cvxpconn - Mathbox for Mario Carneiro

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Theorem cvxpconn 35440

Description: A convex subset of the complex numbers is path-connected. (Contributed by Mario Carneiro, 12-Feb-2015.) Avoid ax-mulf 11109. (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 24758	. . . 4 ⊢ 𝐽 ∈ Top
4		cvxpconn.1	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
5		cnex 11110	. . . . 5 ⊢ ℂ ∈ V
6		ssexg 5260	. . . . 5 ⊢ ((𝑆 ⊆ ℂ ∧ ℂ ∈ V) → 𝑆 ∈ V)
7	4, 5, 6	sylancl 587	. . . 4 ⊢ (𝜑 → 𝑆 ∈ V)
8		resttop 23135	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ V) → (𝐽 ↾_t 𝑆) ∈ Top)
9	3, 7, 8	sylancr 588	. . 3 ⊢ (𝜑 → (𝐽 ↾_t 𝑆) ∈ Top)
10	1, 9	eqeltrid 2841	. 2 ⊢ (𝜑 → 𝐾 ∈ Top)
11	2	dfii3 24860	. . . . . . . 8 ⊢ II = (𝐽 ↾_t (0[,]1))
12	2	cnfldtopon 24757	. . . . . . . . 9 ⊢ 𝐽 ∈ (TopOn‘ℂ)
13	12	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → 𝐽 ∈ (TopOn‘ℂ))
14		unitsscn 13444	. . . . . . . . 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 23636	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑡 ∈ ℂ ↦ 𝑡) ∈ (𝐽 Cn 𝐽))
18	4	sselda 3922	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ ℂ)
19	16, 16, 18	cnmptc 23637	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑡 ∈ ℂ ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
20	2	mpomulcn 24844	. . . . . . . . . . . 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 7369	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑡 ∧ 𝑣 = 𝑥) → (𝑢 · 𝑣) = (𝑡 · 𝑥))
23	16, 17, 19, 16, 16, 21, 22	cnmpt12 23642	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑡 ∈ ℂ ↦ (𝑡 · 𝑥)) ∈ (𝐽 Cn 𝐽))
24	23	adantrl 717	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ ℂ ↦ (𝑡 · 𝑥)) ∈ (𝐽 Cn 𝐽))
25	12	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘ℂ))
26		1cnd 11130	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℂ)
27	25, 25, 26	cnmptc 23637	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑡 ∈ ℂ ↦ 1) ∈ (𝐽 Cn 𝐽))
28	2	cncfcn1 24888	. . . . . . . . . . . . . 14 ⊢ (ℂ–cn→ℂ) = (𝐽 Cn 𝐽)
29	27, 28	eleqtrrdi 2848	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑡 ∈ ℂ ↦ 1) ∈ (ℂ–cn→ℂ))
30	25	cnmptid 23636	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑡 ∈ ℂ ↦ 𝑡) ∈ (𝐽 Cn 𝐽))
31	30, 28	eleqtrrdi 2848	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑡 ∈ ℂ ↦ 𝑡) ∈ (ℂ–cn→ℂ))
32	29, 31	subcncf 25422	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑡 ∈ ℂ ↦ (1 − 𝑡)) ∈ (ℂ–cn→ℂ))
33	32, 28	eleqtrdi 2847	. . . . . . . . . . 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 771	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → 𝑦 ∈ 𝑆)
37	35, 36	sseldd 3923	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → 𝑦 ∈ ℂ)
38	13, 13, 37	cnmptc 23637	. . . . . . . . . 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 7369	. . . . . . . . . 10 ⊢ ((𝑢 = (1 − 𝑡) ∧ 𝑣 = 𝑦) → (𝑢 · 𝑣) = ((1 − 𝑡) · 𝑦))
41	13, 34, 38, 13, 13, 39, 40	cnmpt12 23642	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ ℂ ↦ ((1 − 𝑡) · 𝑦)) ∈ (𝐽 Cn 𝐽))
42	2	addcn 24841	. . . . . . . . . 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 23641	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ ℂ ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ∈ (𝐽 Cn 𝐽))
45	11, 13, 15, 44	cnmpt1res 23651	. . . . . . 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 1356	. . . . . . . . . . . 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 7061	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))):(0[,]1)⟶𝑆)
51	50	frnd 6670	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ran (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ⊆ 𝑆)
52		cnrest2 23261	. . . . . . . 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 1469	. . . . . . 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 7371	. . . . . 6 ⊢ (II Cn 𝐾) = (II Cn (𝐽 ↾_t 𝑆))
56	54, 55	eleqtrrdi 2848	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ∈ (II Cn 𝐾))
57		0elunit 13413	. . . . . . 7 ⊢ 0 ∈ (0[,]1)
58		oveq1 7367	. . . . . . . . 9 ⊢ (𝑡 = 0 → (𝑡 · 𝑥) = (0 · 𝑥))
59		oveq2 7368	. . . . . . . . . . 11 ⊢ (𝑡 = 0 → (1 − 𝑡) = (1 − 0))
60		1m0e1 12288	. . . . . . . . . . 11 ⊢ (1 − 0) = 1
61	59, 60	eqtrdi 2788	. . . . . . . . . 10 ⊢ (𝑡 = 0 → (1 − 𝑡) = 1)
62	61	oveq1d 7375	. . . . . . . . 9 ⊢ (𝑡 = 0 → ((1 − 𝑡) · 𝑦) = (1 · 𝑦))
63	58, 62	oveq12d 7378	. . . . . . . 8 ⊢ (𝑡 = 0 → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) = ((0 · 𝑥) + (1 · 𝑦)))
64		eqid 2737	. . . . . . . 8 ⊢ (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) = (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))
65		ovex 7393	. . . . . . . 8 ⊢ ((0 · 𝑥) + (1 · 𝑦)) ∈ V
