cvxsconn - Mathbox for Mario Carneiro

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Theorem cvxsconn 35211

Description: A convex subset of the complex numbers is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.) Avoid ax-mulf 11207. (Revised by GG, 19-Apr-2025.)

**Hypotheses**
Ref	Expression
cvxpconn.1	⊢ (𝜑 → 𝑆 ⊆ ℂ)
cvxpconn.2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑡 ∈ (0[,]1))) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆)
cvxpconn.3	⊢ 𝐽 = (TopOpen‘ℂ_fld)
cvxpconn.4	⊢ 𝐾 = (𝐽 ↾_t 𝑆)

**Assertion**
Ref	Expression
cvxsconn	⊢ (𝜑 → 𝐾 ∈ SConn)

Distinct variable groups: 𝑡,𝐽 𝑥,𝑦,𝐾,𝑡 𝜑,𝑡,𝑥,𝑦 𝑡,𝑆,𝑥 𝑡,𝐾 𝑦,𝑆 Allowed substitution hints: 𝐽(𝑥,𝑦)

Proof of Theorem cvxsconn Dummy variables 𝑧 𝑢 𝑣 𝑓 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvxpconn.1	. . 3 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
2		cvxpconn.2	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑡 ∈ (0[,]1))) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆)
3		cvxpconn.3	. . 3 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
4		cvxpconn.4	. . 3 ⊢ 𝐾 = (𝐽 ↾_t 𝑆)
5	1, 2, 3, 4	cvxpconn 35210	. 2 ⊢ (𝜑 → 𝐾 ∈ PConn)
6		simprl 770	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓 ∈ (II Cn 𝐾))
7		pconntop 35193	. . . . . . . . . 10 ⊢ (𝐾 ∈ PConn → 𝐾 ∈ Top)
8	5, 7	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Top)
9	8	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝐾 ∈ Top)
10		toptopon2 22854	. . . . . . . 8 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
11	9, 10	sylib 218	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝐾 ∈ (TopOn‘∪ 𝐾))
12		iiuni 24823	. . . . . . . . . 10 ⊢ (0[,]1) = ∪ II
13		eqid 2735	. . . . . . . . . 10 ⊢ ∪ 𝐾 = ∪ 𝐾
14	12, 13	cnf 23182	. . . . . . . . 9 ⊢ (𝑓 ∈ (II Cn 𝐾) → 𝑓:(0[,]1)⟶∪ 𝐾)
15	6, 14	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓:(0[,]1)⟶∪ 𝐾)
16		0elunit 13484	. . . . . . . 8 ⊢ 0 ∈ (0[,]1)
17		ffvelcdm 7070	. . . . . . . 8 ⊢ ((𝑓:(0[,]1)⟶∪ 𝐾 ∧ 0 ∈ (0[,]1)) → (𝑓‘0) ∈ ∪ 𝐾)
18	15, 16, 17	sylancl 586	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑓‘0) ∈ ∪ 𝐾)
19		eqid 2735	. . . . . . . 8 ⊢ ((0[,]1) × {(𝑓‘0)}) = ((0[,]1) × {(𝑓‘0)})
20	19	pcoptcl 24970	. . . . . . 7 ⊢ ((𝐾 ∈ (TopOn‘∪ 𝐾) ∧ (𝑓‘0) ∈ ∪ 𝐾) → (((0[,]1) × {(𝑓‘0)}) ∈ (II Cn 𝐾) ∧ (((0[,]1) × {(𝑓‘0)})‘0) = (𝑓‘0) ∧ (((0[,]1) × {(𝑓‘0)})‘1) = (𝑓‘0)))
21	11, 18, 20	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (((0[,]1) × {(𝑓‘0)}) ∈ (II Cn 𝐾) ∧ (((0[,]1) × {(𝑓‘0)})‘0) = (𝑓‘0) ∧ (((0[,]1) × {(𝑓‘0)})‘1) = (𝑓‘0)))
22	21	simp1d 1142	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → ((0[,]1) × {(𝑓‘0)}) ∈ (II Cn 𝐾))
23		iitopon 24821	. . . . . . . . . . 11 ⊢ II ∈ (TopOn‘(0[,]1))
24	23	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → II ∈ (TopOn‘(0[,]1)))
25	3	dfii3 24825	. . . . . . . . . . . 12 ⊢ II = (𝐽 ↾_t (0[,]1))
26	3	cnfldtopon 24719	. . . . . . . . . . . . 13 ⊢ 𝐽 ∈ (TopOn‘ℂ)
27	26	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝐽 ∈ (TopOn‘ℂ))
28		unitsscn 13515	. . . . . . . . . . . . 13 ⊢ (0[,]1) ⊆ ℂ
29	28	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (0[,]1) ⊆ ℂ)
30	27, 27	cnmpt2nd 23605	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑧 ∈ ℂ, 𝑡 ∈ ℂ ↦ 𝑡) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
