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Theorem jensenlem2 25170

Description: Lemma for jensen 25171. (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 20170	. . . . . . 7 ⊢ 0 = (0_g‘ℂ_fld)
2		cnring 20168	. . . . . . . 8 ⊢ ℂ_fld ∈ Ring
3		ringabl 18971	. . . . . . . 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 4013	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
8		ssfi 8470	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin)
9	5, 7, 8	syl2anc 579	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ Fin)
10		resubdrg 20355	. . . . . . . . 9 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) ∧ ℝ_fld ∈ DivRing)
11	10	simpli 478	. . . . . . . 8 ⊢ ℝ ∈ (SubRing‘ℂ_fld)
12		subrgsubg 19182	. . . . . . . 8 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) → ℝ ∈ (SubGrp‘ℂ_fld))
13	11, 12	mp1i 13	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ (SubGrp‘ℂ_fld))
14		remulcl 10359	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 𝑦) ∈ ℝ)
15	14	adantl 475	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 · 𝑦) ∈ ℝ)
16		jensen.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇:𝐴⟶(0[,)+∞))
17		rge0ssre 12598	. . . . . . . . . 10 ⊢ (0[,)+∞) ⊆ ℝ
18		fss 6306	. . . . . . . . . 10 ⊢ ((𝑇:𝐴⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝑇:𝐴⟶ℝ)
19	16, 17, 18	sylancl 580	. . . . . . . . 9 ⊢ (𝜑 → 𝑇:𝐴⟶ℝ)
20		jensen.6	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋:𝐴⟶𝐷)
21		jensen.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ⊆ ℝ)
22	20, 21	fssd 6307	. . . . . . . . 9 ⊢ (𝜑 → 𝑋:𝐴⟶ℝ)
23		inidm 4043	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐴) = 𝐴
24	15, 19, 22, 5, 5, 23	off 7191	. . . . . . . 8 ⊢ (𝜑 → (𝑇 ∘_𝑓 · 𝑋):𝐴⟶ℝ)
25	24, 7	fssresd 6323	. . . . . . 7 ⊢ (𝜑 → ((𝑇 ∘_𝑓 · 𝑋) ↾ 𝐵):𝐵⟶ℝ)
26		c0ex 10372	. . . . . . . . 9 ⊢ 0 ∈ V
27	26	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ V)
28	25, 9, 27	fdmfifsupp 8575	. . . . . . 7 ⊢ (𝜑 → ((𝑇 ∘_𝑓 · 𝑋) ↾ 𝐵) finSupp 0)
29	1, 4, 9, 13, 25, 28	gsumsubgcl 18710	. . . . . 6 ⊢ (𝜑 → (ℂ_fld Σ_g ((𝑇 ∘_𝑓 · 𝑋) ↾ 𝐵)) ∈ ℝ)
30	29	recnd 10407	. . . . 5 ⊢ (𝜑 → (ℂ_fld Σ_g ((𝑇 ∘_𝑓 · 𝑋) ↾ 𝐵)) ∈ ℂ)
31		ax-resscn 10331	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
32	17, 31	sstri 3830	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℂ
33	6	unssbd 4014	. . . . . . . . 9 ⊢ (𝜑 → {𝑧} ⊆ 𝐴)
34		vex 3401	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
35	34	snss 4549	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 ↔ {𝑧} ⊆ 𝐴)
36	33, 35	sylibr 226	. . . . . . . 8 ⊢ (𝜑 → 𝑧 ∈ 𝐴)
37	16, 36	ffvelrnd 6626	. . . . . . 7 ⊢ (𝜑 → (𝑇‘𝑧) ∈ (0[,)+∞))
38	32, 37	sseldi 3819	. . . . . 6 ⊢ (𝜑 → (𝑇‘𝑧) ∈ ℂ)
39	20, 36	ffvelrnd 6626	. . . . . . . 8 ⊢ (𝜑 → (𝑋‘𝑧) ∈ 𝐷)
40	21, 39	sseldd 3822	. . . . . . 7 ⊢ (𝜑 → (𝑋‘𝑧) ∈ ℝ)
41	40	recnd 10407	. . . . . 6 ⊢ (𝜑 → (𝑋‘𝑧) ∈ ℂ)
42	38, 41	mulcld 10399	. . . . 5 ⊢ (𝜑 → ((𝑇‘𝑧) · (𝑋‘𝑧)) ∈ ℂ)
43		jensen.2	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐷⟶ℝ)
