cvxsconn - Mathbox for Mario Carneiro

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Theorem cvxsconn 35223

Description: A convex subset of the complex numbers is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.) Avoid ax-mulf 11124. (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 35222	. 2 ⊢ (𝜑 → 𝐾 ∈ PConn)
6		simprl 770	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓 ∈ (II Cn 𝐾))
7		pconntop 35205	. . . . . . . . . 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 22838	. . . . . . . 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 24807	. . . . . . . . . 10 ⊢ (0[,]1) = ∪ II
13		eqid 2729	. . . . . . . . . 10 ⊢ ∪ 𝐾 = ∪ 𝐾
14	12, 13	cnf 23166	. . . . . . . . 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 13406	. . . . . . . 8 ⊢ 0 ∈ (0[,]1)
17		ffvelcdm 7035	. . . . . . . 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 2729	. . . . . . . 8 ⊢ ((0[,]1) × {(𝑓‘0)}) = ((0[,]1) × {(𝑓‘0)})
20	19	pcoptcl 24954	. . . . . . 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 24805	. . . . . . . . . . 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 24809	. . . . . . . . . . . 12 ⊢ II = (𝐽 ↾_t (0[,]1))
26	3	cnfldtopon 24703	. . . . . . . . . . . . 13 ⊢ 𝐽 ∈ (TopOn‘ℂ)
27	26	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝐽 ∈ (TopOn‘ℂ))
28		unitsscn 13437	. . . . . . . . . . . . 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 23589	. . . . . . . . . . . 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 23597	. . . . . . . . . . 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 23081	. . . . . . . . . . . . . . . . . 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 2832	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑆))
36		toponuni 22834	. . . . . . . . . . . . . . . 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 2831	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑓‘0) ∈ 𝑆)
40	32, 39	sseldd 3944	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑓‘0) ∈ ℂ)
41	24, 24, 27, 40	cnmpt2c 23590	. . . . . . . . . . 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 24791	. . . . . . . . . . . 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 7378	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑡 ∧ 𝑣 = (𝑓‘0)) → (𝑢 · 𝑣) = (𝑡 · (𝑓‘0)))
45	24, 24, 31, 41, 27, 27, 43, 44	cnmpt22 23594	. . . . . . . . . 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 11102	. . . . . . . . . . . . . 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 23590	. . . . . . . . . . . 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 24788	. . . . . . . . . . . . 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 23595	. . . . . . . . . . 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 23588	. . . . . . . . . . . 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 24704	. . . . . . . . . . . . . 14 ⊢ 𝐽 ∈ Top
54		cnrest2r 23207	. . . . . . . . . . . . . 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 7380	. . . . . . . . . . . . . 14 ⊢ (II Cn 𝐾) = (II Cn (𝐽 ↾_t 𝑆))
57	6, 56	eleqtrdi 2838	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓 ∈ (II Cn (𝐽 ↾_t 𝑆)))
58	55, 57	sselid 3941	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓 ∈ (II Cn 𝐽))
59	24, 24, 52, 58	cnmpt21f 23592	. . . . . . . . . . 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 7378	. . . . . . . . . . 11 ⊢ ((𝑢 = (1 − 𝑡) ∧ 𝑣 = (𝑓‘𝑧)) → (𝑢 · 𝑣) = ((1 − 𝑡) · (𝑓‘𝑧)))
61	24, 24, 51, 59, 27, 27, 43, 60	cnmpt22 23594	. . . . . . . . . 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 24787	. . . . . . . . . . 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 23595	. . . . . . . . 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 7377	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑓‘0) → (𝑡 · 𝑥) = (𝑡 · (𝑓‘0)))
66	65	oveq1d 7384	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑓‘0) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) = ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · 𝑦)))
67	66	eleq1d 2813	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑓‘0) → (((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆 ↔ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆))
68		oveq2 7377	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑓‘𝑧) → ((1 − 𝑡) · 𝑦) = ((1 − 𝑡) · (𝑓‘𝑧)))
69	68	oveq2d 7385	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑓‘𝑧) → ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · 𝑦)) = ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))))
