Theorem txsconn 32776

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 32762	. . 3 ⊢ (𝑅 ∈ SConn → 𝑅 ∈ PConn)
2		sconnpconn 32762	. . 3 ⊢ (𝑆 ∈ SConn → 𝑆 ∈ PConn)
3		txpconn 32767	. . 3 ⊢ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (𝑅 ×t 𝑆) ∈ PConn)
4	1, 2, 3	syl2an 599	. 2 ⊢ ((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) → (𝑅 ×t 𝑆) ∈ PConn)
5		simpll 767	. . . . . . . . 9 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑅 ∈ SConn)
6		simprl 771	. . . . . . . . . 10 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓 ∈ (II Cn (𝑅 ×t 𝑆)))
7		sconntop 32763	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ SConn → 𝑅 ∈ Top)
8	7	ad2antrr 726	. . . . . . . . . . . 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 21670	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘∪ 𝑅))
11	8, 10	sylib 221	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑅 ∈ (TopOn‘∪ 𝑅))
12		sconntop 32763	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ SConn → 𝑆 ∈ Top)
13	12	ad2antlr 727	. . . . . . . . . . . 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 21670	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘∪ 𝑆))
16	13, 15	sylib 221	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑆 ∈ (TopOn‘∪ 𝑆))
17		tx1cn 22362	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ (TopOn‘∪ 𝑅) ∧ 𝑆 ∈ (TopOn‘∪ 𝑆)) → (1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
18	11, 16, 17	syl2anc 587	. . . . . . . . . 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 22019	. . . . . . . . . 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 587	. . . . . . . . 9 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → ((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓) ∈ (II Cn 𝑅))
21		simprr 773	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑓‘0) = (𝑓‘1))
22	21	fveq2d 6680	. . . . . . . . . 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 23633	. . . . . . . . . . . . 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 22344	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ (TopOn‘∪ 𝑅) ∧ 𝑆 ∈ (TopOn‘∪ 𝑆)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(∪ 𝑅 × ∪ 𝑆)))
26	11, 16, 25	syl2anc 587	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑅 ×t 𝑆) ∈ (TopOn‘(∪ 𝑅 × ∪ 𝑆)))
27		cnf2 22002	. . . . . . . . . . . 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 1372	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆))
29		0elunit 12945	. . . . . . . . . . 11 ⊢ 0 ∈ (0[,]1)
30		fvco3 6769	. . . . . . . . . . 11 ⊢ ((𝑓:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 0 ∈ (0[,]1)) → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝑓‘0)))
31	28, 29, 30	sylancl 589	. . . . . . . . . 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 12946	. . . . . . . . . . 11 ⊢ 1 ∈ (0[,]1)
33		fvco3 6769	. . . . . . . . . . 11 ⊢ ((𝑓:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 1 ∈ (0[,]1)) → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘1) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝑓‘1)))
34	28, 32, 33	sylancl 589	. . . . . . . . . 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 2783	. . . . . . . . 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 32764	. . . . . . . . 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 1372	. . . . . . . 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 23748	. . . . . . . 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 221	. . . . . . 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 1145	. . . . . 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 4235	. . . . . 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 221	. . . . 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 769	. . . . . . . . 9 ⊢ (((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) ∧ (𝑓 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑆 ∈ SConn)
44		tx2cn 22363	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ (TopOn‘∪ 𝑅) ∧ 𝑆 ∈ (TopOn‘∪ 𝑆)) → (2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
45	11, 16, 44	syl2anc 587	. . . . . . . . . 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 22019	. . . . . . . . . 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 587	. . . . . . . . 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 6680	. . . . . . . . . 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 6769	. . . . . . . . . . 11 ⊢ ((𝑓:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 0 ∈ (0[,]1)) → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘0) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝑓‘0)))
50	28, 29, 49	sylancl 589	. . . . . . . . . 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 6769	. . . . . . . . . . 11 ⊢ ((𝑓:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 1 ∈ (0[,]1)) → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝑓)‘1) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝑓‘1)))
52	28, 32, 51	sylancl 589	. . . . . . . . . 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 2783	. . . . . . . . 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 32764	. . . . . . . . 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 1372	. . . . . . . 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 23748	. . . . . . . 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 221	. . . . . . 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 1145	. . . . . 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 4235	. . . . . 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 221	. . . . 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 1963	. . . . . 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 484	. . . . . . . . 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 484	. . . . . . . . 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 484	. . . . . . . . 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 771	. . . . . . . . 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 773	. . . . . . . . 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 32775	. . . . . . . 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 416	. . . . . . 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 1941	. . . . . 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 246	. . . . 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 699	. . . 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 460	. . 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 3096	. 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 32761	. 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 586	1 ⊢ ((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) → (𝑅 ×t 𝑆) ∈ SConn)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1088   = wceq 1542  ∃wex 1786   ∈ wcel 2114   ≠ wne 2934  ∀wral 3053  ∅c0 4211  {csn 4516  ∪ cuni 4796   class class class wbr 5030   × cxp 5523   ↾ cres 5527   ∘ ccom 5529  ⟶wf 6335  ‘cfv 6339  (class class class)co 7172  1^st c1st 7714  2^nd c2nd 7715  0cc0 10617  1c1 10618  [,]cicc 12826  Topctop 21646  TopOnctopon 21663   Cn ccn 21977   ×_t ctx 22313  IIcii 23629  PHtpycphtpy 23722   ≃_phcphtpc 23723  PConncpconn 32754  SConncsconn 32755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7481  ax-cnex 10673  ax-resscn 10674  ax-1cn 10675  ax-icn 10676  ax-addcl 10677  ax-addrcl 10678  ax-mulcl 10679  ax-mulrcl 10680  ax-mulcom 10681  ax-addass 10682  ax-mulass 10683  ax-distr 10684  ax-i2m1 10685  ax-1ne0 10686  ax-1rid 10687  ax-rnegex 10688  ax-rrecex 10689  ax-cnre 10690  ax-pre-lttri 10691  ax-pre-lttrn 10692  ax-pre-ltadd 10693  ax-pre-mulgt0 10694  ax-pre-sup 10695

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7129  df-ov 7175  df-oprab 7176  df-mpo 7177  df-om 7602  df-1st 7716  df-2nd 7717  df-wrecs 7978  df-recs 8039  df-rdg 8077  df-er 8322  df-map 8441  df-en 8558  df-dom 8559  df-sdom 8560  df-sup 8981  df-inf 8982  df-pnf 10757  df-mnf 10758  df-xr 10759  df-ltxr 10760  df-le 10761  df-sub 10952  df-neg 10953  df-div 11378  df-nn 11719  df-2 11781  df-3 11782  df-n0 11979  df-z 12065  df-uz 12327  df-q 12433  df-rp 12475  df-xneg 12592  df-xadd 12593  df-xmul 12594  df-icc 12830  df-seq 13463  df-exp 13524  df-cj 14550  df-re 14551  df-im 14552  df-sqrt 14686  df-abs 14687  df-topgen 16822  df-psmet 20211  df-xmet 20212  df-met 20213  df-bl 20214  df-mopn 20215  df-top 21647  df-topon 21664  df-bases 21699  df-cn 21980  df-cnp 21981  df-tx 22315  df-ii 23631  df-htpy 23724  df-phtpy 23725  df-phtpc 23746  df-pconn 32756  df-sconn 32757

This theorem is referenced by: (None)