44		jensen.3	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐷 ∧ 𝑏 ∈ 𝐷)) → (𝑎[,]𝑏) ⊆ 𝐷)
45		jensen.7	. . . . . . . 8 ⊢ (𝜑 → 0 < (ℂ_fld Σ_g 𝑇))
46		jensen.8	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷 ∧ 𝑡 ∈ (0[,]1))) → (𝐹‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹‘𝑥)) + ((1 − 𝑡) · (𝐹‘𝑦))))
47		jensenlem.1	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝐵)
48		jensenlem.s	. . . . . . . 8 ⊢ 𝑆 = (ℂ_fld Σ_g (𝑇 ↾ 𝐵))
49		jensenlem.l	. . . . . . . 8 ⊢ 𝐿 = (ℂ_fld Σ_g (𝑇 ↾ (𝐵 ∪ {𝑧})))
50	21, 43, 44, 5, 16, 20, 45, 46, 47, 6, 48, 49	jensenlem1 25169	. . . . . . 7 ⊢ (𝜑 → 𝐿 = (𝑆 + (𝑇‘𝑧)))
51		jensenlem.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ ℝ⁺)
52	51	rpred 12185	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℝ)
53		elrege0 12596	. . . . . . . . . 10 ⊢ ((𝑇‘𝑧) ∈ (0[,)+∞) ↔ ((𝑇‘𝑧) ∈ ℝ ∧ 0 ≤ (𝑇‘𝑧)))
54	53	simplbi 493	. . . . . . . . 9 ⊢ ((𝑇‘𝑧) ∈ (0[,)+∞) → (𝑇‘𝑧) ∈ ℝ)
55	37, 54	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑇‘𝑧) ∈ ℝ)
56	52, 55	readdcld 10408	. . . . . . 7 ⊢ (𝜑 → (𝑆 + (𝑇‘𝑧)) ∈ ℝ)
57	50, 56	eqeltrd 2859	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ ℝ)
58	57	recnd 10407	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ ℂ)
59		0red 10382	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ)
60	51	rpgt0d 12188	. . . . . . 7 ⊢ (𝜑 → 0 < 𝑆)
61	53	simprbi 492	. . . . . . . . . 10 ⊢ ((𝑇‘𝑧) ∈ (0[,)+∞) → 0 ≤ (𝑇‘𝑧))
62	37, 61	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (𝑇‘𝑧))
63	52, 55	addge01d 10965	. . . . . . . . 9 ⊢ (𝜑 → (0 ≤ (𝑇‘𝑧) ↔ 𝑆 ≤ (𝑆 + (𝑇‘𝑧))))
64	62, 63	mpbid 224	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ≤ (𝑆 + (𝑇‘𝑧)))
65	64, 50	breqtrrd 4916	. . . . . . 7 ⊢ (𝜑 → 𝑆 ≤ 𝐿)
66	59, 52, 57, 60, 65	ltletrd 10538	. . . . . 6 ⊢ (𝜑 → 0 < 𝐿)
67	66	gt0ne0d 10941	. . . . 5 ⊢ (𝜑 → 𝐿 ≠ 0)
68	30, 42, 58, 67	divdird 11191	. . . 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)))
69		cnfldbas 20150	. . . . . . 7 ⊢ ℂ = (Base‘ℂ_fld)
70		cnfldadd 20151	. . . . . . 7 ⊢ + = (+_g‘ℂ_fld)
71		ringcmn 18972	. . . . . . . 8 ⊢ (ℂ_fld ∈ Ring → ℂ_fld ∈ CMnd)
72	2, 71	mp1i 13	. . . . . . 7 ⊢ (𝜑 → ℂ_fld ∈ CMnd)
73	7	sselda 3821	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴)
74	16	ffvelrnda 6625	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑇‘𝑥) ∈ (0[,)+∞))
75	73, 74	syldan 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑇‘𝑥) ∈ (0[,)+∞))
76	32, 75	sseldi 3819	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑇‘𝑥) ∈ ℂ)
77	21	adantr 474	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐷 ⊆ ℝ)
78	20	ffvelrnda 6625	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑋‘𝑥) ∈ 𝐷)
79	73, 78	syldan 585	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑋‘𝑥) ∈ 𝐷)
80	77, 79	sseldd 3822	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑋‘𝑥) ∈ ℝ)
81	80	recnd 10407	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑋‘𝑥) ∈ ℂ)
82	76, 81	mulcld 10399	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑇‘𝑥) · (𝑋‘𝑥)) ∈ ℂ)
