cvxsconn - Mathbox for Mario Carneiro

	Mathbox for Mario Carneiro		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > cvxsconn		Structured version Visualization version GIF version

Theorem cvxsconn 36132

Description: A convex subset of the complex numbers is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.) Avoid ax-mulf 11293. (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 36131	. 2 ⊢ (𝜑 → 𝐾 ∈ PConn)
6		simprl 783	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓 ∈ (II Cn 𝐾))
7		pconntop 36114	. . . . . . . . . 10 ⊢ (𝐾 ∈ PConn → 𝐾 ∈ Top)
8	5, 7	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Top)
9	8	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝐾 ∈ Top)
10		toptopon2 23251	. . . . . . . 8 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
11	9, 10	sylib 221	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝐾 ∈ (TopOn‘∪ 𝐾))
12		iiuni 25217	. . . . . . . . . 10 ⊢ (0[,]1) = ∪ II
13		eqid 2763	. . . . . . . . . 10 ⊢ ∪ 𝐾 = ∪ 𝐾
14	12, 13	cnf 23579	. . . . . . . . 9 ⊢ (𝑓 ∈ (II Cn 𝐾) → 𝑓:(0[,]1)⟶∪ 𝐾)
15	6, 14	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓:(0[,]1)⟶∪ 𝐾)
16		0elunit 13613	. . . . . . . 8 ⊢ 0 ∈ (0[,]1)
17		ffvelcdm 7092	. . . . . . . 8 ⊢ ((𝑓:(0[,]1)⟶∪ 𝐾 ∧ 0 ∈ (0[,]1)) → (𝑓‘0) ∈ ∪ 𝐾)
18	15, 16, 17	sylancl 598	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑓‘0) ∈ ∪ 𝐾)
19		eqid 2763	. . . . . . . 8 ⊢ ((0[,]1) × {(𝑓‘0)}) = ((0[,]1) × {(𝑓‘0)})
20	19	pcoptcl 25357	. . . . . . 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 596	. . . . . 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 1160	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → ((0[,]1) × {(𝑓‘0)}) ∈ (II Cn 𝐾))
23		iitopon 25215	. . . . . . . . . . 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 25219	. . . . . . . . . . . 12 ⊢ II = (𝐽 ↾_t (0[,]1))
26	3	cnfldtopon 25116	. . . . . . . . . . . . 13 ⊢ 𝐽 ∈ (TopOn‘ℂ)
27	26	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝐽 ∈ (TopOn‘ℂ))
28		unitsscn 13644	. . . . . . . . . . . . 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 24003	. . . . . . . . . . . 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 24011	. . . . . . . . . . 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 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑆 ⊆ ℂ)
33		resttopon 23494	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → (𝐽 ↾_t 𝑆) ∈ (TopOn‘𝑆))
34	26, 1, 33	sylancr 599	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐽 ↾_t 𝑆) ∈ (TopOn‘𝑆))
35	4, 34	eqeltrid 2868	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑆))
36		toponuni 23247	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ (TopOn‘𝑆) → 𝑆 = ∪ 𝐾)
37	35, 36	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 = ∪ 𝐾)
38	37	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑆 = ∪ 𝐾)
39	18, 38	eleqtrrd 2867	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑓‘0) ∈ 𝑆)
40	32, 39	sseldd 3914	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑓‘0) ∈ ℂ)
41	24, 24, 27, 40	cnmpt2c 24004	. . . . . . . . . . 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 25203	. . . . . . . . . . . 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 7439	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑡 ∧ 𝑣 = (𝑓‘0)) → (𝑢 · 𝑣) = (𝑡 · (𝑓‘0)))
45	24, 24, 31, 41, 27, 27, 43, 44	cnmpt22 24008	. . . . . . . . . 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 11271	. . . . . . . . . . . . . 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 24004	. . . . . . . . . . . 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 25201	. . . . . . . . . . . . 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 24009	. . . . . . . . . . 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 24002	. . . . . . . . . . . 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 25117	. . . . . . . . . . . . . 14 ⊢ 𝐽 ∈ Top
