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Theorem cvxcl 26895

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 3180	. . . . 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 7394	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥[,]𝑦) = (𝑋[,]𝑦))
7	6	sseq1d 3978	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝑥[,]𝑦) ⊆ 𝐷 ↔ (𝑋[,]𝑦) ⊆ 𝐷))
8		oveq2 7395	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝑋[,]𝑦) = (𝑋[,]𝑌))
9	8	sseq1d 3978	. . . . . . 7 ⊢ (𝑦 = 𝑌 → ((𝑋[,]𝑦) ⊆ 𝐷 ↔ (𝑋[,]𝑌) ⊆ 𝐷))
10	7, 9	rspc2v 3599	. . . . . 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 11126	. . . . . . . 8 ⊢ 1 ∈ ℂ
15		unitssre 13460	. . . . . . . . . 10 ⊢ (0[,]1) ⊆ ℝ
16		simpr3 1197	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑇 ∈ (0[,]1))
17	15, 16	sselid 3944	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑇 ∈ ℝ)
18	17	recnd 11202	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑇 ∈ ℂ)
19		nncan 11451	. . . . . . . 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 7402	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → ((1 − (1 − 𝑇)) · 𝑋) = (𝑇 · 𝑋))
22	21	oveq1d 7402	. . . . 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 3947	. . . . . 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 3947	. . . . . 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 24823	. . . . . 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 13456	. . . . 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 3947	. 2 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑋 < 𝑌) → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) ∈ 𝐷)
38		oveq2 7395	. . . . 5 ⊢ (𝑋 = 𝑌 → (𝑇 · 𝑋) = (𝑇 · 𝑌))
39	38	oveq1d 7402	. . . 4 ⊢ (𝑋 = 𝑌 → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) = ((𝑇 · 𝑌) + ((1 − 𝑇) · 𝑌)))
40		pncan3 11429	. . . . . . 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 7402	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → ((𝑇 + (1 − 𝑇)) · 𝑌) = (1 · 𝑌))
43		1re 11174	. . . . . . . 8 ⊢ 1 ∈ ℝ
44		resubcl 11486	. . . . . . . 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 11202	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → (1 − 𝑇) ∈ ℂ)
47	28	recnd 11202	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑌 ∈ ℂ)
48	18, 46, 47	adddird 11199	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → ((𝑇 + (1 − 𝑇)) · 𝑌) = ((𝑇 · 𝑌) + ((1 − 𝑇) · 𝑌)))
49	47	mullidd 11192	. . . . 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 7394	. . . . . . . 8 ⊢ (𝑥 = 𝑌 → (𝑥[,]𝑦) = (𝑌[,]𝑦))
56	55	sseq1d 3978	. . . . . . 7 ⊢ (𝑥 = 𝑌 → ((𝑥[,]𝑦) ⊆ 𝐷 ↔ (𝑌[,]𝑦) ⊆ 𝐷))
57		oveq2 7395	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → (𝑌[,]𝑦) = (𝑌[,]𝑋))
58	57	sseq1d 3978	. . . . . . 7 ⊢ (𝑦 = 𝑋 → ((𝑌[,]𝑦) ⊆ 𝐷 ↔ (𝑌[,]𝑋) ⊆ 𝐷))
59	56, 58	rspc2v 3599	. . . . . 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 11202	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → 𝑋 ∈ ℂ)
64	18, 63	mulcld 11194	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → (𝑇 · 𝑋) ∈ ℂ)
65	46, 47	mulcld 11194	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) → ((1 − 𝑇) · 𝑌) ∈ ℂ)
66	64, 65	addcomd 11376	. . . . 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 13456	. . . . 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 3947	. 2 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑇 ∈ (0[,]1))) ∧ 𝑌 < 𝑋) → ((𝑇 · 𝑋) + ((1 − 𝑇) · 𝑌)) ∈ 𝐷)
76	26, 28	lttri4d 11315	. 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 3914 class class class wbr 5107 (class class class)co 7387 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 · cmul 11073 < clt 11208 − cmin 11405 [,]cicc 13309

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 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

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 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-po 5546 df-so 5547 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-rp 12952 df-icc 13313

This theorem is referenced by: scvxcvx 26896 jensenlem2 26898 amgmlem 26900

W3C validator