83		fveq2 6448	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑇‘𝑥) = (𝑇‘𝑧))
84		fveq2 6448	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑋‘𝑥) = (𝑋‘𝑧))
85	83, 84	oveq12d 6942	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝑇‘𝑥) · (𝑋‘𝑥)) = ((𝑇‘𝑧) · (𝑋‘𝑧)))
86	69, 70, 72, 9, 82, 36, 47, 42, 85	gsumunsn 18749	. . . . . 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))))
87	16	feqmptd 6511	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 = (𝑥 ∈ 𝐴 ↦ (𝑇‘𝑥)))
88	20	feqmptd 6511	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 = (𝑥 ∈ 𝐴 ↦ (𝑋‘𝑥)))
89	5, 74, 78, 87, 88	offval2 7193	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ∘_𝑓 · 𝑋) = (𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥))))
90	89	reseq1d 5643	. . . . . . . 8 ⊢ (𝜑 → ((𝑇 ∘_𝑓 · 𝑋) ↾ (𝐵 ∪ {𝑧})) = ((𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥))) ↾ (𝐵 ∪ {𝑧})))
91	6	resmptd 5704	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥))) ↾ (𝐵 ∪ {𝑧})) = (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ ((𝑇‘𝑥) · (𝑋‘𝑥))))
92	90, 91	eqtrd 2814	. . . . . . 7 ⊢ (𝜑 → ((𝑇 ∘_𝑓 · 𝑋) ↾ (𝐵 ∪ {𝑧})) = (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ ((𝑇‘𝑥) · (𝑋‘𝑥))))
93	92	oveq2d 6940	. . . . . 6 ⊢ (𝜑 → (ℂ_fld Σ_g ((𝑇 ∘_𝑓 · 𝑋) ↾ (𝐵 ∪ {𝑧}))) = (ℂ_fld Σ_g (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ ((𝑇‘𝑥) · (𝑋‘𝑥)))))
94	89	reseq1d 5643	. . . . . . . . 9 ⊢ (𝜑 → ((𝑇 ∘_𝑓 · 𝑋) ↾ 𝐵) = ((𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥))) ↾ 𝐵))
95	7	resmptd 5704	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥))) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥))))
96	94, 95	eqtrd 2814	. . . . . . . 8 ⊢ (𝜑 → ((𝑇 ∘_𝑓 · 𝑋) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥))))
97	96	oveq2d 6940	. . . . . . 7 ⊢ (𝜑 → (ℂ_fld Σ_g ((𝑇 ∘_𝑓 · 𝑋) ↾ 𝐵)) = (ℂ_fld Σ_g (𝑥 ∈ 𝐵 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥)))))
98	97	oveq1d 6939	. . . . . 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))))
99	86, 93, 98	3eqtr4d 2824	. . . . 5 ⊢ (𝜑 → (ℂ_fld Σ_g ((𝑇 ∘_𝑓 · 𝑋) ↾ (𝐵 ∪ {𝑧}))) = ((ℂ_fld Σ_g ((𝑇 ∘_𝑓 · 𝑋) ↾ 𝐵)) + ((𝑇‘𝑧) · (𝑋‘𝑧))))
100	99	oveq1d 6939	. . . 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))
101	52	recnd 10407	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ℂ)
102	51	rpne0d 12190	. . . . . 6 ⊢ (𝜑 → 𝑆 ≠ 0)
103	30, 101, 58, 102, 67	dmdcand 11182	. . . . 5 ⊢ (𝜑 → ((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_𝑓 · 𝑋) ↾ 𝐵)) / 𝑆)) = ((ℂ_fld Σ_g ((𝑇 ∘_𝑓 · 𝑋) ↾ 𝐵)) / 𝐿))
104	58, 101, 58, 67	divsubdird 11192	. . . . . . . 8 ⊢ (𝜑 → ((𝐿 − 𝑆) / 𝐿) = ((𝐿 / 𝐿) − (𝑆 / 𝐿)))
105	101, 38, 50	mvrladdd 10790	. . . . . . . . 9 ⊢ (𝜑 → (𝐿 − 𝑆) = (𝑇‘𝑧))
106	105	oveq1d 6939	. . . . . . . 8 ⊢ (𝜑 → ((𝐿 − 𝑆) / 𝐿) = ((𝑇‘𝑧) / 𝐿))
107	58, 67	dividd 11151	. . . . . . . . 9 ⊢ (𝜑 → (𝐿 / 𝐿) = 1)
108	107	oveq1d 6939	. . . . . . . 8 ⊢ (𝜑 → ((𝐿 / 𝐿) − (𝑆 / 𝐿)) = (1 − (𝑆 / 𝐿)))
109	104, 106, 108	3eqtr3rd 2823	. . . . . . 7 ⊢ (𝜑 → (1 − (𝑆 / 𝐿)) = ((𝑇‘𝑧) / 𝐿))