70	69	eleq1d 2813	. . . . . . . . . . . . . 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 3178	. . . . . . . . . . . . . . 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 7039	. . . . . . . . . . . . . . 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 2831	. . . . . . . . . . . . . 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 3600	. . . . . . . . . . . . 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 3178	. . . . . . . . . . . 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 2729	. . . . . . . . . . . . 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 8026	. . . . . . . . . . . 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 6678	. . . . . . . . . 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 23206	. . . . . . . . . 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 7380	. . . . . . . 8 ⊢ ((II ×_t II) Cn 𝐾) = ((II ×_t II) Cn (𝐽 ↾_t 𝑆))
92	90, 91	eleqtrrdi 2839	. . . . . . 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 7384	. . . . . . . . . . 11 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 0) → (𝑡 · (𝑓‘0)) = (0 · (𝑓‘0)))
96	94	oveq2d 7385	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 0) → (1 − 𝑡) = (1 − 0))
97		1m0e1 12278	. . . . . . . . . . . . 13 ⊢ (1 − 0) = 1
98	96, 97	eqtrdi 2780	. . . . . . . . . . . 12 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 0) → (1 − 𝑡) = 1)
99		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 0) → 𝑧 = 𝑠)
100	99	fveq2d 6844	. . . . . . . . . . . 12 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 0) → (𝑓‘𝑧) = (𝑓‘𝑠))
101	98, 100	oveq12d 7387	. . . . . . . . . . 11 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 0) → ((1 − 𝑡) · (𝑓‘𝑧)) = (1 · (𝑓‘𝑠)))
102	95, 101	oveq12d 7387	. . . . . . . . . 10 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 0) → ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))) = ((0 · (𝑓‘0)) + (1 · (𝑓‘𝑠))))
103		ovex 7402	. . . . . . . . . 10 ⊢ ((0 · (𝑓‘0)) + (1 · (𝑓‘𝑠))) ∈ V
104	102, 84, 103	ovmpoa 7524	. . . . . . . . 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 11348	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (0 · (𝑓‘0)) = 0)
108	26	toponunii 22836	. . . . . . . . . . . . 13 ⊢ ℂ = ∪ 𝐽
109	12, 108	cnf 23166	. . . . . . . . . . . 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 7038	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (𝑓‘𝑠) ∈ ℂ)
112	111	mullidd 11168	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (1 · (𝑓‘𝑠)) = (𝑓‘𝑠))
113	107, 112	oveq12d 7387	. . . . . . . 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 11351	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (0 + (𝑓‘𝑠)) = (𝑓‘𝑠))
115	105, 113, 114	3eqtrd 2768	. . . . . . 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 13407	. . . . . . . . 9 ⊢ 1 ∈ (0[,]1)
117		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 1) → 𝑡 = 1)
118	117	oveq1d 7384	. . . . . . . . . . 11 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 1) → (𝑡 · (𝑓‘0)) = (1 · (𝑓‘0)))
119	117	oveq2d 7385	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 1) → (1 − 𝑡) = (1 − 1))
120		1m1e0 12234	. . . . . . . . . . . . 13 ⊢ (1 − 1) = 0
121	119, 120	eqtrdi 2780	. . . . . . . . . . . 12 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 1) → (1 − 𝑡) = 0)
122		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 1) → 𝑧 = 𝑠)
123	122	fveq2d 6844	. . . . . . . . . . . 12 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 1) → (𝑓‘𝑧) = (𝑓‘𝑠))
124	121, 123	oveq12d 7387	. . . . . . . . . . 11 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 1) → ((1 − 𝑡) · (𝑓‘𝑧)) = (0 · (𝑓‘𝑠)))
125	118, 124	oveq12d 7387	. . . . . . . . . 10 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 1) → ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))) = ((1 · (𝑓‘0)) + (0 · (𝑓‘𝑠))))
126		ovex 7402	. . . . . . . . . 10 ⊢ ((1 · (𝑓‘0)) + (0 · (𝑓‘𝑠))) ∈ V
127	125, 84, 126	ovmpoa 7524	. . . . . . . . 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 11168	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (1 · (𝑓‘0)) = (𝑓‘0))
130	111	mul02d 11348	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (0 · (𝑓‘𝑠)) = 0)
131	129, 130	oveq12d 7387	. . . . . . . 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 11350	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → ((𝑓‘0) + 0) = (𝑓‘0))