31	25, 27, 29, 25, 27, 29, 30	cnmpt2res 23613	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ 𝑡) ∈ ((II ×_t II) Cn 𝐽))
32	1	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑆 ⊆ ℂ)
33		resttopon 23097	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → (𝐽 ↾_t 𝑆) ∈ (TopOn‘𝑆))
34	26, 1, 33	sylancr 587	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐽 ↾_t 𝑆) ∈ (TopOn‘𝑆))
35	4, 34	eqeltrid 2838	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑆))
36		toponuni 22850	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ (TopOn‘𝑆) → 𝑆 = ∪ 𝐾)
37	35, 36	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 = ∪ 𝐾)
38	37	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑆 = ∪ 𝐾)
39	18, 38	eleqtrrd 2837	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑓‘0) ∈ 𝑆)
40	32, 39	sseldd 3959	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑓‘0) ∈ ℂ)
41	24, 24, 27, 40	cnmpt2c 23606	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ (𝑓‘0)) ∈ ((II ×_t II) Cn 𝐽))
42	3	mpomulcn 24807	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽)
43	42	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
44		oveq12 7412	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑡 ∧ 𝑣 = (𝑓‘0)) → (𝑢 · 𝑣) = (𝑡 · (𝑓‘0)))
45	24, 24, 31, 41, 27, 27, 43, 44	cnmpt22 23610	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ (𝑡 · (𝑓‘0))) ∈ ((II ×_t II) Cn 𝐽))
46		ax-1cn 11185	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
47	46	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 1 ∈ ℂ)
48	24, 24, 27, 47	cnmpt2c 23606	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ 1) ∈ ((II ×_t II) Cn 𝐽))
49	3	subcn 24804	. . . . . . . . . . . . 13 ⊢ − ∈ ((𝐽 ×_t 𝐽) Cn 𝐽)
50	49	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → − ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
51	24, 24, 48, 31, 50	cnmpt22f 23611	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ (1 − 𝑡)) ∈ ((II ×_t II) Cn 𝐽))
52	24, 24	cnmpt1st 23604	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ 𝑧) ∈ ((II ×_t II) Cn II))
53	3	cnfldtop 24720	. . . . . . . . . . . . . 14 ⊢ 𝐽 ∈ Top
54		cnrest2r 23223	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ Top → (II Cn (𝐽 ↾_t 𝑆)) ⊆ (II Cn 𝐽))
55	53, 54	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (II Cn (𝐽 ↾_t 𝑆)) ⊆ (II Cn 𝐽)
56	4	oveq2i 7414	. . . . . . . . . . . . . 14 ⊢ (II Cn 𝐾) = (II Cn (𝐽 ↾_t 𝑆))
57	6, 56	eleqtrdi 2844	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓 ∈ (II Cn (𝐽 ↾_t 𝑆)))
58	55, 57	sselid 3956	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓 ∈ (II Cn 𝐽))
59	24, 24, 52, 58	cnmpt21f 23608	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ (𝑓‘𝑧)) ∈ ((II ×_t II) Cn 𝐽))
60		oveq12 7412	. . . . . . . . . . 11 ⊢ ((𝑢 = (1 − 𝑡) ∧ 𝑣 = (𝑓‘𝑧)) → (𝑢 · 𝑣) = ((1 − 𝑡) · (𝑓‘𝑧)))
61	24, 24, 51, 59, 27, 27, 43, 60	cnmpt22 23610	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((1 − 𝑡) · (𝑓‘𝑧))) ∈ ((II ×_t II) Cn 𝐽))
62	3	addcn 24803	. . . . . . . . . . 11 ⊢ + ∈ ((𝐽 ×_t 𝐽) Cn 𝐽)
63	62	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → + ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
64	24, 24, 45, 61, 63	cnmpt22f 23611	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧)))) ∈ ((II ×_t II) Cn 𝐽))