110	109	oveq1d 6939	. . . . . 6 ⊢ (𝜑 → ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧)) = (((𝑇‘𝑧) / 𝐿) · (𝑋‘𝑧)))
111	38, 41, 58, 67	div23d 11190	. . . . . 6 ⊢ (𝜑 → (((𝑇‘𝑧) · (𝑋‘𝑧)) / 𝐿) = (((𝑇‘𝑧) / 𝐿) · (𝑋‘𝑧)))
112	110, 111	eqtr4d 2817	. . . . 5 ⊢ (𝜑 → ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧)) = (((𝑇‘𝑧) · (𝑋‘𝑧)) / 𝐿))
113	103, 112	oveq12d 6942	. . . 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)))
114	68, 100, 113	3eqtr4d 2824	. . 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))))
115		jensenlem.4	. . . . 5 ⊢ (𝜑 → ((ℂ_fld Σ_g ((𝑇 ∘_𝑓 · 𝑋) ↾ 𝐵)) / 𝑆) ∈ 𝐷)
116	52, 57, 67	redivcld 11205	. . . . . 6 ⊢ (𝜑 → (𝑆 / 𝐿) ∈ ℝ)
117	51	rpge0d 12189	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝑆)
118		divge0 11248	. . . . . . 7 ⊢ (((𝑆 ∈ ℝ ∧ 0 ≤ 𝑆) ∧ (𝐿 ∈ ℝ ∧ 0 < 𝐿)) → 0 ≤ (𝑆 / 𝐿))
119	52, 117, 57, 66, 118	syl22anc 829	. . . . . 6 ⊢ (𝜑 → 0 ≤ (𝑆 / 𝐿))
120	58	mulid1d 10396	. . . . . . . 8 ⊢ (𝜑 → (𝐿 · 1) = 𝐿)
121	65, 120	breqtrrd 4916	. . . . . . 7 ⊢ (𝜑 → 𝑆 ≤ (𝐿 · 1))
122		1red 10379	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
123		ledivmul 11255	. . . . . . . 8 ⊢ ((𝑆 ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 0 < 𝐿)) → ((𝑆 / 𝐿) ≤ 1 ↔ 𝑆 ≤ (𝐿 · 1)))
124	52, 122, 57, 66, 123	syl112anc 1442	. . . . . . 7 ⊢ (𝜑 → ((𝑆 / 𝐿) ≤ 1 ↔ 𝑆 ≤ (𝐿 · 1)))
125	121, 124	mpbird 249	. . . . . 6 ⊢ (𝜑 → (𝑆 / 𝐿) ≤ 1)
126		elicc01 12608	. . . . . 6 ⊢ ((𝑆 / 𝐿) ∈ (0[,]1) ↔ ((𝑆 / 𝐿) ∈ ℝ ∧ 0 ≤ (𝑆 / 𝐿) ∧ (𝑆 / 𝐿) ≤ 1))
127	116, 119, 125, 126	syl3anbrc 1400	. . . . 5 ⊢ (𝜑 → (𝑆 / 𝐿) ∈ (0[,]1))
128	115, 39, 127	3jca 1119	. . . 4 ⊢ (𝜑 → (((ℂ_fld Σ_g ((𝑇 ∘_𝑓 · 𝑋) ↾ 𝐵)) / 𝑆) ∈ 𝐷 ∧ (𝑋‘𝑧) ∈ 𝐷 ∧ (𝑆 / 𝐿) ∈ (0[,]1)))
129	21, 44	cvxcl 25167	. . . 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)
130	128, 129	mpdan 677	. . 3 ⊢ (𝜑 → (((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_𝑓 · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧))) ∈ 𝐷)
131	114, 130	eqeltrd 2859	. 2 ⊢ (𝜑 → ((ℂ_fld Σ_g ((𝑇 ∘_𝑓 · 𝑋) ↾ (𝐵 ∪ {𝑧}))) / 𝐿) ∈ 𝐷)
132	43, 130	ffvelrnd 6626	. . . 4 ⊢ (𝜑 → (𝐹‘(((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_𝑓 · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧)))) ∈ ℝ)
133	43, 115	ffvelrnd 6626	. . . . . 6 ⊢ (𝜑 → (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_𝑓 · 𝑋) ↾ 𝐵)) / 𝑆)) ∈ ℝ)
134	116, 133	remulcld 10409	. . . . 5 ⊢ (𝜑 → ((𝑆 / 𝐿) · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_𝑓 · 𝑋) ↾ 𝐵)) / 𝑆))) ∈ ℝ)
135	43, 39	ffvelrnd 6626	. . . . . . 7 ⊢ (𝜑 → (𝐹‘(𝑋‘𝑧)) ∈ ℝ)
136	55, 135	remulcld 10409	. . . . . 6 ⊢ (𝜑 → ((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) ∈ ℝ)
137	136, 57, 67	redivcld 11205	. . . . 5 ⊢ (𝜑 → (((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) / 𝐿) ∈ ℝ)
138	134, 137	readdcld 10408	. . . 4 ⊢ (𝜑 → (((𝑆 / 𝐿) · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_𝑓 · 𝑋) ↾ 𝐵)) / 𝑆))) + (((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) / 𝐿)) ∈ ℝ)
139		fco 6310	. . . . . . . . . . 11 ⊢ ((𝐹:𝐷⟶ℝ ∧ 𝑋:𝐴⟶𝐷) → (𝐹 ∘ 𝑋):𝐴⟶ℝ)
