jensenlem2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > jensenlem2		Structured version Visualization version GIF version

Theorem jensenlem2 26874

Description: Lemma for jensen 26875. (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 21280	. . . . . . 7 ⊢ 0 = (0_g‘ℂ_fld)
2		cnring 21278	. . . . . . . 8 ⊢ ℂ_fld ∈ Ring
3		ringabl 20166	. . . . . . . 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 4152	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
8	5, 7	ssfid 9188	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ Fin)
9		resubdrg 21493	. . . . . . . . 9 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) ∧ ℝ_fld ∈ DivRing)
10	9	simpli 483	. . . . . . . 8 ⊢ ℝ ∈ (SubRing‘ℂ_fld)
11		subrgsubg 20462	. . . . . . . 8 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) → ℝ ∈ (SubGrp‘ℂ_fld))
12	10, 11	mp1i 13	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ (SubGrp‘ℂ_fld))
13		remulcl 11129	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 𝑦) ∈ ℝ)
14	13	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 · 𝑦) ∈ ℝ)
15		jensen.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇:𝐴⟶(0[,)+∞))
16		rge0ssre 13393	. . . . . . . . . 10 ⊢ (0[,)+∞) ⊆ ℝ
17		fss 6686	. . . . . . . . . 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 6687	. . . . . . . . 9 ⊢ (𝜑 → 𝑋:𝐴⟶ℝ)
22		inidm 4186	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐴) = 𝐴
23	14, 18, 21, 5, 5, 22	off 7651	. . . . . . . 8 ⊢ (𝜑 → (𝑇 ∘_f · 𝑋):𝐴⟶ℝ)
24	23, 7	fssresd 6709	. . . . . . 7 ⊢ (𝜑 → ((𝑇 ∘_f · 𝑋) ↾ 𝐵):𝐵⟶ℝ)
25		c0ex 11144	. . . . . . . . 9 ⊢ 0 ∈ V
26	25	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ V)
27	24, 8, 26	fdmfifsupp 9302	. . . . . . 7 ⊢ (𝜑 → ((𝑇 ∘_f · 𝑋) ↾ 𝐵) finSupp 0)
28	1, 4, 8, 12, 24, 27	gsumsubgcl 19826	. . . . . 6 ⊢ (𝜑 → (ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) ∈ ℝ)
29	28	recnd 11178	. . . . 5 ⊢ (𝜑 → (ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) ∈ ℂ)
30		ax-resscn 11101	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
31	16, 30	sstri 3953	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℂ
32	6	unssbd 4153	. . . . . . . . 9 ⊢ (𝜑 → {𝑧} ⊆ 𝐴)
33		vex 3448	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
34	33	snss 4745	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 ↔ {𝑧} ⊆ 𝐴)
35	32, 34	sylibr 234	. . . . . . . 8 ⊢ (𝜑 → 𝑧 ∈ 𝐴)
36	15, 35	ffvelcdmd 7039	. . . . . . 7 ⊢ (𝜑 → (𝑇‘𝑧) ∈ (0[,)+∞))
37	31, 36	sselid 3941	. . . . . 6 ⊢ (𝜑 → (𝑇‘𝑧) ∈ ℂ)
38	19, 35	ffvelcdmd 7039	. . . . . . . 8 ⊢ (𝜑 → (𝑋‘𝑧) ∈ 𝐷)
39	20, 38	sseldd 3944	. . . . . . 7 ⊢ (𝜑 → (𝑋‘𝑧) ∈ ℝ)
40	39	recnd 11178	. . . . . 6 ⊢ (𝜑 → (𝑋‘𝑧) ∈ ℂ)
41	37, 40	mulcld 11170	. . . . 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 26873	. . . . . . 7 ⊢ (𝜑 → 𝐿 = (𝑆 + (𝑇‘𝑧)))
50		jensenlem.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ ℝ⁺)
51	50	rpred 12971	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℝ)
52		elrege0 13391	. . . . . . . . . 10 ⊢ ((𝑇‘𝑧) ∈ (0[,)+∞) ↔ ((𝑇‘𝑧) ∈ ℝ ∧ 0 ≤ (𝑇‘𝑧)))
53	52	simplbi 497	. . . . . . . . 9 ⊢ ((𝑇‘𝑧) ∈ (0[,)+∞) → (𝑇‘𝑧) ∈ ℝ)
54	36, 53	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑇‘𝑧) ∈ ℝ)
55	51, 54	readdcld 11179	. . . . . . 7 ⊢ (𝜑 → (𝑆 + (𝑇‘𝑧)) ∈ ℝ)
56	49, 55	eqeltrd 2828	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ ℝ)
57	56	recnd 11178	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ ℂ)
58		0red 11153	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ)
59	50	rpgt0d 12974	. . . . . . 7 ⊢ (𝜑 → 0 < 𝑆)
60	52	simprbi 496	. . . . . . . . . 10 ⊢ ((𝑇‘𝑧) ∈ (0[,)+∞) → 0 ≤ (𝑇‘𝑧))
