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Theorem jensenlem2 26147

Description: Lemma for jensen 26148. (Contributed by Mario Carneiro, 21-Jun-2015.)

**Hypotheses**
Ref	Expression
jensen.1	⊢ (𝜑 → 𝐷 ⊆ ℝ)
jensen.2	⊢ (𝜑 → 𝐹:𝐷⟶ℝ)
jensen.3	⊢ ((𝜑 ∧ (𝑎 ∈ 𝐷 ∧ 𝑏 ∈ 𝐷)) → (𝑎[,]𝑏) ⊆ 𝐷)
jensen.4	⊢ (𝜑 → 𝐴 ∈ Fin)
jensen.5	⊢ (𝜑 → 𝑇:𝐴⟶(0[,)+∞))
jensen.6	⊢ (𝜑 → 𝑋:𝐴⟶𝐷)
jensen.7	⊢ (𝜑 → 0 < (ℂ_fld Σ_g 𝑇))
jensen.8	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷 ∧ 𝑡 ∈ (0[,]1))) → (𝐹‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹‘𝑥)) + ((1 − 𝑡) · (𝐹‘𝑦))))
jensenlem.1	⊢ (𝜑 → ¬ 𝑧 ∈ 𝐵)
jensenlem.2	⊢ (𝜑 → (𝐵 ∪ {𝑧}) ⊆ 𝐴)
jensenlem.s	⊢ 𝑆 = (ℂ_fld Σ_g (𝑇 ↾ 𝐵))
jensenlem.l	⊢ 𝐿 = (ℂ_fld Σ_g (𝑇 ↾ (𝐵 ∪ {𝑧})))
jensenlem.3	⊢ (𝜑 → 𝑆 ∈ ℝ⁺)
jensenlem.4	⊢ (𝜑 → ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆) ∈ 𝐷)
jensenlem.5	⊢ (𝜑 → (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) ≤ ((ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆))

**Assertion**
Ref	Expression
jensenlem2	⊢ (𝜑 → (((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ (𝐵 ∪ {𝑧}))) / 𝐿) ∈ 𝐷 ∧ (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ (𝐵 ∪ {𝑧}))) / 𝐿)) ≤ ((ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ (𝐵 ∪ {𝑧}))) / 𝐿)))

Distinct variable groups: 𝑎,𝑏,𝑡,𝑥,𝑦,𝐴 𝐷,𝑎,𝑏,𝑡,𝑥,𝑦 𝜑,𝑎,𝑏,𝑡,𝑥,𝑦 𝐹,𝑎,𝑏,𝑡,𝑥,𝑦 𝑇,𝑎,𝑏,𝑡,𝑥,𝑦 𝑋,𝑎,𝑏,𝑡,𝑥,𝑦 𝑧,𝑎,𝐵,𝑏,𝑡,𝑥,𝑦 𝑡,𝐿,𝑥,𝑦 𝑆,𝑎,𝑏,𝑡,𝑥,𝑦 Allowed substitution hints: 𝜑(𝑧) 𝐴(𝑧) 𝐷(𝑧) 𝑆(𝑧) 𝑇(𝑧) 𝐹(𝑧) 𝐿(𝑧,𝑎,𝑏) 𝑋(𝑧)

**Proof of Theorem jensenlem2**
Step	Hyp	Ref	Expression
1		cnfld0 20632	. . . . . . 7 ⊢ 0 = (0_g‘ℂ_fld)
2		cnring 20630	. . . . . . . 8 ⊢ ℂ_fld ∈ Ring
3		ringabl 19829	. . . . . . . 8 ⊢ (ℂ_fld ∈ Ring → ℂ_fld ∈ Abel)
4	2, 3	mp1i 13	. . . . . . 7 ⊢ (𝜑 → ℂ_fld ∈ Abel)
5		jensen.4	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ Fin)
6		jensenlem.2	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ∪ {𝑧}) ⊆ 𝐴)
7	6	unssad 4120	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
8	5, 7	ssfid 9029	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ Fin)
9		resubdrg 20823	. . . . . . . . 9 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) ∧ ℝ_fld ∈ DivRing)
10	9	simpli 484	. . . . . . . 8 ⊢ ℝ ∈ (SubRing‘ℂ_fld)
11		subrgsubg 20040	. . . . . . . 8 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) → ℝ ∈ (SubGrp‘ℂ_fld))
12	10, 11	mp1i 13	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ (SubGrp‘ℂ_fld))
13		remulcl 10966	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 𝑦) ∈ ℝ)
14	13	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 · 𝑦) ∈ ℝ)
15		jensen.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇:𝐴⟶(0[,)+∞))
16		rge0ssre 13198	. . . . . . . . . 10 ⊢ (0[,)+∞) ⊆ ℝ
17		fss 6609	. . . . . . . . . 10 ⊢ ((𝑇:𝐴⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝑇:𝐴⟶ℝ)
18	15, 16, 17	sylancl 586	. . . . . . . . 9 ⊢ (𝜑 → 𝑇:𝐴⟶ℝ)
19		jensen.6	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋:𝐴⟶𝐷)
20		jensen.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ⊆ ℝ)
21	19, 20	fssd 6610	. . . . . . . . 9 ⊢ (𝜑 → 𝑋:𝐴⟶ℝ)
22		inidm 4152	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐴) = 𝐴
23	14, 18, 21, 5, 5, 22	off 7541	. . . . . . . 8 ⊢ (𝜑 → (𝑇 ∘_f · 𝑋):𝐴⟶ℝ)
24	23, 7	fssresd 6633	. . . . . . 7 ⊢ (𝜑 → ((𝑇 ∘_f · 𝑋) ↾ 𝐵):𝐵⟶ℝ)
25		c0ex 10979	. . . . . . . . 9 ⊢ 0 ∈ V
26	25	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ V)
27	24, 8, 26	fdmfifsupp 9125	. . . . . . 7 ⊢ (𝜑 → ((𝑇 ∘_f · 𝑋) ↾ 𝐵) finSupp 0)
28	1, 4, 8, 12, 24, 27	gsumsubgcl 19531	. . . . . 6 ⊢ (𝜑 → (ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) ∈ ℝ)
29	28	recnd 11013	. . . . 5 ⊢ (𝜑 → (ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) ∈ ℂ)
30		ax-resscn 10938	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
31	16, 30	sstri 3929	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℂ
32	6	unssbd 4121	. . . . . . . . 9 ⊢ (𝜑 → {𝑧} ⊆ 𝐴)
33		vex 3433	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
34	33	snss 4719	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 ↔ {𝑧} ⊆ 𝐴)