133		fvex 6853	. . . . . . . . . . 11 ⊢ (𝑓‘0) ∈ V
134	133	fvconst2 7160	. . . . . . . . . 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 2767	. . . . . . . 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 2768	. . . . . . 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 7384	. . . . . . . . . . 11 ⊢ ((𝑧 = 0 ∧ 𝑡 = 𝑠) → (𝑡 · (𝑓‘0)) = (𝑠 · (𝑓‘0)))
140	138	oveq2d 7385	. . . . . . . . . . . 12 ⊢ ((𝑧 = 0 ∧ 𝑡 = 𝑠) → (1 − 𝑡) = (1 − 𝑠))
141		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 0 ∧ 𝑡 = 𝑠) → 𝑧 = 0)
142	141	fveq2d 6844	. . . . . . . . . . . 12 ⊢ ((𝑧 = 0 ∧ 𝑡 = 𝑠) → (𝑓‘𝑧) = (𝑓‘0))
143	140, 142	oveq12d 7387	. . . . . . . . . . 11 ⊢ ((𝑧 = 0 ∧ 𝑡 = 𝑠) → ((1 − 𝑡) · (𝑓‘𝑧)) = ((1 − 𝑠) · (𝑓‘0)))
144	139, 143	oveq12d 7387	. . . . . . . . . 10 ⊢ ((𝑧 = 0 ∧ 𝑡 = 𝑠) → ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))) = ((𝑠 · (𝑓‘0)) + ((1 − 𝑠) · (𝑓‘0))))
145		ovex 7402	. . . . . . . . . 10 ⊢ ((𝑠 · (𝑓‘0)) + ((1 − 𝑠) · (𝑓‘0))) ∈ V
146	144, 84, 145	ovmpoa 7524	. . . . . . . . 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 3941	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → 𝑠 ∈ ℂ)
149		pncan3 11405	. . . . . . . . . . 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 7384	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → ((𝑠 + (1 − 𝑠)) · (𝑓‘0)) = (1 · (𝑓‘0)))
152		subcl 11396	. . . . . . . . . . 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 11175	. . . . . . . . 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 2772	. . . . . . . 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 2764	. . . . . . 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 7384	. . . . . . . . . . 11 ⊢ ((𝑧 = 1 ∧ 𝑡 = 𝑠) → (𝑡 · (𝑓‘0)) = (𝑠 · (𝑓‘0)))
159	157	oveq2d 7385	. . . . . . . . . . . 12 ⊢ ((𝑧 = 1 ∧ 𝑡 = 𝑠) → (1 − 𝑡) = (1 − 𝑠))
160		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 1 ∧ 𝑡 = 𝑠) → 𝑧 = 1)
161	160	fveq2d 6844	. . . . . . . . . . . 12 ⊢ ((𝑧 = 1 ∧ 𝑡 = 𝑠) → (𝑓‘𝑧) = (𝑓‘1))
162	159, 161	oveq12d 7387	. . . . . . . . . . 11 ⊢ ((𝑧 = 1 ∧ 𝑡 = 𝑠) → ((1 − 𝑡) · (𝑓‘𝑧)) = ((1 − 𝑠) · (𝑓‘1)))
163	158, 162	oveq12d 7387	. . . . . . . . . 10 ⊢ ((𝑧 = 1 ∧ 𝑡 = 𝑠) → ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))) = ((𝑠 · (𝑓‘0)) + ((1 − 𝑠) · (𝑓‘1))))
164		ovex 7402	. . . . . . . . . 10 ⊢ ((𝑠 · (𝑓‘0)) + ((1 − 𝑠) · (𝑓‘1))) ∈ V
165	163, 84, 164	ovmpoa 7524	. . . . . . . . 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 7385	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) · (𝑓‘0)) = ((1 − 𝑠) · (𝑓‘1)))
169	168	oveq2d 7385	. . . . . . . . 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 2772	. . . . . . . 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 2764	. . . . . . 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 24919	. . . . . 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 4301	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑓(PHtpy‘𝐾)((0[,]1) × {(𝑓‘0)})) ≠ ∅)
174		isphtpc 24926	. . . . 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 3125	. 2 ⊢ (𝜑 → ∀𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃_ph‘𝐾)((0[,]1) × {(𝑓‘0)})))
178		issconn 35206	. 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 2109 ≠ wne 2925 ∀wral 3044 ⊆ wss 3911 ∅c0 4292 {csn 4585 ∪ cuni 4867 class class class wbr 5102 × cxp 5629 ran crn 5632 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 ∈ cmpo 7371 ℂcc 11042 0cc0 11044 1c1 11045 + caddc 11047 · cmul 11049 − cmin 11381 [,]cicc 13285 ↾_t crest 17359 TopOpenctopn 17360 ℂ_fldccnfld 21296 Topctop 22813 TopOnctopon 22830 Cn ccn 23144 ×_t ctx 23480 IIcii 24801 PHtpycphtpy 24900 ≃_phcphtpc 24901 PConncpconn 35199 SConncsconn 35200

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123

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 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-icc 13289 df-fz 13445 df-fzo 13592 df-seq 13943 df-exp 14003 df-hash 14272 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-submnd 18693 df-mulg 18982 df-cntz 19231 df-cmn 19696 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-cnfld 21297 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22866 df-cn 23147 df-cnp 23148 df-tx 23482 df-hmeo 23675 df-xms 24241 df-ms 24242 df-tms 24243 df-ii 24803 df-cncf 24804 df-htpy 24902 df-phtpy 24903 df-phtpc 24924 df-pconn 35201 df-sconn 35202

This theorem is referenced by: blsconn 35224 resconn 35226

W3C validator