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Theorem cvxcl 26925

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 3176	. . . . 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 7361	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥[,]𝑦) = (𝑋[,]𝑦))
7	6	sseq1d 3962	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝑥[,]𝑦) ⊆ 𝐷 ↔ (𝑋[,]𝑦) ⊆ 𝐷))
8		oveq2 7362	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝑋[,]𝑦) = (𝑋[,]𝑌))
9	8	sseq1d 3962	. . . . . . 7 ⊢ (𝑦 = 𝑌 → ((𝑋[,]𝑦) ⊆ 𝐷 ↔ (𝑋[,]𝑌) ⊆ 𝐷))
10	7, 9	rspc2v 3584	. . . . . 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 11073	. . . . . . . 8 ⊢ 1 ∈ ℂ
15		unitssre 13403	. . . . . . . . . 10 ⊢ (0[,]1) ⊆ ℝ
16		simpr3 1197	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑇 ∈ (0[,]1))
17	15, 16	sselid 3928	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑇 ∈ ℝ)
18	17	recnd 11149	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑇 ∈ ℂ)
19		nncan 11399	. . . . . . . 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 7369	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → ((1 − (1 − 𝑇)) · 𝑋) = (𝑇 · 𝑋))
22	21	oveq1d 7369	. . . . 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 3931	. . . . . 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 3931	. . . . . 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 24853	. . . . . 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 13399	. . . . 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 2834	. . 3 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 < 𝑌) → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) ∈ (𝑋[,]𝑌))
37	13, 36	sseldd 3931	. 2 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 < 𝑌) → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) ∈ 𝐷)
38		oveq2 7362	. . . . 5 ⊢ (𝑋 = 𝑌 → (𝑇 · 𝑋) = (𝑇 · 𝑌))
39	38	oveq1d 7369	. . . 4 ⊢ (𝑋 = 𝑌 → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) = ((𝑇 · 𝑌) + ((1 − 𝑇) · 𝑌)))
40		pncan3 11377	. . . . . . 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 7369	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → ((𝑇 + (1 − 𝑇)) · 𝑌) = (1 · 𝑌))
43		1re 11121	. . . . . . . 8 ⊢ 1 ∈ ℝ
44		resubcl 11434	. . . . . . . 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 11149	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → (1 − 𝑇) ∈ ℂ)
47	28	recnd 11149	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑌 ∈ ℂ)
48	18, 46, 47	adddird 11146	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → ((𝑇 + (1 − 𝑇)) · 𝑌) = ((𝑇 · 𝑌) + ((1 − 𝑇) · 𝑌)))
49	47	mullidd 11139	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → (1 · 𝑌) = 𝑌)
50	42, 48, 49	3eqtr3d 2776	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → ((𝑇 · 𝑌) + ((1 − 𝑇) · 𝑌)) = 𝑌)
51	39, 50	sylan9eqr 2790	. . 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 2833	. 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 7361	. . . . . . . 8 ⊢ (𝑥 = 𝑌 → (𝑥[,]𝑦) = (𝑌[,]𝑦))
56	55	sseq1d 3962	. . . . . . 7 ⊢ (𝑥 = 𝑌 → ((𝑥[,]𝑦) ⊆ 𝐷 ↔ (𝑌[,]𝑦) ⊆ 𝐷))
57		oveq2 7362	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → (𝑌[,]𝑦) = (𝑌[,]𝑋))
58	57	sseq1d 3962	. . . . . . 7 ⊢ (𝑦 = 𝑋 → ((𝑌[,]𝑦) ⊆ 𝐷 ↔ (𝑌[,]𝑋) ⊆ 𝐷))
59	56, 58	rspc2v 3584	. . . . . 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 11149	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑋 ∈ ℂ)
64	18, 63	mulcld 11141	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → (𝑇 · 𝑋) ∈ ℂ)
65	46, 47	mulcld 11141	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → ((1 − 𝑇) · 𝑌) ∈ ℂ)
66	64, 65	addcomd 11324	. . . . 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 13399	. . . . 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 2833	. . 3 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑌 < 𝑋) → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) ∈ (𝑌[,]𝑋))
75	62, 74	sseldd 3931	. 2 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑌 < 𝑋) → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) ∈ 𝐷)
76	26, 28	lttri4d 11263	. 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 1541 ∈ wcel 2113 ∀wral 3048 ⊆ wss 3898 class class class wbr 5095 (class class class)co 7354 ℂcc 11013 ℝcr 11014 0cc0 11015 1c1 11016 + caddc 11018 · cmul 11020 < clt 11155 − cmin 11353 [,]cicc 13252

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7676 ax-cnex 11071 ax-resscn 11072 ax-1cn 11073 ax-icn 11074 ax-addcl 11075 ax-addrcl 11076 ax-mulcl 11077 ax-mulrcl 11078 ax-mulcom 11079 ax-addass 11080 ax-mulass 11081 ax-distr 11082 ax-i2m1 11083 ax-1ne0 11084 ax-1rid 11085 ax-rnegex 11086 ax-rrecex 11087 ax-cnre 11088 ax-pre-lttri 11089 ax-pre-lttrn 11090 ax-pre-ltadd 11091 ax-pre-mulgt0 11092

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5516 df-po 5529 df-so 5530 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-riota 7311 df-ov 7357 df-oprab 7358 df-mpo 7359 df-er 8630 df-en 8878 df-dom 8879 df-sdom 8880 df-pnf 11157 df-mnf 11158 df-xr 11159 df-ltxr 11160 df-le 11161 df-sub 11355 df-neg 11356 df-rp 12895 df-icc 13256

This theorem is referenced by: scvxcvx 26926 jensenlem2 26928 amgmlem 26930

W3C validator