35	32, 34	sylibr 233	. . . . . . . 8 ⊢ (𝜑 → 𝑧 ∈ 𝐴)
36	15, 35	ffvelrnd 6954	. . . . . . 7 ⊢ (𝜑 → (𝑇‘𝑧) ∈ (0[,)+∞))
37	31, 36	sselid 3918	. . . . . 6 ⊢ (𝜑 → (𝑇‘𝑧) ∈ ℂ)
38	19, 35	ffvelrnd 6954	. . . . . . . 8 ⊢ (𝜑 → (𝑋‘𝑧) ∈ 𝐷)
39	20, 38	sseldd 3921	. . . . . . 7 ⊢ (𝜑 → (𝑋‘𝑧) ∈ ℝ)
40	39	recnd 11013	. . . . . 6 ⊢ (𝜑 → (𝑋‘𝑧) ∈ ℂ)
41	37, 40	mulcld 11005	. . . . 5 ⊢ (𝜑 → ((𝑇‘𝑧) · (𝑋‘𝑧)) ∈ ℂ)
42		jensen.2	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐷⟶ℝ)
43		jensen.3	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐷 ∧ 𝑏 ∈ 𝐷)) → (𝑎[,]𝑏) ⊆ 𝐷)
44		jensen.7	. . . . . . . 8 ⊢ (𝜑 → 0 < (ℂ_fld Σ_g 𝑇))
45		jensen.8	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷 ∧ 𝑡 ∈ (0[,]1))) → (𝐹‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹‘𝑥)) + ((1 − 𝑡) · (𝐹‘𝑦))))
46		jensenlem.1	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝐵)
47		jensenlem.s	. . . . . . . 8 ⊢ 𝑆 = (ℂ_fld Σ_g (𝑇 ↾ 𝐵))
48		jensenlem.l	. . . . . . . 8 ⊢ 𝐿 = (ℂ_fld Σ_g (𝑇 ↾ (𝐵 ∪ {𝑧})))
49	20, 42, 43, 5, 15, 19, 44, 45, 46, 6, 47, 48	jensenlem1 26146	. . . . . . 7 ⊢ (𝜑 → 𝐿 = (𝑆 + (𝑇‘𝑧)))
50		jensenlem.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ ℝ⁺)
51	50	rpred 12782	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℝ)
52		elrege0 13196	. . . . . . . . . 10 ⊢ ((𝑇‘𝑧) ∈ (0[,)+∞) ↔ ((𝑇‘𝑧) ∈ ℝ ∧ 0 ≤ (𝑇‘𝑧)))
53	52	simplbi 498	. . . . . . . . 9 ⊢ ((𝑇‘𝑧) ∈ (0[,)+∞) → (𝑇‘𝑧) ∈ ℝ)
54	36, 53	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑇‘𝑧) ∈ ℝ)
55	51, 54	readdcld 11014	. . . . . . 7 ⊢ (𝜑 → (𝑆 + (𝑇‘𝑧)) ∈ ℝ)
56	49, 55	eqeltrd 2839	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ ℝ)
57	56	recnd 11013	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ ℂ)
58		0red 10988	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ)
59	50	rpgt0d 12785	. . . . . . 7 ⊢ (𝜑 → 0 < 𝑆)
60	52	simprbi 497	. . . . . . . . . 10 ⊢ ((𝑇‘𝑧) ∈ (0[,)+∞) → 0 ≤ (𝑇‘𝑧))
61	36, 60	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (𝑇‘𝑧))
62	51, 54	addge01d 11573	. . . . . . . . 9 ⊢ (𝜑 → (0 ≤ (𝑇‘𝑧) ↔ 𝑆 ≤ (𝑆 + (𝑇‘𝑧))))
63	61, 62	mpbid 231	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ≤ (𝑆 + (𝑇‘𝑧)))
64	63, 49	breqtrrd 5101	. . . . . . 7 ⊢ (𝜑 → 𝑆 ≤ 𝐿)
65	58, 51, 56, 59, 64	ltletrd 11145	. . . . . 6 ⊢ (𝜑 → 0 < 𝐿)
66	65	gt0ne0d 11549	. . . . 5 ⊢ (𝜑 → 𝐿 ≠ 0)
67	29, 41, 57, 66	divdird 11799	. . . 4 ⊢ (𝜑 → (((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) + ((𝑇‘𝑧) · (𝑋‘𝑧))) / 𝐿) = (((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝐿) + (((𝑇‘𝑧) · (𝑋‘𝑧)) / 𝐿)))
68		cnfldbas 20611	. . . . . . 7 ⊢ ℂ = (Base‘ℂ_fld)
69		cnfldadd 20612	. . . . . . 7 ⊢ + = (+_g‘ℂ_fld)
70		ringcmn 19830	. . . . . . . 8 ⊢ (ℂ_fld ∈ Ring → ℂ_fld ∈ CMnd)
71	2, 70	mp1i 13	. . . . . . 7 ⊢ (𝜑 → ℂ_fld ∈ CMnd)
72	7	sselda 3920	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴)
73	15	ffvelrnda 6953	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑇‘𝑥) ∈ (0[,)+∞))
74	72, 73	syldan 591	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑇‘𝑥) ∈ (0[,)+∞))
75	31, 74	sselid 3918	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑇‘𝑥) ∈ ℂ)
76	20	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐷 ⊆ ℝ)
77	19	ffvelrnda 6953	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑋‘𝑥) ∈ 𝐷)
78	72, 77	syldan 591	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑋‘𝑥) ∈ 𝐷)
79	76, 78	sseldd 3921	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑋‘𝑥) ∈ ℝ)
80	79	recnd 11013	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑋‘𝑥) ∈ ℂ)
81	75, 80	mulcld 11005	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑇‘𝑥) · (𝑋‘𝑥)) ∈ ℂ)
82		fveq2 6766	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑇‘𝑥) = (𝑇‘𝑧))
83		fveq2 6766	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑋‘𝑥) = (𝑋‘𝑧))
84	82, 83	oveq12d 7285	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝑇‘𝑥) · (𝑋‘𝑥)) = ((𝑇‘𝑧) · (𝑋‘𝑧)))
85	68, 69, 71, 8, 81, 35, 46, 41, 84	gsumunsn 19571	. . . . . 6 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ ((𝑇‘𝑥) · (𝑋‘𝑥)))) = ((ℂ_fld Σ_g (𝑥 ∈ 𝐵 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥)))) + ((𝑇‘𝑧) · (𝑋‘𝑧))))