54		cnrest2r 23620	. . . . . . . . . . . . . 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 7441	. . . . . . . . . . . . . 14 ⊢ (II Cn 𝐾) = (II Cn (𝐽 ↾_t 𝑆))
57	6, 56	eleqtrdi 2874	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓 ∈ (II Cn (𝐽 ↾_t 𝑆)))
58	55, 57	sselid 3911	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓 ∈ (II Cn 𝐽))
59	24, 24, 52, 58	cnmpt21f 24006	. . . . . . . . . . 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 7439	. . . . . . . . . . 11 ⊢ ((𝑢 = (1 − 𝑡) ∧ 𝑣 = (𝑓‘𝑧)) → (𝑢 · 𝑣) = ((1 − 𝑡) · (𝑓‘𝑧)))
61	24, 24, 51, 59, 27, 27, 43, 60	cnmpt22 24008	. . . . . . . . . 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 25200	. . . . . . . . . . 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 24009	. . . . . . . . 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 7438	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑓‘0) → (𝑡 · 𝑥) = (𝑡 · (𝑓‘0)))
66	65	oveq1d 7445	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑓‘0) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) = ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · 𝑦)))
67	66	eleq1d 2849	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑓‘0) → (((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆 ↔ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆))
68		oveq2 7438	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑓‘𝑧) → ((1 − 𝑡) · 𝑦) = ((1 − 𝑡) · (𝑓‘𝑧)))
69	68	oveq2d 7446	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑓‘𝑧) → ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · 𝑦)) = ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))))
70	69	eleq1d 2849	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑓‘𝑧) → (((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆 ↔ ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))) ∈ 𝑆))
71	2	3exp2 1373	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑥 ∈ 𝑆 → (𝑦 ∈ 𝑆 → (𝑡 ∈ (0[,]1) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆))))
72	71	imp42 432	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) ∧ 𝑡 ∈ (0[,]1)) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆)
73	72	an32s 665	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑡 ∈ (0[,]1)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆)
74	73	ralrimivva 3209	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑡 ∈ (0[,]1)) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆)
75	74	ad2ant2rl 762	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑧 ∈ (0[,]1) ∧ 𝑡 ∈ (0[,]1))) → (𝑓‘0) ∈ 𝑆)
77	15	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑧 ∈ (0[,]1) ∧ 𝑡 ∈ (0[,]1))) → 𝑓:(0[,]1)⟶∪ 𝐾)
78		simprl 783	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑧 ∈ (0[,]1) ∧ 𝑡 ∈ (0[,]1))) → 𝑧 ∈ (0[,]1))
79	77, 78	ffvelcdmd 7096	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑧 ∈ (0[,]1) ∧ 𝑡 ∈ (0[,]1))) → (𝑓‘𝑧) ∈ ∪ 𝐾)
80	38	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ (𝑧 ∈ (0[,]1) ∧ 𝑡 ∈ (0[,]1))) → 𝑆 = ∪ 𝐾)
81	79, 80	eleqtrrd 2867	. . . . . . . . . . . . . 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 3594	. . . . . . . . . . . . 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 3209	. . . . . . . . . . . 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 2763	. . . . . . . . . . . . 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 8090	. . . . . . . . . . . 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 221	. . . . . . . . . . 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 6729	. . . . . . . . . 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 23619	. . . . . . . . . 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 1495	. . . . . . . . 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 235	. . . . . . . 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 7441	. . . . . . . 8 ⊢ ((II ×_t II) Cn 𝐾) = ((II ×_t II) Cn (𝐽 ↾_t 𝑆))
