cvxpconn - Mathbox for Mario Carneiro

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Theorem cvxpconn 35755

Description: A convex subset of the complex numbers is path-connected. (Contributed by Mario Carneiro, 12-Feb-2015.) Avoid ax-mulf 11198. (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 24977	. . . 4 ⊢ 𝐽 ∈ Top
4		cvxpconn.1	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
5		cnex 11199	. . . . 5 ⊢ ℂ ∈ V
6		ssexg 5295	. . . . 5 ⊢ ((𝑆 ⊆ ℂ ∧ ℂ ∈ V) → 𝑆 ∈ V)
7	4, 5, 6	sylancl 598	. . . 4 ⊢ (𝜑 → 𝑆 ∈ V)
8		resttop 23354	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ V) → (𝐽 ↾_t 𝑆) ∈ Top)
9	3, 7, 8	sylancr 599	. . 3 ⊢ (𝜑 → (𝐽 ↾_t 𝑆) ∈ Top)
10	1, 9	eqeltrid 2870	. 2 ⊢ (𝜑 → 𝐾 ∈ Top)
11	2	dfii3 25079	. . . . . . . 8 ⊢ II = (𝐽 ↾_t (0[,]1))
12	2	cnfldtopon 24976	. . . . . . . . 9 ⊢ 𝐽 ∈ (TopOn‘ℂ)
13	12	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → 𝐽 ∈ (TopOn‘ℂ))
14		unitsscn 13545	. . . . . . . . 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 23855	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑡 ∈ ℂ ↦ 𝑡) ∈ (𝐽 Cn 𝐽))
18	4	sselda 3940	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ ℂ)
19	16, 16, 18	cnmptc 23856	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑡 ∈ ℂ ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
20	2	mpomulcn 25063	. . . . . . . . . . . 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 7432	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑡 ∧ 𝑣 = 𝑥) → (𝑢 · 𝑣) = (𝑡 · 𝑥))
23	16, 17, 19, 16, 16, 21, 22	cnmpt12 23861	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑡 ∈ ℂ ↦ (𝑡 · 𝑥)) ∈ (𝐽 Cn 𝐽))
24	23	adantrl 729	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ ℂ ↦ (𝑡 · 𝑥)) ∈ (𝐽 Cn 𝐽))
25	12	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘ℂ))
26		1cnd 11220	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℂ)
27	25, 25, 26	cnmptc 23856	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑡 ∈ ℂ ↦ 1) ∈ (𝐽 Cn 𝐽))
28	2	cncfcn1 25107	. . . . . . . . . . . . . 14 ⊢ (ℂ–cn→ℂ) = (𝐽 Cn 𝐽)
29	27, 28	eleqtrrdi 2877	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑡 ∈ ℂ ↦ 1) ∈ (ℂ–cn→ℂ))
30	25	cnmptid 23855	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑡 ∈ ℂ ↦ 𝑡) ∈ (𝐽 Cn 𝐽))
31	30, 28	eleqtrrdi 2877	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑡 ∈ ℂ ↦ 𝑡) ∈ (ℂ–cn→ℂ))
32	29, 31	subcncf 25641	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑡 ∈ ℂ ↦ (1 − 𝑡)) ∈ (ℂ–cn→ℂ))
33	32, 28	eleqtrdi 2876	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑡 ∈ ℂ ↦ (1 − 𝑡)) ∈ (𝐽 Cn 𝐽))
34	33	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ ℂ ↦ (1 − 𝑡)) ∈ (𝐽 Cn 𝐽))
35	4	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → 𝑆 ⊆ ℂ)
36		simprl 783	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → 𝑦 ∈ 𝑆)
37	35, 36	sseldd 3941	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → 𝑦 ∈ ℂ)
38	13, 13, 37	cnmptc 23856	. . . . . . . . . 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 7432	. . . . . . . . . 10 ⊢ ((𝑢 = (1 − 𝑡) ∧ 𝑣 = 𝑦) → (𝑢 · 𝑣) = ((1 − 𝑡) · 𝑦))
41	13, 34, 38, 13, 13, 39, 40	cnmpt12 23861	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ ℂ ↦ ((1 − 𝑡) · 𝑦)) ∈ (𝐽 Cn 𝐽))
42	2	addcn 25060	. . . . . . . . . 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 23860	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ ℂ ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ∈ (𝐽 Cn 𝐽))
45	11, 13, 15, 44	cnmpt1res 23870	. . . . . . 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 1373	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ 𝑆 → (𝑦 ∈ 𝑆 → (𝑡 ∈ (0[,]1) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆))))
48	47	com23 87	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦 ∈ 𝑆 → (𝑥 ∈ 𝑆 → (𝑡 ∈ (0[,]1) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆))))
49	48	imp42 432	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) ∧ 𝑡 ∈ (0[,]1)) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆)
50	49	fmpttd 7117	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))):(0[,]1)⟶𝑆)
51	50	frnd 6721	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ran (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ⊆ 𝑆)
52		cnrest2 23480	. . . . . . . 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 1495	. . . . . . 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 235	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ∈ (II Cn (𝐽 ↾_t 𝑆)))
55	1	oveq2i 7434	. . . . . 6 ⊢ (II Cn 𝐾) = (II Cn (𝐽 ↾_t 𝑆))
56	54, 55	eleqtrrdi 2877	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ∈ (II Cn 𝐾))
57		0elunit 13514	. . . . . . 7 ⊢ 0 ∈ (0[,]1)
