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Theorem cvxcl 26911

Description: Closure of a 0-1 linear combination in a convex set. (Contributed by Mario Carneiro, 21-Jun-2015.)

**Hypotheses**
Ref	Expression
cvxcl.1	⊢ (𝜑 → 𝐷 ⊆ ℝ)
cvxcl.2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥[,]𝑦) ⊆ 𝐷)

**Assertion**
Ref	Expression
cvxcl	⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) ∈ 𝐷)

Distinct variable groups: 𝑥,𝑦,𝐷 𝜑,𝑥,𝑦 𝑥,𝑋,𝑦 𝑥,𝑌,𝑦 Allowed substitution hints: 𝑇(𝑥,𝑦)

**Proof of Theorem cvxcl**
Step	Hyp	Ref	Expression
1		cvxcl.2	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥[,]𝑦) ⊆ 𝐷)
2	1	ralrimivva 3172	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥[,]𝑦) ⊆ 𝐷)
3	2	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 < 𝑌) → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥[,]𝑦) ⊆ 𝐷)
4		simpr1 1195	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑋 ∈ 𝐷)
5		simpr2 1196	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑌 ∈ 𝐷)
6		oveq1 7360	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥[,]𝑦) = (𝑋[,]𝑦))
7	6	sseq1d 3969	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝑥[,]𝑦) ⊆ 𝐷 ↔ (𝑋[,]𝑦) ⊆ 𝐷))
8		oveq2 7361	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝑋[,]𝑦) = (𝑋[,]𝑌))
9	8	sseq1d 3969	. . . . . . 7 ⊢ (𝑦 = 𝑌 → ((𝑋[,]𝑦) ⊆ 𝐷 ↔ (𝑋[,]𝑌) ⊆ 𝐷))
10	7, 9	rspc2v 3590	. . . . . 6 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷) → (∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥[,]𝑦) ⊆ 𝐷 → (𝑋[,]𝑌) ⊆ 𝐷))
11	4, 5, 10	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → (∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥[,]𝑦) ⊆ 𝐷 → (𝑋[,]𝑌) ⊆ 𝐷))
12	11	adantr 480	. . . 4 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 < 𝑌) → (∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥[,]𝑦) ⊆ 𝐷 → (𝑋[,]𝑌) ⊆ 𝐷))
13	3, 12	mpd 15	. . 3 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 < 𝑌) → (𝑋[,]𝑌) ⊆ 𝐷)
14		ax-1cn 11086	. . . . . . . 8 ⊢ 1 ∈ ℂ
15		unitssre 13420	. . . . . . . . . 10 ⊢ (0[,]1) ⊆ ℝ
16		simpr3 1197	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑇 ∈ (0[,]1))
17	15, 16	sselid 3935	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑇 ∈ ℝ)
18	17	recnd 11162	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑇 ∈ ℂ)
19		nncan 11411	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 𝑇 ∈ ℂ) → (1 − (1 − 𝑇)) = 𝑇)
20	14, 18, 19	sylancr 587	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → (1 − (1 − 𝑇)) = 𝑇)
21	20	oveq1d 7368	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → ((1 − (1 − 𝑇)) · 𝑋) = (𝑇 · 𝑋))
22	21	oveq1d 7368	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → (((1 − (1 − 𝑇)) · 𝑋) + ((1 − 𝑇) · 𝑌)) = ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)))
23	22	adantr 480	. . . 4 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 < 𝑌) → (((1 − (1 − 𝑇)) · 𝑋) + ((1 − 𝑇) · 𝑌)) = ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)))
24		cvxcl.1	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ⊆ ℝ)
25	24	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝐷 ⊆ ℝ)
26	25, 4	sseldd 3938	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑋 ∈ ℝ)
27	26	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 < 𝑌) → 𝑋 ∈ ℝ)
28	25, 5	sseldd 3938	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑌 ∈ ℝ)
29	28	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 < 𝑌) → 𝑌 ∈ ℝ)
30		simpr 484	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 < 𝑌) → 𝑋 < 𝑌)
31		simplr3 1218	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 < 𝑌) → 𝑇 ∈ (0[,]1))
32		iirev 24839	. . . . . 6 ⊢ (𝑇 ∈ (0[,]1) → (1 − 𝑇) ∈ (0[,]1))
33	31, 32	syl 17	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 < 𝑌) → (1 − 𝑇) ∈ (0[,]1))
34		lincmb01cmp 13416	. . . . 5 ⊢ (((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ ∧ 𝑋 < 𝑌) ∧ (1 − 𝑇) ∈ (0[,]1)) → (((1 − (1 − 𝑇)) · 𝑋) + ((1 − 𝑇) · 𝑌)) ∈ (𝑋[,]𝑌))
35	27, 29, 30, 33, 34	syl31anc 1375	. . . 4 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 < 𝑌) → (((1 − (1 − 𝑇)) · 𝑋) + ((1 − 𝑇) · 𝑌)) ∈ (𝑋[,]𝑌))
36	23, 35	eqeltrrd 2829	. . 3 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 < 𝑌) → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) ∈ (𝑋[,]𝑌))
37	13, 36	sseldd 3938	. 2 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 < 𝑌) → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) ∈ 𝐷)
38		oveq2 7361	. . . . 5 ⊢ (𝑋 = 𝑌 → (𝑇 · 𝑋) = (𝑇 · 𝑌))
39	38	oveq1d 7368	. . . 4 ⊢ (𝑋 = 𝑌 → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) = ((𝑇 · 𝑌) + ((1 − 𝑇) · 𝑌)))
40		pncan3 11389	. . . . . . 7 ⊢ ((𝑇 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑇 + (1 − 𝑇)) = 1)
41	18, 14, 40	sylancl 586	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → (𝑇 + (1 − 𝑇)) = 1)