92	90, 91	eleqtrrdi 2875	. . . . . . 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 490	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → 𝑠 ∈ (0[,]1))
94		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 0) → 𝑡 = 0)
95	94	oveq1d 7445	. . . . . . . . . . 11 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 0) → (𝑡 · (𝑓‘0)) = (0 · (𝑓‘0)))
96	94	oveq2d 7446	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 0) → (1 − 𝑡) = (1 − 0))
97		1m0e1 12475	. . . . . . . . . . . . 13 ⊢ (1 − 0) = 1
98	96, 97	eqtrdi 2814	. . . . . . . . . . . 12 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 0) → (1 − 𝑡) = 1)
99		simpl 488	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 0) → 𝑧 = 𝑠)
100	99	fveq2d 6900	. . . . . . . . . . . 12 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 0) → (𝑓‘𝑧) = (𝑓‘𝑠))
101	98, 100	oveq12d 7448	. . . . . . . . . . 11 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 0) → ((1 − 𝑡) · (𝑓‘𝑧)) = (1 · (𝑓‘𝑠)))
102	95, 101	oveq12d 7448	. . . . . . . . . 10 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 0) → ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))) = ((0 · (𝑓‘0)) + (1 · (𝑓‘𝑠))))
103		ovex 7463	. . . . . . . . . 10 ⊢ ((0 · (𝑓‘0)) + (1 · (𝑓‘𝑠))) ∈ V
104	102, 84, 103	ovmpoa 7585	. . . . . . . . 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 598	. . . . . . . 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 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (𝑓‘0) ∈ ℂ)
107	106	mul02d 11521	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (0 · (𝑓‘0)) = 0)
108	26	toponunii 23249	. . . . . . . . . . . . 13 ⊢ ℂ = ∪ 𝐽
109	12, 108	cnf 23579	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (II Cn 𝐽) → 𝑓:(0[,]1)⟶ℂ)
110	58, 109	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓:(0[,]1)⟶ℂ)
111	110	ffvelcdmda 7095	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (𝑓‘𝑠) ∈ ℂ)
112	111	mullidd 11340	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (1 · (𝑓‘𝑠)) = (𝑓‘𝑠))
113	107, 112	oveq12d 7448	. . . . . . . 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 11524	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (0 + (𝑓‘𝑠)) = (𝑓‘𝑠))
115	105, 113, 114	3eqtrd 2802	. . . . . . 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 13614	. . . . . . . . 9 ⊢ 1 ∈ (0[,]1)
117		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 1) → 𝑡 = 1)
118	117	oveq1d 7445	. . . . . . . . . . 11 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 1) → (𝑡 · (𝑓‘0)) = (1 · (𝑓‘0)))
119	117	oveq2d 7446	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 1) → (1 − 𝑡) = (1 − 1))
120		1m1e0 12428	. . . . . . . . . . . . 13 ⊢ (1 − 1) = 0
121	119, 120	eqtrdi 2814	. . . . . . . . . . . 12 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 1) → (1 − 𝑡) = 0)
122		simpl 488	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 1) → 𝑧 = 𝑠)
123	122	fveq2d 6900	. . . . . . . . . . . 12 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 1) → (𝑓‘𝑧) = (𝑓‘𝑠))
124	121, 123	oveq12d 7448	. . . . . . . . . . 11 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 1) → ((1 − 𝑡) · (𝑓‘𝑧)) = (0 · (𝑓‘𝑠)))
125	118, 124	oveq12d 7448	. . . . . . . . . 10 ⊢ ((𝑧 = 𝑠 ∧ 𝑡 = 1) → ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))) = ((1 · (𝑓‘0)) + (0 · (𝑓‘𝑠))))
126		ovex 7463	. . . . . . . . . 10 ⊢ ((1 · (𝑓‘0)) + (0 · (𝑓‘𝑠))) ∈ V
127	125, 84, 126	ovmpoa 7585	. . . . . . . . 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 598	. . . . . . . 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 11340	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (1 · (𝑓‘0)) = (𝑓‘0))