58		oveq1 7430	. . . . . . . . 9 ⊢ (𝑡 = 0 → (𝑡 · 𝑥) = (0 · 𝑥))
59		oveq2 7431	. . . . . . . . . . 11 ⊢ (𝑡 = 0 → (1 − 𝑡) = (1 − 0))
60		1m0e1 12378	. . . . . . . . . . 11 ⊢ (1 − 0) = 1
61	59, 60	eqtrdi 2817	. . . . . . . . . 10 ⊢ (𝑡 = 0 → (1 − 𝑡) = 1)
62	61	oveq1d 7438	. . . . . . . . 9 ⊢ (𝑡 = 0 → ((1 − 𝑡) · 𝑦) = (1 · 𝑦))
63	58, 62	oveq12d 7441	. . . . . . . 8 ⊢ (𝑡 = 0 → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) = ((0 · 𝑥) + (1 · 𝑦)))
64		eqid 2766	. . . . . . . 8 ⊢ (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) = (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))
65		ovex 7456	. . . . . . . 8 ⊢ ((0 · 𝑥) + (1 · 𝑦)) ∈ V
66	63, 64, 65	fvmpt 6996	. . . . . . 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 729	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → 𝑥 ∈ ℂ)
69	68	mul02d 11426	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (0 · 𝑥) = 0)
70	37	mullidd 11245	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (1 · 𝑦) = 𝑦)
71	69, 70	oveq12d 7441	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ((0 · 𝑥) + (1 · 𝑦)) = (0 + 𝑦))
72	37	addlidd 11429	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (0 + 𝑦) = 𝑦)
73	71, 72	eqtrd 2801	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ((0 · 𝑥) + (1 · 𝑦)) = 𝑦)
74	67, 73	eqtrid 2813	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘0) = 𝑦)
75		1elunit 13515	. . . . . . 7 ⊢ 1 ∈ (0[,]1)
76		oveq1 7430	. . . . . . . . 9 ⊢ (𝑡 = 1 → (𝑡 · 𝑥) = (1 · 𝑥))
77		oveq2 7431	. . . . . . . . . . 11 ⊢ (𝑡 = 1 → (1 − 𝑡) = (1 − 1))
78		1m1e0 12331	. . . . . . . . . . 11 ⊢ (1 − 1) = 0
79	77, 78	eqtrdi 2817	. . . . . . . . . 10 ⊢ (𝑡 = 1 → (1 − 𝑡) = 0)
80	79	oveq1d 7438	. . . . . . . . 9 ⊢ (𝑡 = 1 → ((1 − 𝑡) · 𝑦) = (0 · 𝑦))
81	76, 80	oveq12d 7441	. . . . . . . 8 ⊢ (𝑡 = 1 → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) = ((1 · 𝑥) + (0 · 𝑦)))
82		ovex 7456	. . . . . . . 8 ⊢ ((1 · 𝑥) + (0 · 𝑦)) ∈ V
83	81, 64, 82	fvmpt 6996	. . . . . . 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 11245	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (1 · 𝑥) = 𝑥)
86	37	mul02d 11426	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (0 · 𝑦) = 0)
87	85, 86	oveq12d 7441	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ((1 · 𝑥) + (0 · 𝑦)) = (𝑥 + 0))
88	68	addridd 11428	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑥 + 0) = 𝑥)
89	87, 88	eqtrd 2801	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ((1 · 𝑥) + (0 · 𝑦)) = 𝑥)
90	84, 89	eqtrid 2813	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘1) = 𝑥)
91		fveq1 6887	. . . . . . . 8 ⊢ (𝑓 = (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) → (𝑓‘0) = ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘0))
92	91	eqeq1d 2768	. . . . . . 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 6887	. . . . . . . 8 ⊢ (𝑓 = (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) → (𝑓‘1) = ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘1))
94	93	eqeq1d 2768	. . . . . . 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 644	. . . . . 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 3584	. . . . 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 850	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑥))
98	97	ralrimivva 3211	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝑆 ∀𝑥 ∈ 𝑆 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑥))
99		resttopon 23355	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → (𝐽 ↾_t 𝑆) ∈ (TopOn‘𝑆))
100	12, 4, 99	sylancr 599	. . . . . 6 ⊢ (𝜑 → (𝐽 ↾_t 𝑆) ∈ (TopOn‘𝑆))
101	1, 100	eqeltrid 2870	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑆))
102		toponuni 23108	. . . . 5 ⊢ (𝐾 ∈ (TopOn‘𝑆) → 𝑆 = ∪ 𝐾)
103	101, 102	syl 18	. . . 4 ⊢ (𝜑 → 𝑆 = ∪ 𝐾)
104	103	raleqdv 3326	. . . 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 3341	. . 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 235	. 2 ⊢ (𝜑 → ∀𝑦 ∈ ∪ 𝐾∀𝑥 ∈ ∪ 𝐾∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑥))
107		eqid 2766	. . 3 ⊢ ∪ 𝐾 = ∪ 𝐾
108	107	ispconn 35736	. 2 ⊢ (𝐾 ∈ PConn ↔ (𝐾 ∈ Top ∧ ∀𝑦 ∈ ∪ 𝐾∀𝑥 ∈ ∪ 𝐾∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑥)))
109	10, 106, 108	sylanbrc 595	1 ⊢ (𝜑 → 𝐾 ∈ PConn)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ∀wral 3082 ∃wrex 3092 Vcvv 3458 ⊆ wss 3908 ∪ cuni 4877 ↦ cmpt 5197 ran crn 5667 ‘cfv 6543 (class class class)co 7423 ∈ cmpo 7425 ℂcc 11116 0cc0 11118 1c1 11119 + caddc 11121 · cmul 11123 − cmin 11459 [,]cicc 13393 ↾_t crest 17498 TopOpenctopn 17499 ℂ_fldccnfld 21559 Topctop 23087 TopOnctopon 23104 Cn ccn 23418 ×_t ctx 23754 IIcii 25071 –cn→ccncf 25072 PConncpconn 35732