140	43, 20, 139	syl2anc 579	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∘ 𝑋):𝐴⟶ℝ)
141	15, 19, 140, 5, 5, 23	off 7191	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ∘_𝑓 · (𝐹 ∘ 𝑋)):𝐴⟶ℝ)
142	141, 7	fssresd 6323	. . . . . . . 8 ⊢ (𝜑 → ((𝑇 ∘_𝑓 · (𝐹 ∘ 𝑋)) ↾ 𝐵):𝐵⟶ℝ)
143	142, 9, 27	fdmfifsupp 8575	. . . . . . . 8 ⊢ (𝜑 → ((𝑇 ∘_𝑓 · (𝐹 ∘ 𝑋)) ↾ 𝐵) finSupp 0)
144	1, 4, 9, 13, 142, 143	gsumsubgcl 18710	. . . . . . 7 ⊢ (𝜑 → (ℂ_fld Σ_g ((𝑇 ∘_𝑓 · (𝐹 ∘ 𝑋)) ↾ 𝐵)) ∈ ℝ)
145	144, 52, 102	redivcld 11205	. . . . . 6 ⊢ (𝜑 → ((ℂ_fld Σ_g ((𝑇 ∘_𝑓 · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆) ∈ ℝ)
146	116, 145	remulcld 10409	. . . . 5 ⊢ (𝜑 → ((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_𝑓 · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆)) ∈ ℝ)
147		1re 10378	. . . . . . 7 ⊢ 1 ∈ ℝ
148		resubcl 10689	. . . . . . 7 ⊢ ((1 ∈ ℝ ∧ (𝑆 / 𝐿) ∈ ℝ) → (1 − (𝑆 / 𝐿)) ∈ ℝ)
149	147, 116, 148	sylancr 581	. . . . . 6 ⊢ (𝜑 → (1 − (𝑆 / 𝐿)) ∈ ℝ)
150	149, 135	remulcld 10409	. . . . 5 ⊢ (𝜑 → ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧))) ∈ ℝ)
151	146, 150	readdcld 10408	. . . 4 ⊢ (𝜑 → (((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_𝑓 · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧)))) ∈ ℝ)
152		oveq2 6932	. . . . . . . . . . 11 ⊢ (𝑥 = ((ℂ_fld Σ_g ((𝑇 ∘_𝑓 · 𝑋) ↾ 𝐵)) / 𝑆) → (𝑡 · 𝑥) = (𝑡 · ((ℂ_fld Σ_g ((𝑇 ∘_𝑓 · 𝑋) ↾ 𝐵)) / 𝑆)))
153	152	fvoveq1d 6946	. . . . . . . . . 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))))
154		fveq2 6448	. . . . . . . . . . . 12 ⊢ (𝑥 = ((ℂ_fld Σ_g ((𝑇 ∘_𝑓 · 𝑋) ↾ 𝐵)) / 𝑆) → (𝐹‘𝑥) = (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_𝑓 · 𝑋) ↾ 𝐵)) / 𝑆)))
155	154	oveq2d 6940	. . . . . . . . . . 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))))
156	155	oveq1d 6939	. . . . . . . . . 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))))
157	153, 156	breq12d 4901	. . . . . . . . 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)))))
158	157	imbi2d 332	. . . . . . . 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))))))
159		oveq2 6932	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑋‘𝑧) → ((1 − 𝑡) · 𝑦) = ((1 − 𝑡) · (𝑋‘𝑧)))
160	159	oveq2d 6940	. . . . . . . . . . 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))))
161	160	fveq2d 6452	. . . . . . . . . 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)))))
162		fveq2 6448	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑋‘𝑧) → (𝐹‘𝑦) = (𝐹‘(𝑋‘𝑧)))
163	162	oveq2d 6940	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑋‘𝑧) → ((1 − 𝑡) · (𝐹‘𝑦)) = ((1 − 𝑡) · (𝐹‘(𝑋‘𝑧))))
164	163	oveq2d 6940	. . . . . . . . . 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)))))
165	161, 164	breq12d 4901	. . . . . . . . 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))))))
166	165	imbi2d 332	. . . . . . . 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)))))))
167		oveq1 6931	. . . . . . . . . . . 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)))