66	63, 64, 65	fvmpt 6941	. . . . . . 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 717	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → 𝑥 ∈ ℂ)
69	68	mul02d 11335	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (0 · 𝑥) = 0)
70	37	mullidd 11154	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (1 · 𝑦) = 𝑦)
71	69, 70	oveq12d 7378	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ((0 · 𝑥) + (1 · 𝑦)) = (0 + 𝑦))
72	37	addlidd 11338	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (0 + 𝑦) = 𝑦)
73	71, 72	eqtrd 2772	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ((0 · 𝑥) + (1 · 𝑦)) = 𝑦)
74	67, 73	eqtrid 2784	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘0) = 𝑦)
75		1elunit 13414	. . . . . . 7 ⊢ 1 ∈ (0[,]1)
76		oveq1 7367	. . . . . . . . 9 ⊢ (𝑡 = 1 → (𝑡 · 𝑥) = (1 · 𝑥))
77		oveq2 7368	. . . . . . . . . . 11 ⊢ (𝑡 = 1 → (1 − 𝑡) = (1 − 1))
78		1m1e0 12244	. . . . . . . . . . 11 ⊢ (1 − 1) = 0
79	77, 78	eqtrdi 2788	. . . . . . . . . 10 ⊢ (𝑡 = 1 → (1 − 𝑡) = 0)
80	79	oveq1d 7375	. . . . . . . . 9 ⊢ (𝑡 = 1 → ((1 − 𝑡) · 𝑦) = (0 · 𝑦))
81	76, 80	oveq12d 7378	. . . . . . . 8 ⊢ (𝑡 = 1 → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) = ((1 · 𝑥) + (0 · 𝑦)))
82		ovex 7393	. . . . . . . 8 ⊢ ((1 · 𝑥) + (0 · 𝑦)) ∈ V
83	81, 64, 82	fvmpt 6941	. . . . . . 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 11154	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (1 · 𝑥) = 𝑥)
86	37	mul02d 11335	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (0 · 𝑦) = 0)
87	85, 86	oveq12d 7378	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ((1 · 𝑥) + (0 · 𝑦)) = (𝑥 + 0))
88	68	addridd 11337	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑥 + 0) = 𝑥)
89	87, 88	eqtrd 2772	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ((1 · 𝑥) + (0 · 𝑦)) = 𝑥)
90	84, 89	eqtrid 2784	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘1) = 𝑥)
91		fveq1 6833	. . . . . . . 8 ⊢ (𝑓 = (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) → (𝑓‘0) = ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘0))
92	91	eqeq1d 2739	. . . . . . 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 6833	. . . . . . . 8 ⊢ (𝑓 = (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) → (𝑓‘1) = ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘1))
94	93	eqeq1d 2739	. . . . . . 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 633	. . . . . 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 3565	. . . . 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 837	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑥))
98	97	ralrimivva 3181	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝑆 ∀𝑥 ∈ 𝑆 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑥))
99		resttopon 23136	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → (𝐽 ↾_t 𝑆) ∈ (TopOn‘𝑆))
100	12, 4, 99	sylancr 588	. . . . . 6 ⊢ (𝜑 → (𝐽 ↾_t 𝑆) ∈ (TopOn‘𝑆))
101	1, 100	eqeltrid 2841	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑆))
102		toponuni 22889	. . . . 5 ⊢ (𝐾 ∈ (TopOn‘𝑆) → 𝑆 = ∪ 𝐾)
103	101, 102	syl 17	. . . 4 ⊢ (𝜑 → 𝑆 = ∪ 𝐾)
104	103	raleqdv 3296	. . . 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 3312	. . 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 2737	. . 3 ⊢ ∪ 𝐾 = ∪ 𝐾
108	107	ispconn 35421	. 2 ⊢ (𝐾 ∈ PConn ↔ (𝐾 ∈ Top ∧ ∀𝑦 ∈ ∪ 𝐾∀𝑥 ∈ ∪ 𝐾∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑥)))
109	10, 106, 108	sylanbrc 584	1 ⊢ (𝜑 → 𝐾 ∈ PConn)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 Vcvv 3430 ⊆ wss 3890 ∪ cuni 4851 ↦ cmpt 5167 ran crn 5625 ‘cfv 6492 (class class class)co 7360 ∈ cmpo 7362 ℂcc 11027 0cc0 11029 1c1 11030 + caddc 11032 · cmul 11034 − cmin 11368 [,]cicc 13292 ↾_t crest 17374 TopOpenctopn 17375 ℂ_fldccnfld 21344 Topctop 22868 TopOnctopon 22885 Cn ccn 23199 ×_t ctx 23535 IIcii 24852 –cn→ccncf 24853 PConncpconn 35417

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8104 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-er 8636 df-map 8768 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9268 df-fi 9317 df-sup 9348 df-inf 9349 df-oi 9418 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-icc 13296 df-fz 13453 df-fzo 13600 df-seq 13955 df-exp 14015 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-mulg 19035 df-cntz 19283 df-cmn 19748 df-psmet 21336 df-xmet 21337 df-met 21338 df-bl 21339 df-mopn 21340 df-cnfld 21345 df-top 22869 df-topon 22886 df-topsp 22908 df-bases 22921 df-cn 23202 df-cnp 23203 df-tx 23537 df-hmeo 23730 df-xms 24295 df-ms 24296 df-tms 24297 df-ii 24854 df-cncf 24855 df-pconn 35419

This theorem is referenced by: cvxsconn 35441

W3C validator