61	36, 60	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (𝑇‘𝑧))
62	51, 54	addge01d 11742	. . . . . . . . 9 ⊢ (𝜑 → (0 ≤ (𝑇‘𝑧) ↔ 𝑆 ≤ (𝑆 + (𝑇‘𝑧))))
63	61, 62	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ≤ (𝑆 + (𝑇‘𝑧)))
64	63, 49	breqtrrd 5130	. . . . . . 7 ⊢ (𝜑 → 𝑆 ≤ 𝐿)
65	58, 51, 56, 59, 64	ltletrd 11310	. . . . . 6 ⊢ (𝜑 → 0 < 𝐿)
66	65	gt0ne0d 11718	. . . . 5 ⊢ (𝜑 → 𝐿 ≠ 0)
67	29, 41, 57, 66	divdird 11972	. . . 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 21244	. . . . . . 7 ⊢ ℂ = (Base‘ℂ_fld)
69		cnfldadd 21246	. . . . . . 7 ⊢ + = (+_g‘ℂ_fld)
70		ringcmn 20167	. . . . . . . 8 ⊢ (ℂ_fld ∈ Ring → ℂ_fld ∈ CMnd)
71	2, 70	mp1i 13	. . . . . . 7 ⊢ (𝜑 → ℂ_fld ∈ CMnd)
72	7	sselda 3943	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴)
73	15	ffvelcdmda 7038	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑇‘𝑥) ∈ (0[,)+∞))
74	72, 73	syldan 591	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑇‘𝑥) ∈ (0[,)+∞))
75	31, 74	sselid 3941	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑇‘𝑥) ∈ ℂ)
76	20	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐷 ⊆ ℝ)
77	19	ffvelcdmda 7038	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑋‘𝑥) ∈ 𝐷)
78	72, 77	syldan 591	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑋‘𝑥) ∈ 𝐷)
79	76, 78	sseldd 3944	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑋‘𝑥) ∈ ℝ)
80	79	recnd 11178	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑋‘𝑥) ∈ ℂ)
81	75, 80	mulcld 11170	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑇‘𝑥) · (𝑋‘𝑥)) ∈ ℂ)
82		fveq2 6840	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑇‘𝑥) = (𝑇‘𝑧))
83		fveq2 6840	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑋‘𝑥) = (𝑋‘𝑧))
84	82, 83	oveq12d 7387	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝑇‘𝑥) · (𝑋‘𝑥)) = ((𝑇‘𝑧) · (𝑋‘𝑧)))
85	68, 69, 71, 8, 81, 35, 46, 41, 84	gsumunsn 19866	. . . . . 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 6911	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 = (𝑥 ∈ 𝐴 ↦ (𝑇‘𝑥)))
87	19	feqmptd 6911	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 = (𝑥 ∈ 𝐴 ↦ (𝑋‘𝑥)))
88	5, 73, 77, 86, 87	offval2 7653	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ∘_f · 𝑋) = (𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥))))
89	88	reseq1d 5938	. . . . . . . 8 ⊢ (𝜑 → ((𝑇 ∘_f · 𝑋) ↾ (𝐵 ∪ {𝑧})) = ((𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥))) ↾ (𝐵 ∪ {𝑧})))
90	6	resmptd 6000	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥))) ↾ (𝐵 ∪ {𝑧})) = (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ ((𝑇‘𝑥) · (𝑋‘𝑥))))
91	89, 90	eqtrd 2764	. . . . . . 7 ⊢ (𝜑 → ((𝑇 ∘_f · 𝑋) ↾ (𝐵 ∪ {𝑧})) = (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ ((𝑇‘𝑥) · (𝑋‘𝑥))))
92	91	oveq2d 7385	. . . . . 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 5938	. . . . . . . . 9 ⊢ (𝜑 → ((𝑇 ∘_f · 𝑋) ↾ 𝐵) = ((𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥))) ↾ 𝐵))
94	7	resmptd 6000	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥))) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥))))
95	93, 94	eqtrd 2764	. . . . . . . 8 ⊢ (𝜑 → ((𝑇 ∘_f · 𝑋) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥))))
96	95	oveq2d 7385	. . . . . . 7 ⊢ (𝜑 → (ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) = (ℂ_fld Σ_g (𝑥 ∈ 𝐵 ↦ ((𝑇‘𝑥) · (𝑋‘𝑥)))))
97	96	oveq1d 7384	. . . . . 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 2774	. . . . 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 7384	. . . 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 11178	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ℂ)
