32, 33	sylancl 586	. . . . . . . . . 10 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘1) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝑓‘1)))
35	22, 31, 34	3eqtr4d 2788	. . . . . . . . 9 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0) = (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘1))
36		sconnpht 33191	. . . . . . . . 9 ⊢ ((𝑅 ∈ SConn ∧ ((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑅) ∧ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0) = (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘1)) → ((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)( ≃ph‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)}))
37	5, 20, 35, 36	syl3anc 1370	. . . . . . . 8 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)( ≃ph‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)}))
38		isphtpc 24157	. . . . . . . 8 ⊢ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)( ≃ph‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)}) ↔ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑅) ∧ ((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)}) ∈ (II Cn 𝑅) ∧ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ≠ ∅))
39	37, 38	sylib 217	. . . . . . 7 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑅) ∧ ((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)}) ∈ (II Cn 𝑅) ∧ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ≠ ∅))
40	39	simp3d 1143	. . . . . 6 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ≠ ∅)
41		n0 4280	. . . . . 6 ⊢ ((((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ≠ ∅ ↔ ∃𝑔 𝑔 ∈ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})))
42	40, 41	sylib 217	. . . . 5 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ∃𝑔 𝑔 ∈ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})))
43		simplr 766	. . . . . . . . 9 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑆 ∈ SConn)
44		tx2cn 22761	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ (TopOn‘∪ 𝑅) ∧ 𝑆 ∈ (TopOn‘∪ 𝑆)) → (2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
45	11, 16, 44	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
46		cnco 22417	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆)) → ((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑆))
47	6, 45, 46	syl2anc 584	. . . . . . . . 9 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑆))
48	21	fveq2d 6778	. . . . . . . . . 10 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝑓‘0)) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝑓‘1)))
49		fvco3 6867	. . . . . . . . . . 11 ⊢ ((𝑓:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 0 ∈ (0[,]1)) → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝑓‘0)))
50	28, 29, 49	sylancl 586	. . . . . . . . . 10 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝑓‘0)))
51		fvco3 6867	. . . . . . . . . . 11 ⊢ ((𝑓:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 1 ∈ (0[,]1)) → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘1) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝑓‘1)))
52	28, 32, 51	sylancl 586	. . . . . . . . . 10 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘1) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝑓‘1)))
53	48, 50, 52	3eqtr4d 2788	. . . . . . . . 9 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0) = (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘1))
54		sconnpht 33191	. . . . . . . . 9 ⊢ ((𝑆 ∈ SConn ∧ ((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑆) ∧ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0) = (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘1)) → ((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)( ≃ph‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)}))
55	43, 47, 53, 54	syl3anc 1370	. . . . . . . 8 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)( ≃ph‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)}))
56		isphtpc 24157	. . . . . . . 8 ⊢ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)( ≃ph‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)}) ↔ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑆) ∧ ((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)}) ∈ (II Cn 𝑆) ∧ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ≠ ∅))
57	55, 56	sylib 217	. . . . . . 7 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑆) ∧ ((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)}) ∈ (II Cn 𝑆) ∧ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ≠ ∅))
58	57	simp3d 1143	. . . . . 6 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ≠ ∅)
59		n0 4280	. . . . . 6 ⊢ ((((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ≠ ∅ ↔ ∃ℎ ℎ ∈ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})))
60	58, 59	sylib 217	. . . . 5 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ∃ℎ ℎ ∈ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})))
61		exdistrv 1959	. . . . . 6 ⊢ (∃𝑔∃ℎ(𝑔 ∈ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ∧ ℎ ∈ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)}))) ↔ (∃𝑔 𝑔 ∈ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ∧ ∃ℎ ℎ ∈ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)}))))
62	8	adantr 481	. . . . . . . . 9 ⊢ ((((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑔 ∈ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ∧ ℎ ∈ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})))) → 𝑅 ∈ Top)
63	13	adantr 481	. . . . . . . . 9 ⊢ ((((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑔 ∈ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ∧ ℎ ∈ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})))) → 𝑆 ∈ Top)
64	6	adantr 481	. . . . . . . . 9 ⊢ ((((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑔 ∈ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ∧ ℎ ∈ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})))) → 𝑓 ∈ (II Cn (𝑅 ×t 𝑆)))
65		eqid 2738	. . . . . . . . 9 ⊢ ((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)
66		eqid 2738	. . . . . . . . 9 ⊢ ((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)
67		simprl 768	. . . . . . . . 9 ⊢ ((((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑔 ∈ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ∧ ℎ ∈ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})))) → 𝑔 ∈ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})))
68		simprr 770	. . . . . . . . 9 ⊢ ((((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑔 ∈ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ∧ ℎ ∈ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})))) → ℎ ∈ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})))
69	62, 63, 64, 65, 66, 67, 68	txsconnlem 33202	. . . . . . . 8 ⊢ ((((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑔 ∈ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ∧ ℎ ∈ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})))) → 𝑓( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝑓‘0)}))
70	69	ex 413	. . . . . . 7 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ((𝑔 ∈ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ∧ ℎ ∈ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)}))) → 𝑓( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝑓‘0)})))
71	70	exlimdvv 1937	. . . . . 6 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (∃𝑔∃ℎ(𝑔 ∈ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ∧ ℎ ∈ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)}))) → 𝑓( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝑓‘0)})))
72	61, 71	syl5bir 242	. . . . 5 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ((∃𝑔 𝑔 ∈ (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑅)((0[,]1) × {(((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)})) ∧ ∃ℎ ℎ ∈ (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)(PHtpy‘𝑆)((0[,]1) × {(((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0)}))) → 𝑓( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝑓‘0)})))
73	42, 60, 72	mp2and 696	. . . 4 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝑓‘0)}))
74	73	expr 457	. . 3 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ 𝑓 ∈ (II Cn (𝑅 ×t 𝑆))) → ((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝑓‘0)})))
75	74	ralrimiva 3103	. 2 ⊢ ((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) → ∀𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝑓‘0)})))
76		issconn 33188	. 2 ⊢ ((𝑅 ×t 𝑆) ∈ SConn ↔ ((𝑅 ×t 𝑆) ∈ PConn ∧ ∀𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝑓‘0)}))))
77	4, 75, 76	sylanbrc 583	1 ⊢ ((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) → (𝑅 ×t 𝑆) ∈ SConn)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539  ∃wex 1782   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∅c0 4256  {csn 4561  ∪ cuni 4839   class class class wbr 5074   × cxp 5587   ↾ cres 5591   ∘ ccom 5593  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  1^st c1st 7829  2^nd c2nd 7830  0cc0 10871  1c1 10872  [,]cicc 13082  Topctop 22042  TopOnctopon 22059   Cn ccn 22375   ×_t ctx 22711  IIcii 24038  PHtpycphtpy 24131   ≃_phcphtpc 24132  PConncpconn 33181  SConncsconn 33182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-sup 9201  df-inf 9202  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-z 12320  df-uz 12583  df-q 12689  df-rp 12731  df-xneg 12848  df-xadd 12849  df-xmul 12850  df-icc 13086  df-seq 13722  df-exp 13783  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-topgen 17154  df-psmet 20589  df-xmet 20590  df-met 20591  df-bl 20592  df-mopn 20593  df-top 22043  df-topon 22060  df-bases 22096  df-cn 22378  df-cnp 22379  df-tx 22713  df-ii 24040  df-htpy 24133  df-phtpy 24134  df-phtpc 24155  df-pconn 33183  df-sconn 33184

This theorem is referenced by: (None)