86	15	feqmptd 6829	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 = (𝑥 ∈ 𝐴 ↦ (𝑇‘𝑥)))
87	19	feqmptd 6829	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 = (𝑥 ∈ 𝐴 ↦ (𝑋‘𝑥)))
88	5, 73, 77, 86, 87	offval2 7543	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ∘_f · 𝑋) = (𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥))))
89	88	reseq1d 5883	. . . . . . . 8 ⊢ (𝜑 → ((𝑇 ∘_f · 𝑋) ↾ (𝐵 ∪ {𝑧})) = ((𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥))) ↾ (𝐵 ∪ {𝑧})))
90	6	resmptd 5941	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥))) ↾ (𝐵 ∪ {𝑧})) = (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ ((𝑇‘𝑥) · (𝑋‘𝑥))))
91	89, 90	eqtrd 2778	. . . . . . 7 ⊢ (𝜑 → ((𝑇 ∘_f · 𝑋) ↾ (𝐵 ∪ {𝑧})) = (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ ((𝑇‘𝑥) · (𝑋‘𝑥))))
92	91	oveq2d 7283	. . . . . 6 ⊢ (𝜑 → (ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ (𝐵 ∪ {𝑧}))) = (ℂ_fld Σ_g (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ ((𝑇‘𝑥) · (𝑋‘𝑥)))))
93	88	reseq1d 5883	. . . . . . . . 9 ⊢ (𝜑 → ((𝑇 ∘_f · 𝑋) ↾ 𝐵) = ((𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥))) ↾ 𝐵))
94	7	resmptd 5941	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥))) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥))))
95	93, 94	eqtrd 2778	. . . . . . . 8 ⊢ (𝜑 → ((𝑇 ∘_f · 𝑋) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥))))
96	95	oveq2d 7283	. . . . . . 7 ⊢ (𝜑 → (ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) = (ℂ_fld Σ_g (𝑥 ∈ 𝐵 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥)))))
97	96	oveq1d 7282	. . . . . 6 ⊢ (𝜑 → ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) + ((𝑇‘𝑧) · (𝑋‘𝑧))) = ((ℂ_fld Σ_g (𝑥 ∈ 𝐵 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥)))) + ((𝑇‘𝑧) · (𝑋‘𝑧))))
98	85, 92, 97	3eqtr4d 2788	. . . . 5 ⊢ (𝜑 → (ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ (𝐵 ∪ {𝑧}))) = ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) + ((𝑇‘𝑧) · (𝑋‘𝑧))))
99	98	oveq1d 7282	. . . 4 ⊢ (𝜑 → ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ (𝐵 ∪ {𝑧}))) / 𝐿) = (((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) + ((𝑇‘𝑧) · (𝑋‘𝑧))) / 𝐿))
100	51	recnd 11013	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ℂ)
101	50	rpne0d 12787	. . . . . 6 ⊢ (𝜑 → 𝑆 ≠ 0)
102	29, 100, 57, 101, 66	dmdcand 11790	. . . . 5 ⊢ (𝜑 → ((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) = ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝐿))
103	57, 100, 57, 66	divsubdird 11800	. . . . . . . 8 ⊢ (𝜑 → ((𝐿 − 𝑆) / 𝐿) = ((𝐿 / 𝐿) − (𝑆 / 𝐿)))
104	100, 37, 49	mvrladdd 11398	. . . . . . . . 9 ⊢ (𝜑 → (𝐿 − 𝑆) = (𝑇‘𝑧))
105	104	oveq1d 7282	. . . . . . . 8 ⊢ (𝜑 → ((𝐿 − 𝑆) / 𝐿) = ((𝑇‘𝑧) / 𝐿))
106	57, 66	dividd 11759	. . . . . . . . 9 ⊢ (𝜑 → (𝐿 / 𝐿) = 1)
107	106	oveq1d 7282	. . . . . . . 8 ⊢ (𝜑 → ((𝐿 / 𝐿) − (𝑆 / 𝐿)) = (1 − (𝑆 / 𝐿)))
108	103, 105, 107	3eqtr3rd 2787	. . . . . . 7 ⊢ (𝜑 → (1 − (𝑆 / 𝐿)) = ((𝑇‘𝑧) / 𝐿))
109	108	oveq1d 7282	. . . . . 6 ⊢ (𝜑 → ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧)) = (((𝑇‘𝑧) / 𝐿) · (𝑋‘𝑧)))
110	37, 40, 57, 66	div23d 11798	. . . . . 6 ⊢ (𝜑 → (((𝑇‘𝑧) · (𝑋‘𝑧)) / 𝐿) = (((𝑇‘𝑧) / 𝐿) · (𝑋‘𝑧)))
111	109, 110	eqtr4d 2781	. . . . 5 ⊢ (𝜑 → ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧)) = (((𝑇‘𝑧) · (𝑋‘𝑧)) / 𝐿))
112	102, 111	oveq12d 7285	. . . 4 ⊢ (𝜑 → (((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧))) = (((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝐿) + (((𝑇‘𝑧) · (𝑋‘𝑧)) / 𝐿)))
113	67, 99, 112	3eqtr4d 2788	. . 3 ⊢ (𝜑 → ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ (𝐵 ∪ {𝑧}))) / 𝐿) = (((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧))))
114		jensenlem.4	. . . . 5 ⊢ (𝜑 → ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆) ∈ 𝐷)
115	51, 56, 66	redivcld 11813	. . . . . 6 ⊢ (𝜑 → (𝑆 / 𝐿) ∈ ℝ)
116	50	rpge0d 12786	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝑆)
117		divge0 11854	. . . . . . 7 ⊢ (((𝑆 ∈ ℝ ∧ 0 ≤ 𝑆) ∧ (𝐿 ∈ ℝ ∧ 0 < 𝐿)) → 0 ≤ (𝑆 / 𝐿))