65		oveq2 7411	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑓‘0) → (𝑡 · 𝑥) = (𝑡 · (𝑓‘0)))
66	65	oveq1d 7418	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑓‘0) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) = ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · 𝑦)))
67	66	eleq1d 2819	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑓‘0) → (((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆 ↔ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆))
68		oveq2 7411	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑓‘𝑧) → ((1 − 𝑡) · 𝑦) = ((1 − 𝑡) · (𝑓‘𝑧)))
69	68	oveq2d 7419	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑓‘𝑧) → ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · 𝑦)) = ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))))
70	69	eleq1d 2819	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑓‘𝑧) → (((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆 ↔ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))) ∈ 𝑆))
71	2	3exp2 1355	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑥 ∈ 𝑆 → (𝑦 ∈ 𝑆 → (𝑡 ∈ (0[,]1) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆))))
72	71	imp42 426	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ 𝑡 ∈ (0[,]1)) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆)
73	72	an32s 652	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑡 ∈ (0[,]1)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆)
74	73	ralrimivva 3187	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑡 ∈ (0[,]1)) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆)
75	74	ad2ant2rl 749	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑧 ∈ (0[,]1) ∧ 𝑡 ∈ (0[,]1))) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆)
76	39	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑧 ∈ (0[,]1) ∧ 𝑡 ∈ (0[,]1))) → (𝑓‘0) ∈ 𝑆)
77	15	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑧 ∈ (0[,]1) ∧ 𝑡 ∈ (0[,]1))) → 𝑓:(0[,]1)⟶∪ 𝐾)
78		simprl 770	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑧 ∈ (0[,]1) ∧ 𝑡 ∈ (0[,]1))) → 𝑧 ∈ (0[,]1))
79	77, 78	ffvelcdmd 7074	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑧 ∈ (0[,]1) ∧ 𝑡 ∈ (0[,]1))) → (𝑓‘𝑧) ∈ ∪ 𝐾)
80	38	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑧 ∈ (0[,]1) ∧ 𝑡 ∈ (0[,]1))) → 𝑆 = ∪ 𝐾)
81	79, 80	eleqtrrd 2837	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑧 ∈ (0[,]1) ∧ 𝑡 ∈ (0[,]1))) → (𝑓‘𝑧) ∈ 𝑆)
82	67, 70, 75, 76, 81	rspc2dv 3616	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑧 ∈ (0[,]1) ∧ 𝑡 ∈ (0[,]1))) → ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))) ∈ 𝑆)
83	82	ralrimivva 3187	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → ∀𝑧 ∈ (0[,]1)∀𝑡 ∈ (0[,]1)((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))) ∈ 𝑆)
84		eqid 2735	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧)))) = (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))))
85	84	fmpo 8065	. . . . . . . . . . . 12 ⊢ (∀𝑧 ∈ (0[,]1)∀𝑡 ∈ (0[,]1)((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))) ∈ 𝑆 ↔ (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧)))):((0[,]1) × (0[,]1))⟶𝑆)
86	83, 85	sylib 218	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧)))):((0[,]1) × (0[,]1))⟶𝑆)
