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Theorem cvxcl 25264

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 3141	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥[,]𝑦) ⊆ 𝐷)
3	2	ad2antrr 713	. . . 4 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 < 𝑌) → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥[,]𝑦) ⊆ 𝐷)
4		simpr1 1174	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑋 ∈ 𝐷)
5		simpr2 1175	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑌 ∈ 𝐷)
6		oveq1 6983	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥[,]𝑦) = (𝑋[,]𝑦))
7	6	sseq1d 3888	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝑥[,]𝑦) ⊆ 𝐷 ↔ (𝑋[,]𝑦) ⊆ 𝐷))
8		oveq2 6984	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝑋[,]𝑦) = (𝑋[,]𝑌))
9	8	sseq1d 3888	. . . . . . 7 ⊢ (𝑦 = 𝑌 → ((𝑋[,]𝑦) ⊆ 𝐷 ↔ (𝑋[,]𝑌) ⊆ 𝐷))
10	7, 9	rspc2v 3548	. . . . . 6 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷) → (∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥[,]𝑦) ⊆ 𝐷 → (𝑋[,]𝑌) ⊆ 𝐷))
11	4, 5, 10	syl2anc 576	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → (∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥[,]𝑦) ⊆ 𝐷 → (𝑋[,]𝑌) ⊆ 𝐷))
12	11	adantr 473	. . . 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 10393	. . . . . . . 8 ⊢ 1 ∈ ℂ
15		unitssre 12701	. . . . . . . . . 10 ⊢ (0[,]1) ⊆ ℝ
16		simpr3 1176	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑇 ∈ (0[,]1))
17	15, 16	sseldi 3856	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑇 ∈ ℝ)
18	17	recnd 10468	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑇 ∈ ℂ)
19		nncan 10716	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 𝑇 ∈ ℂ) → (1 − (1 − 𝑇)) = 𝑇)
20	14, 18, 19	sylancr 578	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → (1 − (1 − 𝑇)) = 𝑇)
21	20	oveq1d 6991	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → ((1 − (1 − 𝑇)) · 𝑋) = (𝑇 · 𝑋))
22	21	oveq1d 6991	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → (((1 − (1 − 𝑇)) · 𝑋) + ((1 − 𝑇) · 𝑌)) = ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)))
23	22	adantr 473	. . . 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 473	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝐷 ⊆ ℝ)
26	25, 4	sseldd 3859	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑋 ∈ ℝ)
27	26	adantr 473	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 < 𝑌) → 𝑋 ∈ ℝ)
28	25, 5	sseldd 3859	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑌 ∈ ℝ)
29	28	adantr 473	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 < 𝑌) → 𝑌 ∈ ℝ)
30		simpr 477	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 < 𝑌) → 𝑋 < 𝑌)
31		simplr3 1197	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 < 𝑌) → 𝑇 ∈ (0[,]1))
32		iirev 23236	. . . . . 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 12697	. . . . 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 1353	. . . 4 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 < 𝑌) → (((1 − (1 − 𝑇)) · 𝑋) + ((1 − 𝑇) · 𝑌)) ∈ (𝑋[,]𝑌))
36	23, 35	eqeltrrd 2867	. . 3 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 < 𝑌) → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) ∈ (𝑋[,]𝑌))
37	13, 36	sseldd 3859	. 2 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 < 𝑌) → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) ∈ 𝐷)
38		oveq2 6984	. . . . 5 ⊢ (𝑋 = 𝑌 → (𝑇 · 𝑋) = (𝑇 · 𝑌))
39	38	oveq1d 6991	. . . 4 ⊢ (𝑋 = 𝑌 → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) = ((𝑇 · 𝑌) + ((1 − 𝑇) · 𝑌)))
40		pncan3 10694	. . . . . . 7 ⊢ ((𝑇 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑇 + (1 − 𝑇)) = 1)
41	18, 14, 40	sylancl 577	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → (𝑇 + (1 − 𝑇)) = 1)
42	41	oveq1d 6991	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → ((𝑇 + (1 − 𝑇)) · 𝑌) = (1 · 𝑌))
43		1re 10439	. . . . . . . 8 ⊢ 1 ∈ ℝ
44		resubcl 10751	. . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 𝑇 ∈ ℝ) → (1 − 𝑇) ∈ ℝ)