118	51, 116, 56, 65, 117	syl22anc 836	. . . . . 6 ⊢ (𝜑 → 0 ≤ (𝑆 / 𝐿))
119	57	mulid1d 11002	. . . . . . . 8 ⊢ (𝜑 → (𝐿 · 1) = 𝐿)
120	64, 119	breqtrrd 5101	. . . . . . 7 ⊢ (𝜑 → 𝑆 ≤ (𝐿 · 1))
121		1red 10986	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
122		ledivmul 11861	. . . . . . . 8 ⊢ ((𝑆 ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 0 < 𝐿)) → ((𝑆 / 𝐿) ≤ 1 ↔ 𝑆 ≤ (𝐿 · 1)))
123	51, 121, 56, 65, 122	syl112anc 1373	. . . . . . 7 ⊢ (𝜑 → ((𝑆 / 𝐿) ≤ 1 ↔ 𝑆 ≤ (𝐿 · 1)))
124	120, 123	mpbird 256	. . . . . 6 ⊢ (𝜑 → (𝑆 / 𝐿) ≤ 1)
125		elicc01 13208	. . . . . 6 ⊢ ((𝑆 / 𝐿) ∈ (0[,]1) ↔ ((𝑆 / 𝐿) ∈ ℝ ∧ 0 ≤ (𝑆 / 𝐿) ∧ (𝑆 / 𝐿) ≤ 1))
126	115, 118, 124, 125	syl3anbrc 1342	. . . . 5 ⊢ (𝜑 → (𝑆 / 𝐿) ∈ (0[,]1))
127	114, 38, 126	3jca 1127	. . . 4 ⊢ (𝜑 → (((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆) ∈ 𝐷 ∧ (𝑋‘𝑧) ∈ 𝐷 ∧ (𝑆 / 𝐿) ∈ (0[,]1)))
128	20, 43	cvxcl 26144	. . . 4 ⊢ ((𝜑 ∧ (((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆) ∈ 𝐷 ∧ (𝑋‘𝑧) ∈ 𝐷 ∧ (𝑆 / 𝐿) ∈ (0[,]1))) → (((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧))) ∈ 𝐷)
129	127, 128	mpdan 684	. . 3 ⊢ (𝜑 → (((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧))) ∈ 𝐷)
130	113, 129	eqeltrd 2839	. 2 ⊢ (𝜑 → ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ (𝐵 ∪ {𝑧}))) / 𝐿) ∈ 𝐷)
131	42, 129	ffvelrnd 6954	. . . 4 ⊢ (𝜑 → (𝐹‘(((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧)))) ∈ ℝ)
132	42, 114	ffvelrnd 6954	. . . . . 6 ⊢ (𝜑 → (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) ∈ ℝ)
133	115, 132	remulcld 11015	. . . . 5 ⊢ (𝜑 → ((𝑆 / 𝐿) · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))) ∈ ℝ)
134	42, 38	ffvelrnd 6954	. . . . . . 7 ⊢ (𝜑 → (𝐹‘(𝑋‘𝑧)) ∈ ℝ)
135	54, 134	remulcld 11015	. . . . . 6 ⊢ (𝜑 → ((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) ∈ ℝ)
136	135, 56, 66	redivcld 11813	. . . . 5 ⊢ (𝜑 → (((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) / 𝐿) ∈ ℝ)
137	133, 136	readdcld 11014	. . . 4 ⊢ (𝜑 → (((𝑆 / 𝐿) · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))) + (((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) / 𝐿)) ∈ ℝ)
138		fco 6616	. . . . . . . . . . 11 ⊢ ((𝐹:𝐷⟶ℝ ∧ 𝑋:𝐴⟶𝐷) → (𝐹 ∘ 𝑋):𝐴⟶ℝ)
139	42, 19, 138	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∘ 𝑋):𝐴⟶ℝ)
140	14, 18, 139, 5, 5, 22	off 7541	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ∘_f · (𝐹 ∘ 𝑋)):𝐴⟶ℝ)
141	140, 7	fssresd 6633	. . . . . . . 8 ⊢ (𝜑 → ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵):𝐵⟶ℝ)
142	141, 8, 26	fdmfifsupp 9125	. . . . . . . 8 ⊢ (𝜑 → ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵) finSupp 0)
143	1, 4, 8, 12, 141, 142	gsumsubgcl 19531	. . . . . . 7 ⊢ (𝜑 → (ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) ∈ ℝ)
144	143, 51, 101	redivcld 11813	. . . . . 6 ⊢ (𝜑 → ((ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆) ∈ ℝ)
145	115, 144	remulcld 11015	. . . . 5 ⊢ (𝜑 → ((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆)) ∈ ℝ)
146		1re 10985	. . . . . . 7 ⊢ 1 ∈ ℝ
147		resubcl 11295	. . . . . . 7 ⊢ ((1 ∈ ℝ ∧ (𝑆 / 𝐿) ∈ ℝ) → (1 − (𝑆 / 𝐿)) ∈ ℝ)
148	146, 115, 147	sylancr 587	. . . . . 6 ⊢ (𝜑 → (1 − (𝑆 / 𝐿)) ∈ ℝ)
149	148, 134	remulcld 11015	. . . . 5 ⊢ (𝜑 → ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧))) ∈ ℝ)
150	145, 149	readdcld 11014	. . . 4 ⊢ (𝜑 → (((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧)))) ∈ ℝ)
151		oveq2 7275	. . . . . . . . . . 11 ⊢ (𝑥 = ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆) → (𝑡 · 𝑥) = (𝑡 · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)))
152	151	fvoveq1d 7289	. . . . . . . . . 10 ⊢ (𝑥 = ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆) → (𝐹‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) = (𝐹‘((𝑡 · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − 𝑡) · 𝑦))))
153		fveq2 6766	. . . . . . . . . . . 12 ⊢ (𝑥 = ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆) → (𝐹‘𝑥) = (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)))