168		oveq2 6932	. . . . . . . . . . . . 13 ⊢ (𝑡 = (𝑆 / 𝐿) → (1 − 𝑡) = (1 − (𝑆 / 𝐿)))
169	168	oveq1d 6939	. . . . . . . . . . . 12 ⊢ (𝑡 = (𝑆 / 𝐿) → ((1 − 𝑡) · (𝑋‘𝑧)) = ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧)))
170	167, 169	oveq12d 6942	. . . . . . . . . . 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))))
171	170	fveq2d 6452	. . . . . . . . . 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)))))
172		oveq1 6931	. . . . . . . . . . 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))))
173	168	oveq1d 6939	. . . . . . . . . . 11 ⊢ (𝑡 = (𝑆 / 𝐿) → ((1 − 𝑡) · (𝐹‘(𝑋‘𝑧))) = ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧))))
174	172, 173	oveq12d 6942	. . . . . . . . . 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)))))
175	171, 174	breq12d 4901	. . . . . . . . 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))))))
176	175	imbi2d 332	. . . . . . . 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)))))))
177	46	expcom 404	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷 ∧ 𝑡 ∈ (0[,]1)) → (𝜑 → (𝐹‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹‘𝑥)) + ((1 − 𝑡) · (𝐹‘𝑦)))))
178	158, 166, 176, 177	vtocl3ga 3478	. . . . . . 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))))))
179	115, 39, 127, 178	syl3anc 1439	. . . . . 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))))))
180	179	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)))))
181	109	oveq1d 6939	. . . . . . 7 ⊢ (𝜑 → ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧))) = (((𝑇‘𝑧) / 𝐿) · (𝐹‘(𝑋‘𝑧))))
182	135	recnd 10407	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘(𝑋‘𝑧)) ∈ ℂ)
183	38, 182, 58, 67	div23d 11190	. . . . . . 7 ⊢ (𝜑 → (((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) / 𝐿) = (((𝑇‘𝑧) / 𝐿) · (𝐹‘(𝑋‘𝑧))))
184	181, 183	eqtr4d 2817	. . . . . 6 ⊢ (𝜑 → ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧))) = (((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) / 𝐿))
185	184	oveq2d 6940	. . . . 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)))
186	180, 185	breqtrd 4914	. . . 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)))
187	183, 181	eqtr4d 2817	. . . . . 6 ⊢ (𝜑 → (((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) / 𝐿) = ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧))))
188	187	oveq2d 6940	. . . . 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)))))
189		jensenlem.5	. . . . . . 7 ⊢ (𝜑 → (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_𝑓 · 𝑋) ↾ 𝐵)) / 𝑆)) ≤ ((ℂ_fld Σ_g ((𝑇 ∘_𝑓 · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆))
190	52, 57, 60, 66	divgt0d 11315	. . . . . . . 8 ⊢ (𝜑 → 0 < (𝑆 / 𝐿))
191		lemul2 11232	. . . . . . . 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))))
192	133, 145, 116, 190, 191	syl112anc 1442	. . . . . . 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))))
193	189, 192	mpbid 224	. . . . . 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)))