101	50	rpne0d 12976	. . . . . 6 ⊢ (𝜑 → 𝑆 ≠ 0)
102	29, 100, 57, 101, 66	dmdcand 11963	. . . . 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 11973	. . . . . . . 8 ⊢ (𝜑 → ((𝐿 − 𝑆) / 𝐿) = ((𝐿 / 𝐿) − (𝑆 / 𝐿)))
104	100, 37, 49	mvrladdd 11567	. . . . . . . . 9 ⊢ (𝜑 → (𝐿 − 𝑆) = (𝑇‘𝑧))
105	104	oveq1d 7384	. . . . . . . 8 ⊢ (𝜑 → ((𝐿 − 𝑆) / 𝐿) = ((𝑇‘𝑧) / 𝐿))
106	57, 66	dividd 11932	. . . . . . . . 9 ⊢ (𝜑 → (𝐿 / 𝐿) = 1)
107	106	oveq1d 7384	. . . . . . . 8 ⊢ (𝜑 → ((𝐿 / 𝐿) − (𝑆 / 𝐿)) = (1 − (𝑆 / 𝐿)))
108	103, 105, 107	3eqtr3rd 2773	. . . . . . 7 ⊢ (𝜑 → (1 − (𝑆 / 𝐿)) = ((𝑇‘𝑧) / 𝐿))
109	108	oveq1d 7384	. . . . . 6 ⊢ (𝜑 → ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧)) = (((𝑇‘𝑧) / 𝐿) · (𝑋‘𝑧)))
110	37, 40, 57, 66	div23d 11971	. . . . . 6 ⊢ (𝜑 → (((𝑇‘𝑧) · (𝑋‘𝑧)) / 𝐿) = (((𝑇‘𝑧) / 𝐿) · (𝑋‘𝑧)))
111	109, 110	eqtr4d 2767	. . . . 5 ⊢ (𝜑 → ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧)) = (((𝑇‘𝑧) · (𝑋‘𝑧)) / 𝐿))
112	102, 111	oveq12d 7387	. . . 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 2774	. . 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 11986	. . . . . 6 ⊢ (𝜑 → (𝑆 / 𝐿) ∈ ℝ)
116	50	rpge0d 12975	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝑆)
117		divge0 12028	. . . . . . 7 ⊢ (((𝑆 ∈ ℝ ∧ 0 ≤ 𝑆) ∧ (𝐿 ∈ ℝ ∧ 0 < 𝐿)) → 0 ≤ (𝑆 / 𝐿))
118	51, 116, 56, 65, 117	syl22anc 838	. . . . . 6 ⊢ (𝜑 → 0 ≤ (𝑆 / 𝐿))
119	57	mulridd 11167	. . . . . . . 8 ⊢ (𝜑 → (𝐿 · 1) = 𝐿)
120	64, 119	breqtrrd 5130	. . . . . . 7 ⊢ (𝜑 → 𝑆 ≤ (𝐿 · 1))
121		1red 11151	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
122		ledivmul 12035	. . . . . . . 8 ⊢ ((𝑆 ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 0 < 𝐿)) → ((𝑆 / 𝐿) ≤ 1 ↔ 𝑆 ≤ (𝐿 · 1)))
123	51, 121, 56, 65, 122	syl112anc 1376	. . . . . . 7 ⊢ (𝜑 → ((𝑆 / 𝐿) ≤ 1 ↔ 𝑆 ≤ (𝐿 · 1)))
124	120, 123	mpbird 257	. . . . . 6 ⊢ (𝜑 → (𝑆 / 𝐿) ≤ 1)
125		elicc01 13403	. . . . . 6 ⊢ ((𝑆 / 𝐿) ∈ (0[,]1) ↔ ((𝑆 / 𝐿) ∈ ℝ ∧ 0 ≤ (𝑆 / 𝐿) ∧ (𝑆 / 𝐿) ≤ 1))
126	115, 118, 124, 125	syl3anbrc 1344	. . . . 5 ⊢ (𝜑 → (𝑆 / 𝐿) ∈ (0[,]1))
127	114, 38, 126	3jca 1128	. . . 4 ⊢ (𝜑 → (((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆) ∈ 𝐷 ∧ (𝑋‘𝑧) ∈ 𝐷 ∧ (𝑆 / 𝐿) ∈ (0[,]1)))
128	20, 43	cvxcl 26871	. . . 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 687	. . 3 ⊢ (𝜑 → (((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧))) ∈ 𝐷)
130	113, 129	eqeltrd 2828	. 2 ⊢ (𝜑 → ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ (𝐵 ∪ {𝑧}))) / 𝐿) ∈ 𝐷)
131	42, 129	ffvelcdmd 7039	. . . 4 ⊢ (𝜑 → (𝐹‘(((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧)))) ∈ ℝ)
132	42, 114	ffvelcdmd 7039	. . . . . 6 ⊢ (𝜑 → (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆)) ∈ ℝ)
133	115, 132	remulcld 11180	. . . . 5 ⊢ (𝜑 → ((𝑆 / 𝐿) · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))) ∈ ℝ)
134	42, 38	ffvelcdmd 7039	. . . . . . 7 ⊢ (𝜑 → (𝐹‘(𝑋‘𝑧)) ∈ ℝ)
135	54, 134	remulcld 11180	. . . . . 6 ⊢ (𝜑 → ((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) ∈ ℝ)
136	135, 56, 66	redivcld 11986	. . . . 5 ⊢ (𝜑 → (((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) / 𝐿) ∈ ℝ)
137	133, 136	readdcld 11179	. . . 4 ⊢ (𝜑 → (((𝑆 / 𝐿) · (𝐹‘((ℂ_fld Σ_g ((𝑇 ∘_f · 𝑋) ↾ 𝐵)) / 𝑆))) + (((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) / 𝐿)) ∈ ℝ)
138		fco 6694	. . . . . . . . . . 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 7651	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ∘_f · (𝐹 ∘ 𝑋)):𝐴⟶ℝ)
141	140, 7	fssresd 6709	. . . . . . . 8 ⊢ (𝜑 → ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵):𝐵⟶ℝ)