154	153	oveq2d 7283	. . . . . . . . . . 11 ⊢ (𝑥 = ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆) → (𝑡 · (𝐹‘𝑥)) = (𝑡 · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))))
155	154	oveq1d 7282	. . . . . . . . . 10 ⊢ (𝑥 = ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆) → ((𝑡 · (𝐹‘𝑥)) + ((1 − 𝑡) · (𝐹‘𝑦))) = ((𝑡 · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − 𝑡) · (𝐹‘𝑦))))
156	152, 155	breq12d 5086	. . . . . . . . 9 ⊢ (𝑥 = ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆) → ((𝐹‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹‘𝑥)) + ((1 − 𝑡) · (𝐹‘𝑦))) ↔ (𝐹‘((𝑡 · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − 𝑡) · (𝐹‘𝑦)))))
157	156	imbi2d 341	. . . . . . . 8 ⊢ (𝑥 = ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆) → ((𝜑 → (𝐹‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹‘𝑥)) + ((1 − 𝑡) · (𝐹‘𝑦)))) ↔ (𝜑 → (𝐹‘((𝑡 · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − 𝑡) · (𝐹‘𝑦))))))
158		oveq2 7275	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑋‘𝑧) → ((1 − 𝑡) · 𝑦) = ((1 − 𝑡) · (𝑋‘𝑧)))
159	158	oveq2d 7283	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑋‘𝑧) → ((𝑡 · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − 𝑡) · 𝑦)) = ((𝑡 · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − 𝑡) · (𝑋‘𝑧))))
160	159	fveq2d 6770	. . . . . . . . . 10 ⊢ (𝑦 = (𝑋‘𝑧) → (𝐹‘((𝑡 · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − 𝑡) · 𝑦))) = (𝐹‘((𝑡 · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − 𝑡) · (𝑋‘𝑧)))))
161		fveq2 6766	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑋‘𝑧) → (𝐹‘𝑦) = (𝐹‘(𝑋‘𝑧)))
162	161	oveq2d 7283	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑋‘𝑧) → ((1 − 𝑡) · (𝐹‘𝑦)) = ((1 − 𝑡) · (𝐹‘(𝑋‘𝑧))))
163	162	oveq2d 7283	. . . . . . . . . 10 ⊢ (𝑦 = (𝑋‘𝑧) → ((𝑡 · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − 𝑡) · (𝐹‘𝑦))) = ((𝑡 · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − 𝑡) · (𝐹‘(𝑋‘𝑧)))))
164	160, 163	breq12d 5086	. . . . . . . . 9 ⊢ (𝑦 = (𝑋‘𝑧) → ((𝐹‘((𝑡 · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − 𝑡) · (𝐹‘𝑦))) ↔ (𝐹‘((𝑡 · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − 𝑡) · (𝑋‘𝑧)))) ≤ ((𝑡 · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − 𝑡) · (𝐹‘(𝑋‘𝑧))))))
165	164	imbi2d 341	. . . . . . . 8 ⊢ (𝑦 = (𝑋‘𝑧) → ((𝜑 → (𝐹‘((𝑡 · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − 𝑡) · (𝐹‘𝑦)))) ↔ (𝜑 → (𝐹‘((𝑡 · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − 𝑡) · (𝑋‘𝑧)))) ≤ ((𝑡 · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − 𝑡) · (𝐹‘(𝑋‘𝑧)))))))
166		oveq1 7274	. . . . . . . . . . . 12 ⊢ (𝑡 = (𝑆 / 𝐿) → (𝑡 · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) = ((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)))
167		oveq2 7275	. . . . . . . . . . . . 13 ⊢ (𝑡 = (𝑆 / 𝐿) → (1 − 𝑡) = (1 − (𝑆 / 𝐿)))
168	167	oveq1d 7282	. . . . . . . . . . . 12 ⊢ (𝑡 = (𝑆 / 𝐿) → ((1 − 𝑡) · (𝑋‘𝑧)) = ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧)))
169	166, 168	oveq12d 7285	. . . . . . . . . . 11 ⊢ (𝑡 = (𝑆 / 𝐿) → ((𝑡 · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − 𝑡) · (𝑋‘𝑧))) = (((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧))))
170	169	fveq2d 6770	. . . . . . . . . 10 ⊢ (𝑡 = (𝑆 / 𝐿) → (𝐹‘((𝑡 · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − 𝑡) · (𝑋‘𝑧)))) = (𝐹‘(((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧)))))
171		oveq1 7274	. . . . . . . . . . 11 ⊢ (𝑡 = (𝑆 / 𝐿) → (𝑡 · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))) = ((𝑆 / 𝐿) · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))))
172	167	oveq1d 7282	. . . . . . . . . . 11 ⊢ (𝑡 = (𝑆 / 𝐿) → ((1 − 𝑡) · (𝐹‘(𝑋‘𝑧))) = ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧))))
173	171, 172	oveq12d 7285	. . . . . . . . . 10 ⊢ (𝑡 = (𝑆 / 𝐿) → ((𝑡 · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − 𝑡) · (𝐹‘(𝑋‘𝑧)))) = (((𝑆 / 𝐿) · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧)))))
