cvxpconn - Mathbox for Mario Carneiro

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Theorem cvxpconn 35214

Description: A convex subset of the complex numbers is path-connected. (Contributed by Mario Carneiro, 12-Feb-2015.) Avoid ax-mulf 11108. (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
cvxpconn	⊢ (𝜑 → 𝐾 ∈ PConn)

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

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

Step	Hyp	Ref	Expression
1		cvxpconn.4	. . 3 ⊢ 𝐾 = (𝐽 ↾_t 𝑆)
2		cvxpconn.3	. . . . 5 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
3	2	cnfldtop 24687	. . . 4 ⊢ 𝐽 ∈ Top
4		cvxpconn.1	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
5		cnex 11109	. . . . 5 ⊢ ℂ ∈ V
6		ssexg 5265	. . . . 5 ⊢ ((𝑆 ⊆ ℂ ∧ ℂ ∈ V) → 𝑆 ∈ V)
7	4, 5, 6	sylancl 586	. . . 4 ⊢ (𝜑 → 𝑆 ∈ V)
8		resttop 23063	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ V) → (𝐽 ↾_t 𝑆) ∈ Top)
9	3, 7, 8	sylancr 587	. . 3 ⊢ (𝜑 → (𝐽 ↾_t 𝑆) ∈ Top)
10	1, 9	eqeltrid 2832	. 2 ⊢ (𝜑 → 𝐾 ∈ Top)
11	2	dfii3 24792	. . . . . . . 8 ⊢ II = (𝐽 ↾_t (0[,]1))
12	2	cnfldtopon 24686	. . . . . . . . 9 ⊢ 𝐽 ∈ (TopOn‘ℂ)
13	12	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → 𝐽 ∈ (TopOn‘ℂ))
14		unitsscn 13421	. . . . . . . . 9 ⊢ (0[,]1) ⊆ ℂ
15	14	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (0[,]1) ⊆ ℂ)
16	12	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐽 ∈ (TopOn‘ℂ))
17	16	cnmptid 23564	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑡 ∈ ℂ ↦ 𝑡) ∈ (𝐽 Cn 𝐽))
18	4	sselda 3937	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ ℂ)
19	16, 16, 18	cnmptc 23565	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑡 ∈ ℂ ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
20	2	mpomulcn 24774	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽)
21	20	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
22		oveq12 7362	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑡 ∧ 𝑣 = 𝑥) → (𝑢 · 𝑣) = (𝑡 · 𝑥))
23	16, 17, 19, 16, 16, 21, 22	cnmpt12 23570	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑡 ∈ ℂ ↦ (𝑡 · 𝑥)) ∈ (𝐽 Cn 𝐽))
24	23	adantrl 716	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ ℂ ↦ (𝑡 · 𝑥)) ∈ (𝐽 Cn 𝐽))
25	12	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘ℂ))
26		1cnd 11129	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℂ)
27	25, 25, 26	cnmptc 23565	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑡 ∈ ℂ ↦ 1) ∈ (𝐽 Cn 𝐽))
28	2	cncfcn1 24820	. . . . . . . . . . . . . 14 ⊢ (ℂ–cn→ℂ) = (𝐽 Cn 𝐽)
29	27, 28	eleqtrrdi 2839	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑡 ∈ ℂ ↦ 1) ∈ (ℂ–cn→ℂ))
30	25	cnmptid 23564	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑡 ∈ ℂ ↦ 𝑡) ∈ (𝐽 Cn 𝐽))
31	30, 28	eleqtrrdi 2839	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑡 ∈ ℂ ↦ 𝑡) ∈ (ℂ–cn→ℂ))
32	29, 31	subcncf 25361	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑡 ∈ ℂ ↦ (1 − 𝑡)) ∈ (ℂ–cn→ℂ))
33	32, 28	eleqtrdi 2838	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑡 ∈ ℂ ↦ (1 − 𝑡)) ∈ (𝐽 Cn 𝐽))
34	33	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ ℂ ↦ (1 − 𝑡)) ∈ (𝐽 Cn 𝐽))
35	4	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → 𝑆 ⊆ ℂ)
36		simprl 770	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → 𝑦 ∈ 𝑆)