194	134, 146, 150, 193	leadd1dd 10991	. . . . 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)))))
195	188, 194	eqbrtrd 4910	. . . 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)))))
196	132, 138, 151, 186, 195	letrd 10535	. . 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)))))
197	114	fveq2d 6452	. . 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)))))
198	144	recnd 10407	. . . . 5 ⊢ (𝜑 → (ℂ_fld Σ_g ((𝑇 ∘_𝑓 · (𝐹 ∘ 𝑋)) ↾ 𝐵)) ∈ ℂ)
199	136	recnd 10407	. . . . 5 ⊢ (𝜑 → ((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) ∈ ℂ)
200	198, 199, 58, 67	divdird 11191	. . . 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)))
201	17, 74	sseldi 3819	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑇‘𝑥) ∈ ℝ)
202	43	ffvelrnda 6625	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋‘𝑥) ∈ 𝐷) → (𝐹‘(𝑋‘𝑥)) ∈ ℝ)
203	78, 202	syldan 585	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘(𝑋‘𝑥)) ∈ ℝ)
204	201, 203	remulcld 10409	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥))) ∈ ℝ)
205	204	recnd 10407	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥))) ∈ ℂ)
206	73, 205	syldan 585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥))) ∈ ℂ)
207	84	fveq2d 6452	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝐹‘(𝑋‘𝑥)) = (𝐹‘(𝑋‘𝑧)))
208	83, 207	oveq12d 6942	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥))) = ((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))))
209	69, 70, 72, 9, 206, 36, 47, 199, 208	gsumunsn 18749	. . . . . 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)))))
210	43	feqmptd 6511	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ 𝐷 ↦ (𝐹‘𝑦)))
211		fveq2 6448	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑋‘𝑥) → (𝐹‘𝑦) = (𝐹‘(𝑋‘𝑥)))
212	78, 88, 210, 211	fmptco 6663	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∘ 𝑋) = (𝑥 ∈ 𝐴 ↦ (𝐹‘(𝑋‘𝑥))))
213	5, 74, 203, 87, 212	offval2 7193	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ∘_𝑓 · (𝐹 ∘ 𝑋)) = (𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥)))))
214	213	reseq1d 5643	. . . . . . . 8 ⊢ (𝜑 → ((𝑇 ∘_𝑓 · (𝐹 ∘ 𝑋)) ↾ (𝐵 ∪ {𝑧})) = ((𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥)))) ↾ (𝐵 ∪ {𝑧})))
215	6	resmptd 5704	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥)))) ↾ (𝐵 ∪ {𝑧})) = (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥)))))
216	214, 215	eqtrd 2814	. . . . . . 7 ⊢ (𝜑 → ((𝑇 ∘_𝑓 · (𝐹 ∘ 𝑋)) ↾ (𝐵 ∪ {𝑧})) = (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥)))))
217	216	oveq2d 6940	. . . . . 6 ⊢ (𝜑 → (ℂ_fld Σ_g ((𝑇 ∘_𝑓 · (𝐹 ∘ 𝑋)) ↾ (𝐵 ∪ {𝑧}))) = (ℂ_fld Σ_g (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥))))))
218	213	reseq1d 5643	. . . . . . . . 9 ⊢ (𝜑 → ((𝑇 ∘_𝑓 · (𝐹 ∘ 𝑋)) ↾ 𝐵) = ((𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥)))) ↾ 𝐵))
219	7	resmptd 5704	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥)))) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥)))))
220	218, 219	eqtrd 2814	. . . . . . . 8 ⊢ (𝜑 → ((𝑇 ∘_𝑓 · (𝐹 ∘ 𝑋)) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥)))))