174	170, 173	breq12d 5086	. . . . . . . . 9 ⊢ (𝑡 = (𝑆 / 𝐿) → ((𝐹‘((𝑡 · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − 𝑡) · (𝑋‘𝑧)))) ≤ ((𝑡 · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − 𝑡) · (𝐹‘(𝑋‘𝑧)))) ↔ (𝐹‘(((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧)))) ≤ (((𝑆 / 𝐿) · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧))))))
175	174	imbi2d 341	. . . . . . . 8 ⊢ (𝑡 = (𝑆 / 𝐿) → ((𝜑 → (𝐹‘((𝑡 · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − 𝑡) · (𝑋‘𝑧)))) ≤ ((𝑡 · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − 𝑡) · (𝐹‘(𝑋‘𝑧))))) ↔ (𝜑 → (𝐹‘(((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧)))) ≤ (((𝑆 / 𝐿) · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧)))))))
176	45	expcom 414	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷 ∧ 𝑡 ∈ (0[,]1)) → (𝜑 → (𝐹‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹‘𝑥)) + ((1 − 𝑡) · (𝐹‘𝑦)))))
177	157, 165, 175, 176	vtocl3ga 3514	. . . . . . 7 ⊢ ((((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆) ∈ 𝐷 ∧ (𝑋‘𝑧) ∈ 𝐷 ∧ (𝑆 / 𝐿) ∈ (0[,]1)) → (𝜑 → (𝐹‘(((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧)))) ≤ (((𝑆 / 𝐿) · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧))))))
178	114, 38, 126, 177	syl3anc 1370	. . . . . 6 ⊢ (𝜑 → (𝜑 → (𝐹‘(((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧)))) ≤ (((𝑆 / 𝐿) · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧))))))
179	178	pm2.43i 52	. . . . 5 ⊢ (𝜑 → (𝐹‘(((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧)))) ≤ (((𝑆 / 𝐿) · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧)))))
180	108	oveq1d 7282	. . . . . . 7 ⊢ (𝜑 → ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧))) = (((𝑇‘𝑧) / 𝐿) · (𝐹‘(𝑋‘𝑧))))
181	134	recnd 11013	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘(𝑋‘𝑧)) ∈ ℂ)
182	37, 181, 57, 66	div23d 11798	. . . . . . 7 ⊢ (𝜑 → (((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) / 𝐿) = (((𝑇‘𝑧) / 𝐿) · (𝐹‘(𝑋‘𝑧))))
183	180, 182	eqtr4d 2781	. . . . . 6 ⊢ (𝜑 → ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧))) = (((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) / 𝐿))
184	183	oveq2d 7283	. . . . 5 ⊢ (𝜑 → (((𝑆 / 𝐿) · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧)))) = (((𝑆 / 𝐿) · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))) + (((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) / 𝐿)))
185	179, 184	breqtrd 5099	. . . 4 ⊢ (𝜑 → (𝐹‘(((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧)))) ≤ (((𝑆 / 𝐿) · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))) + (((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) / 𝐿)))
186	182, 180	eqtr4d 2781	. . . . . 6 ⊢ (𝜑 → (((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) / 𝐿) = ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧))))
187	186	oveq2d 7283	. . . . 5 ⊢ (𝜑 → (((𝑆 / 𝐿) · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))) + (((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) / 𝐿)) = (((𝑆 / 𝐿) · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧)))))
188		jensenlem.5	. . . . . . 7 ⊢ (𝜑 → (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) ≤ ((ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆))
189	51, 56, 59, 65	divgt0d 11920	. . . . . . . 8 ⊢ (𝜑 → 0 < (𝑆 / 𝐿))
190		lemul2 11838	. . . . . . . 8 ⊢ (((𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) ∈ ℝ ∧ ((ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆) ∈ ℝ ∧ ((𝑆 / 𝐿) ∈ ℝ ∧ 0 < (𝑆 / 𝐿))) → ((𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) ≤ ((ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆) ↔ ((𝑆 / 𝐿) · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))) ≤ ((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆))))
191	132, 144, 115, 189, 190	syl112anc 1373	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) ≤ ((ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆) ↔ ((𝑆 / 𝐿) · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))) ≤ ((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆))))
192	188, 191	mpbid 231	. . . . . 6 ⊢ (𝜑 → ((𝑆 / 𝐿) · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))) ≤ ((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆)))
