cvxpconn - Mathbox for Mario Carneiro

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Theorem cvxpconn 34788

Description: A convex subset of the complex numbers is path-connected. (Contributed by Mario Carneiro, 12-Feb-2015.) Avoid ax-mulf 11210. (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 11211	. . . . 5 ⊢ ℂ ∈ V
6		ssexg 5317	. . . . 5 ⊢ ((𝑆 ⊆ ℂ ∧ ℂ ∈ V) → 𝑆 ∈ V)
7	4, 5, 6	sylancl 585	. . . 4 ⊢ (𝜑 → 𝑆 ∈ V)
8		resttop 23051	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ V) → (𝐽 ↾_t 𝑆) ∈ Top)
9	3, 7, 8	sylancr 586	. . 3 ⊢ (𝜑 → (𝐽 ↾_t 𝑆) ∈ Top)
10	1, 9	eqeltrid 2832	. 2 ⊢ (𝜑 → 𝐾 ∈ Top)
11	2	dfii3 24790	. . . . . . . 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 13501	. . . . . . . . 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 23552	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑡 ∈ ℂ ↦ 𝑡) ∈ (𝐽 Cn 𝐽))
18	4	sselda 3978	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ ℂ)
19	16, 16, 18	cnmptc 23553	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑡 ∈ ℂ ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
20	2	mpomulcn 24772	. . . . . . . . . . . 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 7423	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑡 ∧ 𝑣 = 𝑥) → (𝑢 · 𝑣) = (𝑡 · 𝑥))
23	16, 17, 19, 16, 16, 21, 22	cnmpt12 23558	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑡 ∈ ℂ ↦ (𝑡 · 𝑥)) ∈ (𝐽 Cn 𝐽))
24	23	adantrl 715	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ ℂ ↦ (𝑡 · 𝑥)) ∈ (𝐽 Cn 𝐽))
25	12	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘ℂ))
26		1cnd 11231	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℂ)
27	25, 25, 26	cnmptc 23553	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑡 ∈ ℂ ↦ 1) ∈ (𝐽 Cn 𝐽))
28	2	cncfcn1 24818	. . . . . . . . . . . . . 14 ⊢ (ℂ–cn→ℂ) = (𝐽 Cn 𝐽)
29	27, 28	eleqtrrdi 2839	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑡 ∈ ℂ ↦ 1) ∈ (ℂ–cn→ℂ))
30	25	cnmptid 23552	. . . . . . . . . . . . . 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 25360	. . . . . . . . . . . 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 3979	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → 𝑦 ∈ ℂ)
38	13, 13, 37	cnmptc 23553	. . . . . . . . . 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 7423	. . . . . . . . . 10 ⊢ ((𝑢 = (1 − 𝑡) ∧ 𝑣 = 𝑦) → (𝑢 · 𝑣) = ((1 − 𝑡) · 𝑦))
41	13, 34, 38, 13, 13, 39, 40	cnmpt12 23558	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ ℂ ↦ ((1 − 𝑡) · 𝑦)) ∈ (𝐽 Cn 𝐽))
42	2	addcn 24768	. . . . . . . . . 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 23557	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ ℂ ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ∈ (𝐽 Cn 𝐽))
45	11, 13, 15, 44	cnmpt1res 23567	. . . . . . 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 1352	. . . . . . . . . . . 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 7119	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))):(0[,]1)⟶𝑆)
51	50	frnd 6724	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ran (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ⊆ 𝑆)
52		cnrest2 23177	. . . . . . . 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 1463	. . . . . . 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 231	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ∈ (II Cn (𝐽 ↾_t 𝑆)))
55	1	oveq2i 7425	. . . . . 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 13470	. . . . . . 7 ⊢ 0 ∈ (0[,]1)
58		oveq1 7421	. . . . . . . . 9 ⊢ (𝑡 = 0 → (𝑡 · 𝑥) = (0 · 𝑥))
59		oveq2 7422	. . . . . . . . . . 11 ⊢ (𝑡 = 0 → (1 − 𝑡) = (1 − 0))
60		1m0e1 12355	. . . . . . . . . . 11 ⊢ (1 − 0) = 1
61	59, 60	eqtrdi 2783	. . . . . . . . . 10 ⊢ (𝑡 = 0 → (1 − 𝑡) = 1)
62	61	oveq1d 7429	. . . . . . . . 9 ⊢ (𝑡 = 0 → ((1 − 𝑡) · 𝑦) = (1 · 𝑦))
63	58, 62	oveq12d 7432	. . . . . . . 8 ⊢ (𝑡 = 0 → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) = ((0 · 𝑥) + (1 · 𝑦)))
64		eqid 2727	. . . . . . . 8 ⊢ (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) = (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))
65		ovex 7447	. . . . . . . 8 ⊢ ((0 · 𝑥) + (1 · 𝑦)) ∈ V
66	63, 64, 65	fvmpt 6999	. . . . . . 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 715	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → 𝑥 ∈ ℂ)
69	68	mul02d 11434	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (0 · 𝑥) = 0)
70	37	mullidd 11254	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (1 · 𝑦) = 𝑦)
71	69, 70	oveq12d 7432	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ((0 · 𝑥) + (1 · 𝑦)) = (0 + 𝑦))
72	37	addlidd 11437	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (0 + 𝑦) = 𝑦)
73	71, 72	eqtrd 2767	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ((0 · 𝑥) + (1 · 𝑦)) = 𝑦)
74	67, 73	eqtrid 2779	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘0) = 𝑦)
75		1elunit 13471	. . . . . . 7 ⊢ 1 ∈ (0[,]1)
76		oveq1 7421	. . . . . . . . 9 ⊢ (𝑡 = 1 → (𝑡 · 𝑥) = (1 · 𝑥))
77		oveq2 7422	. . . . . . . . . . 11 ⊢ (𝑡 = 1 → (1 − 𝑡) = (1 − 1))
78		1m1e0 12306	. . . . . . . . . . 11 ⊢ (1 − 1) = 0
79	77, 78	eqtrdi 2783	. . . . . . . . . 10 ⊢ (𝑡 = 1 → (1 − 𝑡) = 0)
80	79	oveq1d 7429	. . . . . . . . 9 ⊢ (𝑡 = 1 → ((1 − 𝑡) · 𝑦) = (0 · 𝑦))
81	76, 80	oveq12d 7432	. . . . . . . 8 ⊢ (𝑡 = 1 → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) = ((1 · 𝑥) + (0 · 𝑦)))
82		ovex 7447	. . . . . . . 8 ⊢ ((1 · 𝑥) + (0 · 𝑦)) ∈ V
83	81, 64, 82	fvmpt 6999	. . . . . . 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 11254	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (1 · 𝑥) = 𝑥)
86	37	mul02d 11434	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (0 · 𝑦) = 0)
87	85, 86	oveq12d 7432	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ((1 · 𝑥) + (0 · 𝑦)) = (𝑥 + 0))
88	68	addridd 11436	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑥 + 0) = 𝑥)
89	87, 88	eqtrd 2767	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ((1 · 𝑥) + (0 · 𝑦)) = 𝑥)
90	84, 89	eqtrid 2779	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘1) = 𝑥)
91		fveq1 6890	. . . . . . . 8 ⊢ (𝑓 = (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) → (𝑓‘0) = ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘0))
92	91	eqeq1d 2729	. . . . . . 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 6890	. . . . . . . 8 ⊢ (𝑓 = (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) → (𝑓‘1) = ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘1))
94	93	eqeq1d 2729	. . . . . . 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 630	. . . . . 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 3607	. . . . 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 3195	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝑆 ∀𝑥 ∈ 𝑆 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑥))
99		resttopon 23052	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → (𝐽 ↾_t 𝑆) ∈ (TopOn‘𝑆))
100	12, 4, 99	sylancr 586	. . . . . 6 ⊢ (𝜑 → (𝐽 ↾_t 𝑆) ∈ (TopOn‘𝑆))
101	1, 100	eqeltrid 2832	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑆))
102		toponuni 22803	. . . . 5 ⊢ (𝐾 ∈ (TopOn‘𝑆) → 𝑆 = ∪ 𝐾)
103	101, 102	syl 17	. . . 4 ⊢ (𝜑 → 𝑆 = ∪ 𝐾)
104	103	raleqdv 3320	. . . 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 3337	. . 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 231	. 2 ⊢ (𝜑 → ∀𝑦 ∈ ∪ 𝐾∀𝑥 ∈ ∪ 𝐾∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑥))
107		eqid 2727	. . 3 ⊢ ∪ 𝐾 = ∪ 𝐾
108	107	ispconn 34769	. 2 ⊢ (𝐾 ∈ PConn ↔ (𝐾 ∈ Top ∧ ∀𝑦 ∈ ∪ 𝐾∀𝑥 ∈ ∪ 𝐾∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑥)))
109	10, 106, 108	sylanbrc 582	1 ⊢ (𝜑 → 𝐾 ∈ PConn)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∀wral 3056 ∃wrex 3065 Vcvv 3469 ⊆ wss 3944 ∪ cuni 4903 ↦ cmpt 5225 ran crn 5673 ‘cfv 6542 (class class class)co 7414 ∈ cmpo 7416 ℂcc 11128 0cc0 11130 1c1 11131 + caddc 11133 · cmul 11135 − cmin 11466 [,]cicc 13351 ↾_t crest 17393 TopOpenctopn 17394 ℂ_fldccnfld 21266 Topctop 22782 TopOnctopon 22799 Cn ccn 23115 ×_t ctx 23451 IIcii 24782 –cn→ccncf 24783 PConncpconn 34765

