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Theorem cvxcl 24511

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 2954	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥[,]𝑦) ⊆ 𝐷)
3	2	ad2antrr 758	. . . 4 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 < 𝑌) → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥[,]𝑦) ⊆ 𝐷)
4		simpr1 1060	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑋 ∈ 𝐷)
5		simpr2 1061	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑌 ∈ 𝐷)
6		oveq1 6556	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥[,]𝑦) = (𝑋[,]𝑦))
7	6	sseq1d 3595	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝑥[,]𝑦) ⊆ 𝐷 ↔ (𝑋[,]𝑦) ⊆ 𝐷))
8		oveq2 6557	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝑋[,]𝑦) = (𝑋[,]𝑌))
9	8	sseq1d 3595	. . . . . . 7 ⊢ (𝑦 = 𝑌 → ((𝑋[,]𝑦) ⊆ 𝐷 ↔ (𝑋[,]𝑌) ⊆ 𝐷))
10	7, 9	rspc2v 3293	. . . . . 6 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷) → (∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥[,]𝑦) ⊆ 𝐷 → (𝑋[,]𝑌) ⊆ 𝐷))
11	4, 5, 10	syl2anc 691	. . . . 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 9873	. . . . . . . 8 ⊢ 1 ∈ ℂ
15		unitssre 12190	. . . . . . . . . 10 ⊢ (0[,]1) ⊆ ℝ
16		simpr3 1062	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑇 ∈ (0[,]1))
17	15, 16	sseldi 3566	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑇 ∈ ℝ)
18	17	recnd 9947	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑇 ∈ ℂ)
19		nncan 10189	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 𝑇 ∈ ℂ) → (1 − (1 − 𝑇)) = 𝑇)
20	14, 18, 19	sylancr 694	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → (1 − (1 − 𝑇)) = 𝑇)
21	20	oveq1d 6564	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → ((1 − (1 − 𝑇)) · 𝑋) = (𝑇 · 𝑋))
22	21	oveq1d 6564	. . . . 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 3569	. . . . . 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 3569	. . . . . 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 476	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 < 𝑌) → 𝑋 < 𝑌)
31		simplr3 1098	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 < 𝑌) → 𝑇 ∈ (0[,]1))
32		iirev 22536	. . . . . 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 12186	. . . . 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 1321	. . . 4 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 < 𝑌) → (((1 − (1 − 𝑇)) · 𝑋) + ((1 − 𝑇) · 𝑌)) ∈ (𝑋[,]𝑌))
36	23, 35	eqeltrrd 2689	. . 3 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 < 𝑌) → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) ∈ (𝑋[,]𝑌))
37	13, 36	sseldd 3569	. 2 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 < 𝑌) → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) ∈ 𝐷)
38		oveq2 6557	. . . . 5 ⊢ (𝑋 = 𝑌 → (𝑇 · 𝑋) = (𝑇 · 𝑌))
39	38	oveq1d 6564	. . . 4 ⊢ (𝑋 = 𝑌 → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) = ((𝑇 · 𝑌) + ((1 − 𝑇) · 𝑌)))
40		pncan3 10168	. . . . . . 7 ⊢ ((𝑇 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑇 + (1 − 𝑇)) = 1)
41	18, 14, 40	sylancl 693	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → (𝑇 + (1 − 𝑇)) = 1)
42	41	oveq1d 6564	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → ((𝑇 + (1 − 𝑇)) · 𝑌) = (1 · 𝑌))
43		1re 9918	. . . . . . . 8 ⊢ 1 ∈ ℝ
44		resubcl 10224	. . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 𝑇 ∈ ℝ) → (1 − 𝑇) ∈ ℝ)
45	43, 17, 44	sylancr 694	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → (1 − 𝑇) ∈ ℝ)
46	45	recnd 9947	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → (1 − 𝑇) ∈ ℂ)
47	28	recnd 9947	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑌 ∈ ℂ)
48	18, 46, 47	adddird 9944	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → ((𝑇 + (1 − 𝑇)) · 𝑌) = ((𝑇 · 𝑌) + ((1 − 𝑇) · 𝑌)))
49	47	mulid2d 9937	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → (1 · 𝑌) = 𝑌)
50	42, 48, 49	3eqtr3d 2652	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → ((𝑇 · 𝑌) + ((1 − 𝑇) · 𝑌)) = 𝑌)
51	39, 50	sylan9eqr 2666	. . 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 2688	. 2 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 = 𝑌) → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) ∈ 𝐷)
54	2	ad2antrr 758	. . . 4 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑌 < 𝑋) → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥[,]𝑦) ⊆ 𝐷)
55		oveq1 6556	. . . . . . . 8 ⊢ (𝑥 = 𝑌 → (𝑥[,]𝑦) = (𝑌[,]𝑦))
56	55	sseq1d 3595	. . . . . . 7 ⊢ (𝑥 = 𝑌 → ((𝑥[,]𝑦) ⊆ 𝐷 ↔ (𝑌[,]𝑦) ⊆ 𝐷))
57		oveq2 6557	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → (𝑌[,]𝑦) = (𝑌[,]𝑋))
58	57	sseq1d 3595	. . . . . . 7 ⊢ (𝑦 = 𝑋 → ((𝑌[,]𝑦) ⊆ 𝐷 ↔ (𝑌[,]𝑋) ⊆ 𝐷))
59	56, 58	rspc2v 3293	. . . . . 6 ⊢ ((𝑌 ∈ 𝐷 ∧ 𝑋 ∈ 𝐷) → (∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥[,]𝑦) ⊆ 𝐷 → (𝑌[,]𝑋) ⊆ 𝐷))
60	5, 4, 59	syl2anc 691	. . . . 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 9947	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑋 ∈ ℂ)
64	18, 63	mulcld 9939	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → (𝑇 · 𝑋) ∈ ℂ)
65	46, 47	mulcld 9939	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → ((1 − 𝑇) · 𝑌) ∈ ℂ)
66	64, 65	addcomd 10117	. . . . 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 476	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑌 < 𝑋) → 𝑌 < 𝑋)
71		simplr3 1098	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑌 < 𝑋) → 𝑇 ∈ (0[,]1))
72		lincmb01cmp 12186	. . . . 5 ⊢ (((𝑌 ∈ ℝ ∧ 𝑋 ∈ ℝ ∧ 𝑌 < 𝑋) ∧ 𝑇 ∈ (0[,]1)) → (((1 − 𝑇) · 𝑌) + (𝑇 · 𝑋)) ∈ (𝑌[,]𝑋))
73	68, 69, 70, 71, 72	syl31anc 1321	. . . 4 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑌 < 𝑋) → (((1 − 𝑇) · 𝑌) + (𝑇 · 𝑋)) ∈ (𝑌[,]𝑋))
74	67, 73	eqeltrd 2688	. . 3 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑌 < 𝑋) → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) ∈ (𝑌[,]𝑋))
75	62, 74	sseldd 3569	. 2 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑌 < 𝑋) → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) ∈ 𝐷)
76	26, 28	lttri4d 10057	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → (𝑋 < 𝑌 ∨ 𝑋 = 𝑌 ∨ 𝑌 < 𝑋))
77	37, 53, 75, 76	mpjao3dan 1387	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 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ⊆ wss 3540 class class class wbr 4583 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 − cmin 10145 [,]cicc 12049

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-rp 11709 df-icc 12053

This theorem is referenced by: scvxcvx 24512 jensenlem2 24514 amgmlem 24516

W3C validator