193	133, 145, 149, 192	leadd1dd 11599	. . . . 5 ⊢ (𝜑 → (((𝑆 / 𝐿) · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))) + ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧)))) ≤ (((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧)))))
194	187, 193	eqbrtrd 5095	. . . 4 ⊢ (𝜑 → (((𝑆 / 𝐿) · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))) + (((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) / 𝐿)) ≤ (((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧)))))
195	131, 137, 150, 185, 194	letrd 11142	. . 3 ⊢ (𝜑 → (𝐹‘(((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧)))) ≤ (((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧)))))
196	113	fveq2d 6770	. . 3 ⊢ (𝜑 → (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ (𝐵 ∪ {𝑧}))) / 𝐿)) = (𝐹‘(((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧)))))
197	143	recnd 11013	. . . . 5 ⊢ (𝜑 → (ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) ∈ ℂ)
198	135	recnd 11013	. . . . 5 ⊢ (𝜑 → ((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) ∈ ℂ)
199	197, 198, 57, 66	divdird 11799	. . . 4 ⊢ (𝜑 → (((ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) + ((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧)))) / 𝐿) = (((ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝐿) + (((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) / 𝐿)))
200	16, 73	sselid 3918	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑇‘𝑥) ∈ ℝ)
201	42	ffvelrnda 6953	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋‘𝑥) ∈ 𝐷) → (𝐹‘(𝑋‘𝑥)) ∈ ℝ)
202	77, 201	syldan 591	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘(𝑋‘𝑥)) ∈ ℝ)
203	200, 202	remulcld 11015	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥))) ∈ ℝ)
204	203	recnd 11013	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥))) ∈ ℂ)
205	72, 204	syldan 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥))) ∈ ℂ)
206	83	fveq2d 6770	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝐹‘(𝑋‘𝑥)) = (𝐹‘(𝑋‘𝑧)))
207	82, 206	oveq12d 7285	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥))) = ((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))))
208	68, 69, 71, 8, 205, 35, 46, 198, 207	gsumunsn 19571	. . . . . 6 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥))))) = ((ℂ_fld Σ_g (𝑥 ∈ 𝐵 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥))))) + ((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧)))))
209	42	feqmptd 6829	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ 𝐷 ↦ (𝐹‘𝑦)))
210		fveq2 6766	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑋‘𝑥) → (𝐹‘𝑦) = (𝐹‘(𝑋‘𝑥)))
211	77, 87, 209, 210	fmptco 6993	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∘ 𝑋) = (𝑥 ∈ 𝐴 ↦ (𝐹‘(𝑋‘𝑥))))
212	5, 73, 202, 86, 211	offval2 7543	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ∘_f · (𝐹 ∘ 𝑋)) = (𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥)))))
213	212	reseq1d 5883	. . . . . . . 8 ⊢ (𝜑 → ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ (𝐵 ∪ {𝑧})) = ((𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥)))) ↾ (𝐵 ∪ {𝑧})))
214	6	resmptd 5941	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥)))) ↾ (𝐵 ∪ {𝑧})) = (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥)))))
215	213, 214	eqtrd 2778	. . . . . . 7 ⊢ (𝜑 → ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ (𝐵 ∪ {𝑧})) = (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥)))))
216	215	oveq2d 7283	. . . . . 6 ⊢ (𝜑 → (ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ (𝐵 ∪ {𝑧}))) = (ℂ_fld Σ_g (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥))))))
217	212	reseq1d 5883	. . . . . . . . 9 ⊢ (𝜑 → ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵) = ((𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥)))) ↾ 𝐵))
218	7	resmptd 5941	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥)))) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥)))))
219	217, 218	eqtrd 2778	. . . . . . . 8 ⊢ (𝜑 → ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥)))))