42	41	oveq1d 7368	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → ((𝑇 + (1 − 𝑇)) · 𝑌) = (1 · 𝑌))
43		1re 11134	. . . . . . . 8 ⊢ 1 ∈ ℝ
44		resubcl 11446	. . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 𝑇 ∈ ℝ) → (1 − 𝑇) ∈ ℝ)
45	43, 17, 44	sylancr 587	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → (1 − 𝑇) ∈ ℝ)
46	45	recnd 11162	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → (1 − 𝑇) ∈ ℂ)
47	28	recnd 11162	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑌 ∈ ℂ)
48	18, 46, 47	adddird 11159	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → ((𝑇 + (1 − 𝑇)) · 𝑌) = ((𝑇 · 𝑌) + ((1 − 𝑇) · 𝑌)))
49	47	mullidd 11152	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → (1 · 𝑌) = 𝑌)
50	42, 48, 49	3eqtr3d 2772	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → ((𝑇 · 𝑌) + ((1 − 𝑇) · 𝑌)) = 𝑌)
51	39, 50	sylan9eqr 2786	. . 3 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 = 𝑌) → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) = 𝑌)
52	5	adantr 480	. . 3 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 = 𝑌) → 𝑌 ∈ 𝐷)
53	51, 52	eqeltrd 2828	. 2 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 = 𝑌) → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) ∈ 𝐷)
54	2	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑌 < 𝑋) → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥[,]𝑦) ⊆ 𝐷)
55		oveq1 7360	. . . . . . . 8 ⊢ (𝑥 = 𝑌 → (𝑥[,]𝑦) = (𝑌[,]𝑦))
56	55	sseq1d 3969	. . . . . . 7 ⊢ (𝑥 = 𝑌 → ((𝑥[,]𝑦) ⊆ 𝐷 ↔ (𝑌[,]𝑦) ⊆ 𝐷))
57		oveq2 7361	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → (𝑌[,]𝑦) = (𝑌[,]𝑋))
58	57	sseq1d 3969	. . . . . . 7 ⊢ (𝑦 = 𝑋 → ((𝑌[,]𝑦) ⊆ 𝐷 ↔ (𝑌[,]𝑋) ⊆ 𝐷))
59	56, 58	rspc2v 3590	. . . . . 6 ⊢ ((𝑌 ∈ 𝐷 ∧ 𝑋 ∈ 𝐷) → (∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥[,]𝑦) ⊆ 𝐷 → (𝑌[,]𝑋) ⊆ 𝐷))
60	5, 4, 59	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → (∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥[,]𝑦) ⊆ 𝐷 → (𝑌[,]𝑋) ⊆ 𝐷))
61	60	adantr 480	. . . 4 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑌 < 𝑋) → (∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥[,]𝑦) ⊆ 𝐷 → (𝑌[,]𝑋) ⊆ 𝐷))
62	54, 61	mpd 15	. . 3 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑌 < 𝑋) → (𝑌[,]𝑋) ⊆ 𝐷)
63	26	recnd 11162	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑋 ∈ ℂ)
64	18, 63	mulcld 11154	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → (𝑇 · 𝑋) ∈ ℂ)
65	46, 47	mulcld 11154	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → ((1 − 𝑇) · 𝑌) ∈ ℂ)
66	64, 65	addcomd 11336	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) = (((1 − 𝑇) · 𝑌) + (𝑇 · 𝑋)))
67	66	adantr 480	. . . 4 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑌 < 𝑋) → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) = (((1 − 𝑇) · 𝑌) + (𝑇 · 𝑋)))
68	28	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑌 < 𝑋) → 𝑌 ∈ ℝ)
69	26	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑌 < 𝑋) → 𝑋 ∈ ℝ)
70		simpr 484	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑌 < 𝑋) → 𝑌 < 𝑋)
71		simplr3 1218	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑌 < 𝑋) → 𝑇 ∈ (0[,]1))
72		lincmb01cmp 13416	. . . . 5 ⊢ (((𝑌 ∈ ℝ ∧ 𝑋 ∈ ℝ ∧ 𝑌 < 𝑋) ∧ 𝑇 ∈ (0[,]1)) → (((1 − 𝑇) · 𝑌) + (𝑇 · 𝑋)) ∈ (𝑌[,]𝑋))
73	68, 69, 70, 71, 72	syl31anc 1375	. . . 4 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑌 < 𝑋) → (((1 − 𝑇) · 𝑌) + (𝑇 · 𝑋)) ∈ (𝑌[,]𝑋))
74	67, 73	eqeltrd 2828	. . 3 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑌 < 𝑋) → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) ∈ (𝑌[,]𝑋))
75	62, 74	sseldd 3938	. 2 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑌 < 𝑋) → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) ∈ 𝐷)
76	26, 28	lttri4d 11275	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → (𝑋 < 𝑌 ∨ 𝑋 = 𝑌 ∨ 𝑌 < 𝑋))
77	37, 53, 75, 76	mpjao3dan 1434	1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) ∈ 𝐷)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ⊆ wss 3905 class class class wbr 5095 (class class class)co 7353 ℂcc 11026 ℝcr 11027 0cc0 11028 1c1 11029 + caddc 11031 · cmul 11033 < clt 11168 − cmin 11365 [,]cicc 13269

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-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

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-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-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5518 df-po 5531 df-so 5532 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-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-er 8632 df-en 8880 df-dom 8881 df-sdom 8882 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-rp 12912 df-icc 13273

This theorem is referenced by: scvxcvx 26912 jensenlem2 26914 amgmlem 26916

W3C validator