87	86	frnd 6713	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → ran (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧)))) ⊆ 𝑆)
88		cnrest2 23222	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ ran (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧)))) ⊆ 𝑆 ∧ 𝑆 ⊆ ℂ) → ((𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧)))) ∈ ((II ×_t II) Cn 𝐽) ↔ (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧)))) ∈ ((II ×_t II) Cn (𝐽 ↾_t 𝑆))))
89	26, 87, 32, 88	mp3an2i 1468	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → ((𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧)))) ∈ ((II ×_t II) Cn 𝐽) ↔ (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧)))) ∈ ((II ×_t II) Cn (𝐽 ↾_t 𝑆))))
90	64, 89	mpbid 232	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧)))) ∈ ((II ×_t II) Cn (𝐽 ↾_t 𝑆)))
91	4	oveq2i 7414	. . . . . . . 8 ⊢ ((II ×_t II) Cn 𝐾) = ((II ×_t II) Cn (𝐽 ↾_t 𝑆))
92	90, 91	eleqtrrdi 2845	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧)))) ∈ ((II ×_t II) Cn 𝐾))
93		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → 𝑠 ∈ (0[,]1))
94		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 0) → 𝑡 = 0)
95	94	oveq1d 7418	. . . . . . . . . . 11 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 0) → (𝑡 · (𝑓‘0)) = (0 · (𝑓‘0)))
96	94	oveq2d 7419	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 0) → (1 − 𝑡) = (1 − 0))
97		1m0e1 12359	. . . . . . . . . . . . 13 ⊢ (1 − 0) = 1
98	96, 97	eqtrdi 2786	. . . . . . . . . . . 12 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 0) → (1 − 𝑡) = 1)
99		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 0) → 𝑧 = 𝑠)
100	99	fveq2d 6879	. . . . . . . . . . . 12 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 0) → (𝑓‘𝑧) = (𝑓‘𝑠))
101	98, 100	oveq12d 7421	. . . . . . . . . . 11 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 0) → ((1 − 𝑡) · (𝑓‘𝑧)) = (1 · (𝑓‘𝑠)))
102	95, 101	oveq12d 7421	. . . . . . . . . 10 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 0) → ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))) = ((0 · (𝑓‘0)) + (1 · (𝑓‘𝑠))))
103		ovex 7436	. . . . . . . . . 10 ⊢ ((0 · (𝑓‘0)) + (1 · (𝑓‘𝑠))) ∈ V
104	102, 84, 103	ovmpoa 7560	. . . . . . . . 9 ⊢ ((𝑠 ∈ (0[,]1) ∧ 0 ∈ (0[,]1)) → (𝑠(𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))))0) = ((0 · (𝑓‘0)) + (1 · (𝑓‘𝑠))))
105	93, 16, 104	sylancl 586	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (𝑠(𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))))0) = ((0 · (𝑓‘0)) + (1 · (𝑓‘𝑠))))
106	40	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (𝑓‘0) ∈ ℂ)
107	106	mul02d 11431	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (0 · (𝑓‘0)) = 0)
108	26	toponunii 22852	. . . . . . . . . . . . 13 ⊢ ℂ = ∪ 𝐽
109	12, 108	cnf 23182	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (II Cn 𝐽) → 𝑓:(0[,]1)⟶ℂ)
110	58, 109	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓:(0[,]1)⟶ℂ)
111	110	ffvelcdmda 7073	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (𝑓‘𝑠) ∈ ℂ)
112	111	mullidd 11251	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (1 · (𝑓‘𝑠)) = (𝑓‘𝑠))
113	107, 112	oveq12d 7421	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → ((0 · (𝑓‘0)) + (1 · (𝑓‘𝑠))) = (0 + (𝑓‘𝑠)))
114	111	addlidd 11434	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (0 + (𝑓‘𝑠)) = (𝑓‘𝑠))