220	219	oveq2d 7283	. . . . . . 7 ⊢ (𝜑 → (ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) = (ℂ_fld Σ_g (𝑥 ∈ 𝐵 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥))))))
221	220	oveq1d 7282	. . . . . 6 ⊢ (𝜑 → ((ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) + ((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧)))) = ((ℂ_fld Σ_g (𝑥 ∈ 𝐵 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥))))) + ((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧)))))
222	208, 216, 221	3eqtr4d 2788	. . . . 5 ⊢ (𝜑 → (ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ (𝐵 ∪ {𝑧}))) = ((ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) + ((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧)))))
223	222	oveq1d 7282	. . . 4 ⊢ (𝜑 → ((ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ (𝐵 ∪ {𝑧}))) / 𝐿) = (((ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) + ((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧)))) / 𝐿))
224	197, 100, 57, 101, 66	dmdcand 11790	. . . . 5 ⊢ (𝜑 → ((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆)) = ((ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝐿))
225	224, 183	oveq12d 7285	. . . 4 ⊢ (𝜑 → (((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧)))) = (((ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝐿) + (((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) / 𝐿)))
226	199, 223, 225	3eqtr4d 2788	. . 3 ⊢ (𝜑 → ((ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ (𝐵 ∪ {𝑧}))) / 𝐿) = (((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧)))))
227	195, 196, 226	3brtr4d 5105	. 2 ⊢ (𝜑 → (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ (𝐵 ∪ {𝑧}))) / 𝐿)) ≤ ((ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ (𝐵 ∪ {𝑧}))) / 𝐿))
228	130, 227	jca 512	1 ⊢ (𝜑 → (((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ (𝐵 ∪ {𝑧}))) / 𝐿) ∈ 𝐷 ∧ (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ (𝐵 ∪ {𝑧}))) / 𝐿)) ≤ ((ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ (𝐵 ∪ {𝑧}))) / 𝐿)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 Vcvv 3429 ∪ cun 3884 ⊆ wss 3886 {csn 4561 class class class wbr 5073 ↦ cmpt 5156 ↾ cres 5586 ∘ ccom 5588 ⟶wf 6422 ‘cfv 6426 (class class class)co 7267 ∘_f cof 7521 Fincfn 8720 ℂcc 10879 ℝcr 10880 0cc0 10881 1c1 10882 + caddc 10884 · cmul 10886 +∞cpnf 11016 < clt 11019 ≤ cle 11020 − cmin 11215 / cdiv 11642 ℝ⁺crp 12740 [,)cico 13091 [,]cicc 13092 Σ_g cgsu 17161 SubGrpcsubg 18759 CMndccmn 19396 Abelcabl 19397 Ringcrg 19793 DivRingcdr 20001 SubRingcsubrg 20030 ℂ_fldccnfld 20607 ℝ_fldcrefld 20819

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 ax-addf 10960 ax-mulf 10961

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-se 5540 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-isom 6435 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-of 7523 df-om 7703 df-1st 7820 df-2nd 7821 df-supp 7965 df-tpos 8029 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-1o 8284 df-er 8485 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 df-fsupp 9116 df-oi 9256 df-card 9707 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-div 11643 df-nn 11984 df-2 12046 df-3 12047 df-4 12048 df-5 12049 df-6 12050 df-7 12051 df-8 12052 df-9 12053 df-n0 12244 df-z 12330 df-dec 12448 df-uz 12593 df-rp 12741 df-ico 13095 df-icc 13096 df-fz 13250 df-fzo 13393 df-seq 13732 df-hash 14055 df-struct 16858 df-sets 16875 df-slot 16893 df-ndx 16905 df-base 16923 df-ress 16952 df-plusg 16985 df-mulr 16986 df-starv 16987 df-tset 16991 df-ple 16992 df-ds 16994 df-unif 16995 df-0g 17162 df-gsum 17163 df-mre 17305 df-mrc 17306 df-acs 17308 df-mgm 18336 df-sgrp 18385 df-mnd 18396 df-submnd 18441 df-grp 18590 df-minusg 18591 df-mulg 18711 df-subg 18762 df-cntz 18933 df-cmn 19398 df-abl 19399 df-mgp 19731 df-ur 19748 df-ring 19795 df-cring 19796 df-oppr 19872 df-dvdsr 19893 df-unit 19894 df-invr 19924 df-dvr 19935 df-drng 20003 df-subrg 20032 df-cnfld 20608 df-refld 20820

This theorem is referenced by: jensen 26148

W3C validator