37	35, 36	sseldd 3938	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → 𝑦 ∈ ℂ)
38	13, 13, 37	cnmptc 23565	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ ℂ ↦ 𝑦) ∈ (𝐽 Cn 𝐽))
39	20	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
40		oveq12 7362	. . . . . . . . . 10 ⊢ ((𝑢 = (1 − 𝑡) ∧ 𝑣 = 𝑦) → (𝑢 · 𝑣) = ((1 − 𝑡) · 𝑦))
41	13, 34, 38, 13, 13, 39, 40	cnmpt12 23570	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ ℂ ↦ ((1 − 𝑡) · 𝑦)) ∈ (𝐽 Cn 𝐽))
42	2	addcn 24770	. . . . . . . . . 10 ⊢ + ∈ ((𝐽 ×_t 𝐽) Cn 𝐽)
43	42	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → + ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
44	13, 24, 41, 43	cnmpt12f 23569	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ ℂ ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ∈ (𝐽 Cn 𝐽))
45	11, 13, 15, 44	cnmpt1res 23579	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ∈ (II Cn 𝐽))
46		cvxpconn.2	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑡 ∈ (0[,]1))) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆)
47	46	3exp2 1355	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ 𝑆 → (𝑦 ∈ 𝑆 → (𝑡 ∈ (0[,]1) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆))))
48	47	com23 86	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦 ∈ 𝑆 → (𝑥 ∈ 𝑆 → (𝑡 ∈ (0[,]1) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆))))
49	48	imp42 426	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) ∧ 𝑡 ∈ (0[,]1)) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆)
50	49	fmpttd 7053	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))):(0[,]1)⟶𝑆)
51	50	frnd 6664	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ran (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ⊆ 𝑆)
52		cnrest2 23189	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ ran (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ⊆ 𝑆 ∧ 𝑆 ⊆ ℂ) → ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ∈ (II Cn 𝐽) ↔ (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ∈ (II Cn (𝐽 ↾_t 𝑆))))
53	12, 51, 35, 52	mp3an2i 1468	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ∈ (II Cn 𝐽) ↔ (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ∈ (II Cn (𝐽 ↾_t 𝑆))))
54	45, 53	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ∈ (II Cn (𝐽 ↾_t 𝑆)))
55	1	oveq2i 7364	. . . . . 6 ⊢ (II Cn 𝐾) = (II Cn (𝐽 ↾_t 𝑆))
56	54, 55	eleqtrrdi 2839	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ∈ (II Cn 𝐾))
57		0elunit 13390	. . . . . . 7 ⊢ 0 ∈ (0[,]1)
58		oveq1 7360	. . . . . . . . 9 ⊢ (𝑡 = 0 → (𝑡 · 𝑥) = (0 · 𝑥))
59		oveq2 7361	. . . . . . . . . . 11 ⊢ (𝑡 = 0 → (1 − 𝑡) = (1 − 0))
60		1m0e1 12262	. . . . . . . . . . 11 ⊢ (1 − 0) = 1
61	59, 60	eqtrdi 2780	. . . . . . . . . 10 ⊢ (𝑡 = 0 → (1 − 𝑡) = 1)
62	61	oveq1d 7368	. . . . . . . . 9 ⊢ (𝑡 = 0 → ((1 − 𝑡) · 𝑦) = (1 · 𝑦))
63	58, 62	oveq12d 7371	. . . . . . . 8 ⊢ (𝑡 = 0 → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) = ((0 · 𝑥) + (1 · 𝑦)))
64		eqid 2729	. . . . . . . 8 ⊢ (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) = (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))
65		ovex 7386	. . . . . . . 8 ⊢ ((0 · 𝑥) + (1 · 𝑦)) ∈ V