221	220	oveq2d 6940	. . . . . . 7 ⊢ (𝜑 → (ℂ_fld Σ_g ((𝑇 ∘_𝑓 · (𝐹 ∘ 𝑋)) ↾ 𝐵)) = (ℂ_fld Σ_g (𝑥 ∈ 𝐵 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥))))))
222	221	oveq1d 6939	. . . . . 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)))))
223	209, 217, 222	3eqtr4d 2824	. . . . 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)))))
224	223	oveq1d 6939	. . . 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))
225	198, 101, 58, 102, 67	dmdcand 11182	. . . . 5 ⊢ (𝜑 → ((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_𝑓 · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆)) = ((ℂ_fld Σ_g ((𝑇 ∘_𝑓 · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝐿))
226	225, 184	oveq12d 6942	. . . 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)))
227	200, 224, 226	3eqtr4d 2824	. . 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)))))
228	196, 197, 227	3brtr4d 4920	. 2 ⊢ (𝜑 → (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_𝑓 · 𝑋) ↾ (𝐵 ∪ {𝑧}))) / 𝐿)) ≤ ((ℂ_fld Σ_g ((𝑇 ∘_𝑓 · (𝐹 ∘ 𝑋)) ↾ (𝐵 ∪ {𝑧}))) / 𝐿))
229	131, 228	jca 507	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 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 Vcvv 3398 ∪ cun 3790 ⊆ wss 3792 {csn 4398 class class class wbr 4888 ↦ cmpt 4967 ↾ cres 5359 ∘ ccom 5361 ⟶wf 6133 ‘cfv 6137 (class class class)co 6924 ∘_𝑓 cof 7174 Fincfn 8243 ℂcc 10272 ℝcr 10273 0cc0 10274 1c1 10275 + caddc 10277 · cmul 10279 +∞cpnf 10410 < clt 10413 ≤ cle 10414 − cmin 10608 / cdiv 11034 ℝ⁺crp 12141 [,)cico 12493 [,]cicc 12494 Σ_g cgsu 16491 SubGrpcsubg 17976 CMndccmn 18583 Abelcabl 18584 Ringcrg 18938 DivRingcdr 19143 SubRingcsubrg 19172 ℂ_fldccnfld 20146 ℝ_fldcrefld 20351

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-addf 10353 ax-mulf 10354

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-of 7176 df-om 7346 df-1st 7447 df-2nd 7448 df-supp 7579 df-tpos 7636 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fsupp 8566 df-oi 8706 df-card 9100 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-2 11442 df-3 11443 df-4 11444 df-5 11445 df-6 11446 df-7 11447 df-8 11448 df-9 11449 df-n0 11647 df-z 11733 df-dec 11850 df-uz 11997 df-rp 12142 df-ico 12497 df-icc 12498 df-fz 12648 df-fzo 12789 df-seq 13124 df-hash 13440 df-struct 16261 df-ndx 16262 df-slot 16263 df-base 16265 df-sets 16266 df-ress 16267 df-plusg 16355 df-mulr 16356 df-starv 16357 df-tset 16361 df-ple 16362 df-ds 16364 df-unif 16365 df-0g 16492 df-gsum 16493 df-mre 16636 df-mrc 16637 df-acs 16639 df-mgm 17632 df-sgrp 17674 df-mnd 17685 df-submnd 17726 df-grp 17816 df-minusg 17817 df-mulg 17932 df-subg 17979 df-cntz 18137 df-cmn 18585 df-abl 18586 df-mgp 18881 df-ur 18893 df-ring 18940 df-cring 18941 df-oppr 19014 df-dvdsr 19032 df-unit 19033 df-invr 19063 df-dvr 19074 df-drng 19145 df-subrg 19174 df-cnfld 20147 df-refld 20352

This theorem is referenced by: jensen 25171

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