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196 ax-addf 11197

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-iin 4964 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-se 5620 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-of 7687 df-om 7872 df-1st 7995 df-2nd 7996 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8703 df-map 8835 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-fi 9381 df-sup 9412 df-inf 9413 df-oi 9482 df-card 9944 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12523 df-z 12610 df-dec 12730 df-uz 12881 df-q 12991 df-rp 13035 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-icc 13397 df-fz 13554 df-fzo 13702 df-seq 14058 df-exp 14118 df-hash 14387 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-struct 17232 df-sets 17249 df-slot 17267 df-ndx 17279 df-base 17295 df-ress 17316 df-plusg 17348 df-mulr 17349 df-starv 17350 df-sca 17351 df-vsca 17352 df-ip 17353 df-tset 17354 df-ple 17355 df-ds 17357 df-unif 17358 df-hom 17359 df-cco 17360 df-rest 17500 df-topn 17501 df-0g 17519 df-gsum 17520 df-topgen 17521 df-pt 17522 df-prds 17525 df-xrs 17581 df-qtop 17586 df-imas 17587 df-xps 17589 df-mre 17663 df-mrc 17664 df-acs 17666 df-mgm 18723 df-sgrp 18806 df-mnd 18822 df-submnd 18873 df-mulg 19165 df-cntz 19418 df-cmn 19883 df-psmet 21551 df-xmet 21552 df-met 21553 df-bl 21554 df-mopn 21555 df-cnfld 21560 df-top 23088 df-topon 23105 df-topsp 23127 df-bases 23140 df-cn 23421 df-cnp 23422 df-tx 23756 df-hmeo 23949 df-xms 24514 df-ms 24515 df-tms 24516 df-ii 25073 df-cncf 25074 df-pconn 35734

This theorem is used by: cvxsconn 35756

W3C validator