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208 ax-addf 11209

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-map 8838 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9378 df-fi 9426 df-sup 9457 df-inf 9458 df-oi 9525 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-z 12581 df-dec 12700 df-uz 12845 df-q 12955 df-rp 12999 df-xneg 13116 df-xadd 13117 df-xmul 13118 df-icc 13355 df-fz 13509 df-fzo 13652 df-seq 13991 df-exp 14051 df-hash 14314 df-cj 15070 df-re 15071 df-im 15072 df-sqrt 15206 df-abs 15207 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-starv 17239 df-sca 17240 df-vsca 17241 df-ip 17242 df-tset 17243 df-ple 17244 df-ds 17246 df-unif 17247 df-hom 17248 df-cco 17249 df-rest 17395 df-topn 17396 df-0g 17414 df-gsum 17415 df-topgen 17416 df-pt 17417 df-prds 17420 df-xrs 17475 df-qtop 17480 df-imas 17481 df-xps 17483 df-mre 17557 df-mrc 17558 df-acs 17560 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-submnd 18732 df-mulg 19015 df-cntz 19259 df-cmn 19728 df-psmet 21258 df-xmet 21259 df-met 21260 df-bl 21261 df-mopn 21262 df-cnfld 21267 df-top 22783 df-topon 22800 df-topsp 22822 df-bases 22836 df-cn 23118 df-cnp 23119 df-tx 23453 df-hmeo 23646 df-xms 24213 df-ms 24214 df-tms 24215 df-ii 24784 df-cncf 24785 df-pconn 34767

This theorem is referenced by: cvxsconn 34789

W3C validator