115	105, 113, 114	3eqtrd 2774	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (𝑠(𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))))0) = (𝑓‘𝑠))
116		1elunit 13485	. . . . . . . . 9 ⊢ 1 ∈ (0[,]1)
117		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 1) → 𝑡 = 1)
118	117	oveq1d 7418	. . . . . . . . . . 11 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 1) → (𝑡 · (𝑓‘0)) = (1 · (𝑓‘0)))
119	117	oveq2d 7419	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 1) → (1 − 𝑡) = (1 − 1))
120		1m1e0 12310	. . . . . . . . . . . . 13 ⊢ (1 − 1) = 0
121	119, 120	eqtrdi 2786	. . . . . . . . . . . 12 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 1) → (1 − 𝑡) = 0)
122		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 1) → 𝑧 = 𝑠)
123	122	fveq2d 6879	. . . . . . . . . . . 12 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 1) → (𝑓‘𝑧) = (𝑓‘𝑠))
124	121, 123	oveq12d 7421	. . . . . . . . . . 11 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 1) → ((1 − 𝑡) · (𝑓‘𝑧)) = (0 · (𝑓‘𝑠)))
125	118, 124	oveq12d 7421	. . . . . . . . . 10 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 1) → ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))) = ((1 · (𝑓‘0)) + (0 · (𝑓‘𝑠))))
126		ovex 7436	. . . . . . . . . 10 ⊢ ((1 · (𝑓‘0)) + (0 · (𝑓‘𝑠))) ∈ V
127	125, 84, 126	ovmpoa 7560	. . . . . . . . 9 ⊢ ((𝑠 ∈ (0[,]1) ∧ 1 ∈ (0[,]1)) → (𝑠(𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))))1) = ((1 · (𝑓‘0)) + (0 · (𝑓‘𝑠))))
128	93, 116, 127	sylancl 586	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (𝑠(𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))))1) = ((1 · (𝑓‘0)) + (0 · (𝑓‘𝑠))))
129	106	mullidd 11251	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (1 · (𝑓‘0)) = (𝑓‘0))
130	111	mul02d 11431	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (0 · (𝑓‘𝑠)) = 0)
131	129, 130	oveq12d 7421	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → ((1 · (𝑓‘0)) + (0 · (𝑓‘𝑠))) = ((𝑓‘0) + 0))
132	106	addridd 11433	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → ((𝑓‘0) + 0) = (𝑓‘0))
133		fvex 6888	. . . . . . . . . . 11 ⊢ (𝑓‘0) ∈ V
134	133	fvconst2 7195	. . . . . . . . . 10 ⊢ (𝑠 ∈ (0[,]1) → (((0[,]1) × {(𝑓‘0)})‘𝑠) = (𝑓‘0))
135	134	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (((0[,]1) × {(𝑓‘0)})‘𝑠) = (𝑓‘0))
136	132, 135	eqtr4d 2773	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → ((𝑓‘0) + 0) = (((0[,]1) × {(𝑓‘0)})‘𝑠))
137	128, 131, 136	3eqtrd 2774	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (𝑠(𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))))1) = (((0[,]1) × {(𝑓‘0)})‘𝑠))
138		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑧 = 0 ∧ 𝑡 = 𝑠) → 𝑡 = 𝑠)
139	138	oveq1d 7418	. . . . . . . . . . 11 ⊢ ((𝑧 = 0 ∧ 𝑡 = 𝑠) → (𝑡 · (𝑓‘0)) = (𝑠 · (𝑓‘0)))
140	138	oveq2d 7419	. . . . . . . . . . . 12 ⊢ ((𝑧 = 0 ∧ 𝑡 = 𝑠) → (1 − 𝑡) = (1 − 𝑠))
141		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 0 ∧ 𝑡 = 𝑠) → 𝑧 = 0)
142	141	fveq2d 6879	. . . . . . . . . . . 12 ⊢ ((𝑧 = 0 ∧ 𝑡 = 𝑠) → (𝑓‘𝑧) = (𝑓‘0))
143	140, 142	oveq12d 7421	. . . . . . . . . . 11 ⊢ ((𝑧 = 0 ∧ 𝑡 = 𝑠) → ((1 − 𝑡) · (𝑓‘𝑧)) = ((1 − 𝑠) · (𝑓‘0)))
144	139, 143	oveq12d 7421	. . . . . . . . . 10 ⊢ ((𝑧 = 0 ∧ 𝑡 = 𝑠) → ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))) = ((𝑠 · (𝑓‘0)) + ((1 − 𝑠) · (𝑓‘0))))