66	63, 64, 65	fvmpt 6934	. . . . . . 7 ⊢ (0 ∈ (0[,]1) → ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘0) = ((0 · 𝑥) + (1 · 𝑦)))
67	57, 66	ax-mp 5	. . . . . 6 ⊢ ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘0) = ((0 · 𝑥) + (1 · 𝑦))
68	18	adantrl 716	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → 𝑥 ∈ ℂ)
69	68	mul02d 11332	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (0 · 𝑥) = 0)
70	37	mullidd 11152	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (1 · 𝑦) = 𝑦)
71	69, 70	oveq12d 7371	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ((0 · 𝑥) + (1 · 𝑦)) = (0 + 𝑦))
72	37	addlidd 11335	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (0 + 𝑦) = 𝑦)
73	71, 72	eqtrd 2764	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ((0 · 𝑥) + (1 · 𝑦)) = 𝑦)
74	67, 73	eqtrid 2776	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘0) = 𝑦)
75		1elunit 13391	. . . . . . 7 ⊢ 1 ∈ (0[,]1)
76		oveq1 7360	. . . . . . . . 9 ⊢ (𝑡 = 1 → (𝑡 · 𝑥) = (1 · 𝑥))
77		oveq2 7361	. . . . . . . . . . 11 ⊢ (𝑡 = 1 → (1 − 𝑡) = (1 − 1))
78		1m1e0 12218	. . . . . . . . . . 11 ⊢ (1 − 1) = 0
79	77, 78	eqtrdi 2780	. . . . . . . . . 10 ⊢ (𝑡 = 1 → (1 − 𝑡) = 0)
80	79	oveq1d 7368	. . . . . . . . 9 ⊢ (𝑡 = 1 → ((1 − 𝑡) · 𝑦) = (0 · 𝑦))
81	76, 80	oveq12d 7371	. . . . . . . 8 ⊢ (𝑡 = 1 → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) = ((1 · 𝑥) + (0 · 𝑦)))
82		ovex 7386	. . . . . . . 8 ⊢ ((1 · 𝑥) + (0 · 𝑦)) ∈ V
83	81, 64, 82	fvmpt 6934	. . . . . . 7 ⊢ (1 ∈ (0[,]1) → ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘1) = ((1 · 𝑥) + (0 · 𝑦)))
84	75, 83	ax-mp 5	. . . . . 6 ⊢ ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘1) = ((1 · 𝑥) + (0 · 𝑦))
85	68	mullidd 11152	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (1 · 𝑥) = 𝑥)
86	37	mul02d 11332	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (0 · 𝑦) = 0)
87	85, 86	oveq12d 7371	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ((1 · 𝑥) + (0 · 𝑦)) = (𝑥 + 0))
88	68	addridd 11334	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑥 + 0) = 𝑥)
89	87, 88	eqtrd 2764	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ((1 · 𝑥) + (0 · 𝑦)) = 𝑥)
90	84, 89	eqtrid 2776	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘1) = 𝑥)
91		fveq1 6825	. . . . . . . 8 ⊢ (𝑓 = (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) → (𝑓‘0) = ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘0))
92	91	eqeq1d 2731	. . . . . . 7 ⊢ (𝑓 = (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) → ((𝑓‘0) = 𝑦 ↔ ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘0) = 𝑦))
93		fveq1 6825	. . . . . . . 8 ⊢ (𝑓 = (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) → (𝑓‘1) = ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘1))
94	93	eqeq1d 2731	. . . . . . 7 ⊢ (𝑓 = (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) → ((𝑓‘1) = 𝑥 ↔ ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘1) = 𝑥))
95	92, 94	anbi12d 632	. . . . . 6 ⊢ (𝑓 = (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) → (((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑥) ↔ (((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘0) = 𝑦 ∧ ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘1) = 𝑥)))