130	111	mul02d 11521	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (0 · (𝑓‘𝑠)) = 0)
131	129, 130	oveq12d 7448	. . . . . . . 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 11523	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → ((𝑓‘0) + 0) = (𝑓‘0))
133		fvex 6909	. . . . . . . . . . 11 ⊢ (𝑓‘0) ∈ V
134	133	fvconst2 7220	. . . . . . . . . 10 ⊢ (𝑠 ∈ (0[,]1) → (((0[,]1) × {(𝑓‘0)})‘𝑠) = (𝑓‘0))
135	134	adantl 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (((0[,]1) × {(𝑓‘0)})‘𝑠) = (𝑓‘0))
136	132, 135	eqtr4d 2801	. . . . . . . 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 2802	. . . . . . 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 490	. . . . . . . . . . . 12 ⊢ ((𝑧 = 0 ∧ 𝑡 = 𝑠) → 𝑡 = 𝑠)
139	138	oveq1d 7445	. . . . . . . . . . 11 ⊢ ((𝑧 = 0 ∧ 𝑡 = 𝑠) → (𝑡 · (𝑓‘0)) = (𝑠 · (𝑓‘0)))
140	138	oveq2d 7446	. . . . . . . . . . . 12 ⊢ ((𝑧 = 0 ∧ 𝑡 = 𝑠) → (1 − 𝑡) = (1 − 𝑠))
141		simpl 488	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 0 ∧ 𝑡 = 𝑠) → 𝑧 = 0)
142	141	fveq2d 6900	. . . . . . . . . . . 12 ⊢ ((𝑧 = 0 ∧ 𝑡 = 𝑠) → (𝑓‘𝑧) = (𝑓‘0))
143	140, 142	oveq12d 7448	. . . . . . . . . . 11 ⊢ ((𝑧 = 0 ∧ 𝑡 = 𝑠) → ((1 − 𝑡) · (𝑓‘𝑧)) = ((1 − 𝑠) · (𝑓‘0)))
144	139, 143	oveq12d 7448	. . . . . . . . . 10 ⊢ ((𝑧 = 0 ∧ 𝑡 = 𝑠) → ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))) = ((𝑠 · (𝑓‘0)) + ((1 − 𝑠) · (𝑓‘0))))
145		ovex 7463	. . . . . . . . . 10 ⊢ ((𝑠 · (𝑓‘0)) + ((1 − 𝑠) · (𝑓‘0))) ∈ V
146	144, 84, 145	ovmpoa 7585	. . . . . . . . 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 599	. . . . . . . 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 3911	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → 𝑠 ∈ ℂ)
149		pncan3 11578	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑠 + (1 − 𝑠)) = 1)
150	148, 46, 149	sylancl 598	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (𝑠 + (1 − 𝑠)) = 1)
151	150	oveq1d 7445	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → ((𝑠 + (1 − 𝑠)) · (𝑓‘0)) = (1 · (𝑓‘0)))
152		subcl 11569	. . . . . . . . . . 11 ⊢ ((1 ∈ ℂ ∧ 𝑠 ∈ ℂ) → (1 − 𝑠) ∈ ℂ)
153	46, 148, 152	sylancr 599	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (1 − 𝑠) ∈ ℂ)
154	148, 153, 106	adddird 11347	. . . . . . . . 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 2806	. . . . . . . 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 2798	. . . . . . 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 490	. . . . . . . . . . . 12 ⊢ ((𝑧 = 1 ∧ 𝑡 = 𝑠) → 𝑡 = 𝑠)
158	157	oveq1d 7445	. . . . . . . . . . 11 ⊢ ((𝑧 = 1 ∧ 𝑡 = 𝑠) → (𝑡 · (𝑓‘0)) = (𝑠 · (𝑓‘0)))
159	157	oveq2d 7446	. . . . . . . . . . . 12 ⊢ ((𝑧 = 1 ∧ 𝑡 = 𝑠) → (1 − 𝑡) = (1 − 𝑠))
160		simpl 488	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 1 ∧ 𝑡 = 𝑠) → 𝑧 = 1)
161	160	fveq2d 6900	. . . . . . . . . . . 12 ⊢ ((𝑧 = 1 ∧ 𝑡 = 𝑠) → (𝑓‘𝑧) = (𝑓‘1))
162	159, 161	oveq12d 7448	. . . . . . . . . . 11 ⊢ ((𝑧 = 1 ∧ 𝑡 = 𝑠) → ((1 − 𝑡) · (𝑓‘𝑧)) = ((1 − 𝑠) · (𝑓‘1)))
163	158, 162	oveq12d 7448	. . . . . . . . . 10 ⊢ ((𝑧 = 1 ∧ 𝑡 = 𝑠) → ((𝑡 · (𝑓‘0)) + ((1 − 𝑡) · (𝑓‘𝑧))) = ((𝑠 · (𝑓‘0)) + ((1 − 𝑠) · (𝑓‘1))))
164		ovex 7463	. . . . . . . . . 10 ⊢ ((𝑠 · (𝑓‘0)) + ((1 − 𝑠) · (𝑓‘1))) ∈ V
165	163, 84, 164	ovmpoa 7585	. . . . . . . . 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 599	. . . . . . . 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 790	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → (𝑓‘0) = (𝑓‘1))
168	167	oveq2d 7446	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) · (𝑓‘0)) = ((1 − 𝑠) · (𝑓‘1)))