145		ovex 7436	. . . . . . . . . 10 ⊢ ((𝑠 · (𝑓‘0)) + ((1 − 𝑠) · (𝑓‘0))) ∈ V
146	144, 84, 145	ovmpoa 7560	. . . . . . . . 9 ⊢ ((0 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (0(𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))))𝑠) = ((𝑠 · (𝑓‘0)) + ((1 − 𝑠) · (𝑓‘0))))
147	16, 93, 146	sylancr 587	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (0(𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))))𝑠) = ((𝑠 · (𝑓‘0)) + ((1 − 𝑠) · (𝑓‘0))))
148	28, 93	sselid 3956	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → 𝑠 ∈ ℂ)
149		pncan3 11488	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑠 + (1 − 𝑠)) = 1)
150	148, 46, 149	sylancl 586	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (𝑠 + (1 − 𝑠)) = 1)
151	150	oveq1d 7418	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → ((𝑠 + (1 − 𝑠)) · (𝑓‘0)) = (1 · (𝑓‘0)))
152		subcl 11479	. . . . . . . . . . 11 ⊢ ((1 ∈ ℂ ∧ 𝑠 ∈ ℂ) → (1 − 𝑠) ∈ ℂ)
153	46, 148, 152	sylancr 587	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (1 − 𝑠) ∈ ℂ)
154	148, 153, 106	adddird 11258	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → ((𝑠 + (1 − 𝑠)) · (𝑓‘0)) = ((𝑠 · (𝑓‘0)) + ((1 − 𝑠) · (𝑓‘0))))
155	151, 154, 129	3eqtr3d 2778	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → ((𝑠 · (𝑓‘0)) + ((1 − 𝑠) · (𝑓‘0))) = (𝑓‘0))
156	147, 155	eqtrd 2770	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (0(𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))))𝑠) = (𝑓‘0))
157		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑧 = 1 ∧ 𝑡 = 𝑠) → 𝑡 = 𝑠)
158	157	oveq1d 7418	. . . . . . . . . . 11 ⊢ ((𝑧 = 1 ∧ 𝑡 = 𝑠) → (𝑡 · (𝑓‘0)) = (𝑠 · (𝑓‘0)))
159	157	oveq2d 7419	. . . . . . . . . . . 12 ⊢ ((𝑧 = 1 ∧ 𝑡 = 𝑠) → (1 − 𝑡) = (1 − 𝑠))
160		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 1 ∧ 𝑡 = 𝑠) → 𝑧 = 1)
161	160	fveq2d 6879	. . . . . . . . . . . 12 ⊢ ((𝑧 = 1 ∧ 𝑡 = 𝑠) → (𝑓‘𝑧) = (𝑓‘1))
162	159, 161	oveq12d 7421	. . . . . . . . . . 11 ⊢ ((𝑧 = 1 ∧ 𝑡 = 𝑠) → ((1 − 𝑡) · (𝑓‘𝑧)) = ((1 − 𝑠) · (𝑓‘1)))
163	158, 162	oveq12d 7421	. . . . . . . . . 10 ⊢ ((𝑧 = 1 ∧ 𝑡 = 𝑠) → ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))) = ((𝑠 · (𝑓‘0)) + ((1 − 𝑠) · (𝑓‘1))))
164		ovex 7436	. . . . . . . . . 10 ⊢ ((𝑠 · (𝑓‘0)) + ((1 − 𝑠) · (𝑓‘1))) ∈ V
165	163, 84, 164	ovmpoa 7560	. . . . . . . . 9 ⊢ ((1 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (1(𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))))𝑠) = ((𝑠 · (𝑓‘0)) + ((1 − 𝑠) · (𝑓‘1))))
166	116, 93, 165	sylancr 587	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (1(𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))))𝑠) = ((𝑠 · (𝑓‘0)) + ((1 − 𝑠) · (𝑓‘1))))
167		simplrr 777	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (𝑓‘0) = (𝑓‘1))
168	167	oveq2d 7419	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) · (𝑓‘0)) = ((1 − 𝑠) · (𝑓‘1)))
169	168	oveq2d 7419	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → ((𝑠 · (𝑓‘0)) + ((1 − 𝑠) · (𝑓‘0))) = ((𝑠 · (𝑓‘0)) + ((1 − 𝑠) · (𝑓‘1))))
170	155, 169, 167	3eqtr3d 2778	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → ((𝑠 · (𝑓‘0)) + ((1 − 𝑠) · (𝑓‘1))) = (𝑓‘1))
171	166, 170	eqtrd 2770	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (1(𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))))𝑠) = (𝑓‘1))