96	95	rspcev 3579	. . . . 5 ⊢ (((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ∈ (II Cn 𝐾) ∧ (((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘0) = 𝑦 ∧ ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘1) = 𝑥)) → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑥))
97	56, 74, 90, 96	syl12anc 836	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑥))
98	97	ralrimivva 3172	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝑆 ∀𝑥 ∈ 𝑆 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑥))
99		resttopon 23064	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → (𝐽 ↾_t 𝑆) ∈ (TopOn‘𝑆))
100	12, 4, 99	sylancr 587	. . . . . 6 ⊢ (𝜑 → (𝐽 ↾_t 𝑆) ∈ (TopOn‘𝑆))
101	1, 100	eqeltrid 2832	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑆))
102		toponuni 22817	. . . . 5 ⊢ (𝐾 ∈ (TopOn‘𝑆) → 𝑆 = ∪ 𝐾)
103	101, 102	syl 17	. . . 4 ⊢ (𝜑 → 𝑆 = ∪ 𝐾)
104	103	raleqdv 3290	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ 𝑆 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑥) ↔ ∀𝑥 ∈ ∪ 𝐾∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑥)))
105	103, 104	raleqbidv 3310	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ 𝑆 ∀𝑥 ∈ 𝑆 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑥) ↔ ∀𝑦 ∈ ∪ 𝐾∀𝑥 ∈ ∪ 𝐾∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑥)))
106	98, 105	mpbid 232	. 2 ⊢ (𝜑 → ∀𝑦 ∈ ∪ 𝐾∀𝑥 ∈ ∪ 𝐾∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑥))
107		eqid 2729	. . 3 ⊢ ∪ 𝐾 = ∪ 𝐾
108	107	ispconn 35195	. 2 ⊢ (𝐾 ∈ PConn ↔ (𝐾 ∈ Top ∧ ∀𝑦 ∈ ∪ 𝐾∀𝑥 ∈ ∪ 𝐾∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑥)))
109	10, 106, 108	sylanbrc 583	1 ⊢ (𝜑 → 𝐾 ∈ PConn)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 Vcvv 3438 ⊆ wss 3905 ∪ cuni 4861 ↦ cmpt 5176 ran crn 5624 ‘cfv 6486 (class class class)co 7353 ∈ cmpo 7355 ℂcc 11026 0cc0 11028 1c1 11029 + caddc 11031 · cmul 11033 − cmin 11365 [,]cicc 13269 ↾_t crest 17342 TopOpenctopn 17343 ℂ_fldccnfld 21279 Topctop 22796 TopOnctopon 22813 Cn ccn 23127 ×_t ctx 23463 IIcii 24784 –cn→ccncf 24785 PConncpconn 35191

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 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107

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 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8632 df-map 8762 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9271 df-fi 9320 df-sup 9351 df-inf 9352 df-oi 9421 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12610 df-uz 12754 df-q 12868 df-rp 12912 df-xneg 13032 df-xadd 13033 df-xmul 13034 df-icc 13273 df-fz 13429 df-fzo 13576 df-seq 13927 df-exp 13987 df-hash 14256 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-rest 17344 df-topn 17345 df-0g 17363 df-gsum 17364 df-topgen 17365 df-pt 17366 df-prds 17369 df-xrs 17424 df-qtop 17429 df-imas 17430 df-xps 17432 df-mre 17506 df-mrc 17507 df-acs 17509 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-submnd 18676 df-mulg 18965 df-cntz 19214 df-cmn 19679 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-cnfld 21280 df-top 22797 df-topon 22814 df-topsp 22836 df-bases 22849 df-cn 23130 df-cnp 23131 df-tx 23465 df-hmeo 23658 df-xms 24224 df-ms 24225 df-tms 24226 df-ii 24786 df-cncf 24787 df-pconn 35193

This theorem is referenced by: cvxsconn 35215

W3C validator