169	168	oveq2d 7446	. . . . . . . . 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 2806	. . . . . . . 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 2798	. . . . . . 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 25323	. . . . . 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 4291	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑓(PHtpy‘𝐾)((0[,]1) × {(𝑓‘0)})) ≠ ∅)
174		isphtpc 25330	. . . . 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 1362	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓( ≃_ph‘𝐾)((0[,]1) × {(𝑓‘0)}))
176	175	expr 462	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (II Cn 𝐾)) → ((𝑓‘0) = (𝑓‘1) → 𝑓( ≃_ph‘𝐾)((0[,]1) × {(𝑓‘0)})))
177	176	ralrimiva 3158	. 2 ⊢ (𝜑 → ∀𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃_ph‘𝐾)((0[,]1) × {(𝑓‘0)})))
178		issconn 36115	. 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 595	1 ⊢ (𝜑 → 𝐾 ∈ SConn)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2146 ≠ wne 2959 ∀wral 3080 ⊆ wss 3900 ∅c0 4281 {csn 4588 ∪ cuni 4870 class class class wbr 5106 × cxp 5653 ran crn 5656 ⟶wf 6543 ‘cfv 6547 (class class class)co 7430 ∈ cmpo 7432 ℂcc 11211 0cc0 11213 1c1 11214 + caddc 11216 · cmul 11218 − cmin 11554 [,]cicc 13492 ↾_t crest 17606 TopOpenctopn 17607 ℂ_fldccnfld 21693 Topctop 23226 TopOnctopon 23243 Cn ccn 23557 ×_t ctx 23894 IIcii 25211 PHtpycphtpy 25304 ≃_phcphtpc 25305 PConncpconn 36108 SConncsconn 36109

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 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7761 ax-cnex 11269 ax-resscn 11270 ax-1cn 11271 ax-icn 11272 ax-addcl 11273 ax-addrcl 11274 ax-mulcl 11275 ax-mulrcl 11276 ax-mulcom 11277 ax-addass 11278 ax-mulass 11279 ax-distr 11280 ax-i2m1 11281 ax-1ne0 11282 ax-1rid 11283 ax-rnegex 11284 ax-rrecex 11285 ax-cnre 11286 ax-pre-lttri 11287 ax-pre-lttrn 11288 ax-pre-ltadd 11289 ax-pre-mulgt0 11290 ax-pre-sup 11291 ax-addf 11292

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 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6311 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6501 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-isom 6556 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7703 df-om 7888 df-1st 8011 df-2nd 8012 df-supp 8191 df-frecs 8313 df-wrecs 8344 df-recs 8393 df-rdg 8432 df-1o 8490 df-2o 8491 df-er 8731 df-map 8863 df-ixp 8940 df-en 8988 df-dom 8989 df-sdom 8990 df-fin 8991 df-fsupp 9368 df-fi 9416 df-sup 9447 df-inf 9448 df-oi 9517 df-card 10033 df-pnf 11358 df-mnf 11359 df-xr 11360 df-ltxr 11361 df-le 11362 df-sub 11556 df-neg 11557 df-div 11987 df-nn 12349 df-2 12418 df-3 12419 df-4 12420 df-5 12421 df-6 12422 df-7 12423 df-8 12424 df-9 12425 df-n0 12620 df-z 12707 df-dec 12828 df-uz 12979 df-q 13089 df-rp 13134 df-xneg 13254 df-xadd 13255 df-xmul 13256 df-icc 13496 df-fz 13653 df-fzo 13802 df-seq 14158 df-exp 14218 df-hash 14488 df-cj 15281 df-re 15282 df-im 15283 df-sqrt 15417 df-abs 15418 df-struct 17340 df-sets 17357 df-slot 17375 df-ndx 17387 df-base 17403 df-ress 17424 df-plusg 17456 df-mulr 17457 df-starv 17458 df-sca 17459 df-vsca 17460 df-ip 17461 df-tset 17462 df-ple 17463 df-ds 17465 df-unif 17466 df-hom 17467 df-cco 17468 df-rest 17608 df-topn 17609 df-0g 17627 df-gsum 17628 df-topgen 17629 df-pt 17630 df-prds 17633 df-xrs 17689 df-qtop 17694 df-imas 17695 df-xps 17697 df-mre 17771 df-mrc 17772 df-acs 17774 df-mgm 18831 df-sgrp 18923 df-mnd 18939 df-submnd 18994 df-mulg 19293 df-cntz 19546 df-cmn 20011 df-psmet 21685 df-xmet 21686 df-met 21687 df-bl 21688 df-mopn 21689 df-cnfld 21694 df-top 23227 df-topon 23244 df-topsp 23266 df-bases 23279 df-cn 23560 df-cnp 23561 df-tx 23896 df-hmeo 24089 df-xms 24654 df-ms 24655 df-tms 24656 df-ii 25213 df-cncf 25214 df-htpy 25306 df-phtpy 25307 df-phtpc 25328 df-pconn 36110 df-sconn 36111

This theorem is used by: blsconn 36133 resconn 36135

W3C validator