45	43, 17, 44	sylancr 578	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → (1 − 𝑇) ∈ ℝ)
46	45	recnd 10468	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → (1 − 𝑇) ∈ ℂ)
47	28	recnd 10468	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑌 ∈ ℂ)
48	18, 46, 47	adddird 10465	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → ((𝑇 + (1 − 𝑇)) · 𝑌) = ((𝑇 · 𝑌) + ((1 − 𝑇) · 𝑌)))
49	47	mulid2d 10458	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → (1 · 𝑌) = 𝑌)
50	42, 48, 49	3eqtr3d 2822	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → ((𝑇 · 𝑌) + ((1 − 𝑇) · 𝑌)) = 𝑌)
51	39, 50	sylan9eqr 2836	. . 3 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 = 𝑌) → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) = 𝑌)
52	5	adantr 473	. . 3 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 = 𝑌) → 𝑌 ∈ 𝐷)
53	51, 52	eqeltrd 2866	. 2 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 = 𝑌) → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) ∈ 𝐷)
54	2	ad2antrr 713	. . . 4 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑌 < 𝑋) → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥[,]𝑦) ⊆ 𝐷)
55		oveq1 6983	. . . . . . . 8 ⊢ (𝑥 = 𝑌 → (𝑥[,]𝑦) = (𝑌[,]𝑦))
56	55	sseq1d 3888	. . . . . . 7 ⊢ (𝑥 = 𝑌 → ((𝑥[,]𝑦) ⊆ 𝐷 ↔ (𝑌[,]𝑦) ⊆ 𝐷))
57		oveq2 6984	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → (𝑌[,]𝑦) = (𝑌[,]𝑋))
58	57	sseq1d 3888	. . . . . . 7 ⊢ (𝑦 = 𝑋 → ((𝑌[,]𝑦) ⊆ 𝐷 ↔ (𝑌[,]𝑋) ⊆ 𝐷))
59	56, 58	rspc2v 3548	. . . . . 6 ⊢ ((𝑌 ∈ 𝐷 ∧ 𝑋 ∈ 𝐷) → (∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥[,]𝑦) ⊆ 𝐷 → (𝑌[,]𝑋) ⊆ 𝐷))
60	5, 4, 59	syl2anc 576	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → (∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥[,]𝑦) ⊆ 𝐷 → (𝑌[,]𝑋) ⊆ 𝐷))
61	60	adantr 473	. . . 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 10468	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑋 ∈ ℂ)
64	18, 63	mulcld 10460	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → (𝑇 · 𝑋) ∈ ℂ)
65	46, 47	mulcld 10460	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → ((1 − 𝑇) · 𝑌) ∈ ℂ)
66	64, 65	addcomd 10642	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) = (((1 − 𝑇) · 𝑌) + (𝑇 · 𝑋)))
67	66	adantr 473	. . . 4 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑌 < 𝑋) → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) = (((1 − 𝑇) · 𝑌) + (𝑇 · 𝑋)))
68	28	adantr 473	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑌 < 𝑋) → 𝑌 ∈ ℝ)
69	26	adantr 473	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑌 < 𝑋) → 𝑋 ∈ ℝ)
70		simpr 477	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑌 < 𝑋) → 𝑌 < 𝑋)
71		simplr3 1197	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑌 < 𝑋) → 𝑇 ∈ (0[,]1))
72		lincmb01cmp 12697	. . . . 5 ⊢ (((𝑌 ∈ ℝ ∧ 𝑋 ∈ ℝ ∧ 𝑌 < 𝑋) ∧ 𝑇 ∈ (0[,]1)) → (((1 − 𝑇) · 𝑌) + (𝑇 · 𝑋)) ∈ (𝑌[,]𝑋))
73	68, 69, 70, 71, 72	syl31anc 1353	. . . 4 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑌 < 𝑋) → (((1 − 𝑇) · 𝑌) + (𝑇 · 𝑋)) ∈ (𝑌[,]𝑋))
74	67, 73	eqeltrd 2866	. . 3 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑌 < 𝑋) → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) ∈ (𝑌[,]𝑋))
75	62, 74	sseldd 3859	. 2 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑌 < 𝑋) → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) ∈ 𝐷)
76	26, 28	lttri4d 10581	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → (𝑋 < 𝑌 ∨ 𝑋 = 𝑌 ∨ 𝑌 < 𝑋))
77	37, 53, 75, 76	mpjao3dan 1411	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 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 ∀wral 3088 ⊆ wss 3829 class class class wbr 4929 (class class class)co 6976 ℂcc 10333 ℝcr 10334 0cc0 10335 1c1 10336 + caddc 10338 · cmul 10340 < clt 10474 − cmin 10670 [,]cicc 12557

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-po 5326 df-so 5327 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-rp 12205 df-icc 12561

This theorem is referenced by: scvxcvx 25265 jensenlem2 25267 amgmlem 25269

W3C validator