172	6, 22, 92, 115, 137, 156, 171	isphtpy2d 24935	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑧 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧)))) ∈ (𝑓(PHtpy‘𝐾)((0[,]1) × {(𝑓‘0)})))
173	172	ne0d 4317	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑓(PHtpy‘𝐾)((0[,]1) × {(𝑓‘0)})) ≠ ∅)
174		isphtpc 24942	. . . . 5 ⊢ (𝑓( ≃_ph‘𝐾)((0[,]1) × {(𝑓‘0)}) ↔ (𝑓 ∈ (II Cn 𝐾) ∧ ((0[,]1) × {(𝑓‘0)}) ∈ (II Cn 𝐾) ∧ (𝑓(PHtpy‘𝐾)((0[,]1) × {(𝑓‘0)})) ≠ ∅))
175	6, 22, 173, 174	syl3anbrc 1344	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓( ≃_ph‘𝐾)((0[,]1) × {(𝑓‘0)}))
176	175	expr 456	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (II Cn 𝐾)) → ((𝑓‘0) = (𝑓‘1) → 𝑓( ≃_ph‘𝐾)((0[,]1) × {(𝑓‘0)})))
177	176	ralrimiva 3132	. 2 ⊢ (𝜑 → ∀𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃_ph‘𝐾)((0[,]1) × {(𝑓‘0)})))
178		issconn 35194	. 2 ⊢ (𝐾 ∈ SConn ↔ (𝐾 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃_ph‘𝐾)((0[,]1) × {(𝑓‘0)}))))
179	5, 177, 178	sylanbrc 583	1 ⊢ (𝜑 → 𝐾 ∈ SConn)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 ∀wral 3051 ⊆ wss 3926 ∅c0 4308 {csn 4601 ∪ cuni 4883 class class class wbr 5119 × cxp 5652 ran crn 5655 ⟶wf 6526 ‘cfv 6530 (class class class)co 7403 ∈ cmpo 7405 ℂcc 11125 0cc0 11127 1c1 11128 + caddc 11130 · cmul 11132 − cmin 11464 [,]cicc 13363 ↾_t crest 17432 TopOpenctopn 17433 ℂ_fldccnfld 21313 Topctop 22829 TopOnctopon 22846 Cn ccn 23160 ×_t ctx 23496 IIcii 24817 PHtpycphtpy 24916 ≃_phcphtpc 24917 PConncpconn 35187 SConncsconn 35188

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-isom 6539 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-of 7669 df-om 7860 df-1st 7986 df-2nd 7987 df-supp 8158 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-er 8717 df-map 8840 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9372 df-fi 9421 df-sup 9452 df-inf 9453 df-oi 9522 df-card 9951 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-div 11893 df-nn 12239 df-2 12301 df-3 12302 df-4 12303 df-5 12304 df-6 12305 df-7 12306 df-8 12307 df-9 12308 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12851 df-q 12963 df-rp 13007 df-xneg 13126 df-xadd 13127 df-xmul 13128 df-icc 13367 df-fz 13523 df-fzo 13670 df-seq 14018 df-exp 14078 df-hash 14347 df-cj 15116 df-re 15117 df-im 15118 df-sqrt 15252 df-abs 15253 df-struct 17164 df-sets 17181 df-slot 17199 df-ndx 17211 df-base 17227 df-ress 17250 df-plusg 17282 df-mulr 17283 df-starv 17284 df-sca 17285 df-vsca 17286 df-ip 17287 df-tset 17288 df-ple 17289 df-ds 17291 df-unif 17292 df-hom 17293 df-cco 17294 df-rest 17434 df-topn 17435 df-0g 17453 df-gsum 17454 df-topgen 17455 df-pt 17456 df-prds 17459 df-xrs 17514 df-qtop 17519 df-imas 17520 df-xps 17522 df-mre 17596 df-mrc 17597 df-acs 17599 df-mgm 18616 df-sgrp 18695 df-mnd 18711 df-submnd 18760 df-mulg 19049 df-cntz 19298 df-cmn 19761 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-cnfld 21314 df-top 22830 df-topon 22847 df-topsp 22869 df-bases 22882 df-cn 23163 df-cnp 23164 df-tx 23498 df-hmeo 23691 df-xms 24257 df-ms 24258 df-tms 24259 df-ii 24819 df-cncf 24820 df-htpy 24918 df-phtpy 24919 df-phtpc 24940 df-pconn 35189 df-sconn 35190

This theorem is referenced by: blsconn 35212 resconn 35214

W3C validator