142	141, 8, 26	fdmfifsupp 9302	. . . . . . . 8 ⊢ (𝜑 → ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵) finSupp 0)
143	1, 4, 8, 12, 141, 142	gsumsubgcl 19826	. . . . . . 7 ⊢ (𝜑 → (ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) ∈ ℝ)
144	143, 51, 101	redivcld 11986	. . . . . 6 ⊢ (𝜑 → ((ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆) ∈ ℝ)
145	115, 144	remulcld 11180	. . . . 5 ⊢ (𝜑 → ((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆)) ∈ ℝ)
146		1re 11150	. . . . . . 7 ⊢ 1 ∈ ℝ
147		resubcl 11462	. . . . . . 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 11180	. . . . 5 ⊢ (𝜑 → ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧))) ∈ ℝ)
150	145, 149	readdcld 11179	. . . 4 ⊢ (𝜑 → (((𝑆 / 𝐿) · ((ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) / 𝑆)) + ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧)))) ∈ ℝ)
151		oveq2 7377	. . . . . . . . . . 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 7391	. . . . . . . . . 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 6840	. . . . . . . . . . . 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 7385	. . . . . . . . . . 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 7384	. . . . . . . . . 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 5115	. . . . . . . . 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 340	. . . . . . . 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 7377	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑋‘𝑧) → ((1 − 𝑡) · 𝑦) = ((1 − 𝑡) · (𝑋‘𝑧)))
159	158	oveq2d 7385	. . . . . . . . . . 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 6844	. . . . . . . . . 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 6840	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑋‘𝑧) → (𝐹‘𝑦) = (𝐹‘(𝑋‘𝑧)))
162	161	oveq2d 7385	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑋‘𝑧) → ((1 − 𝑡) · (𝐹‘𝑦)) = ((1 − 𝑡) · (𝐹‘(𝑋‘𝑧))))
163	162	oveq2d 7385	. . . . . . . . . 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 5115	. . . . . . . . 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 340	. . . . . . . 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 7376	. . . . . . . . . . . 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 7377	. . . . . . . . . . . . 13 ⊢ (𝑡 = (𝑆 / 𝐿) → (1 − 𝑡) = (1 − (𝑆 / 𝐿)))
168	167	oveq1d 7384	. . . . . . . . . . . 12 ⊢ (𝑡 = (𝑆 / 𝐿) → ((1 − 𝑡) · (𝑋‘𝑧)) = ((1 − (𝑆 / 𝐿)) · (𝑋‘𝑧)))
169	166, 168	oveq12d 7387	. . . . . . . . . . 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 6844	. . . . . . . . . 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 7376	. . . . . . . . . . 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 7384	. . . . . . . . . . 11 ⊢ (𝑡 = (𝑆 / 𝐿) → ((1 − 𝑡) · (𝐹‘(𝑋‘𝑧))) = ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧))))
173	171, 172	oveq12d 7387	. . . . . . . . . 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 5115	. . . . . . . . 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 340	. . . . . . . 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 413	. . . . . . . 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 3546	. . . . . . 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 1373	. . . . . 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 7384	. . . . . . 7 ⊢ (𝜑 → ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧))) = (((𝑇‘𝑧) / 𝐿) · (𝐹‘(𝑋‘𝑧))))
181	134	recnd 11178	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘(𝑋‘𝑧)) ∈ ℂ)
182	37, 181, 57, 66	div23d 11971	. . . . . . 7 ⊢ (𝜑 → (((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) / 𝐿) = (((𝑇‘𝑧) / 𝐿) · (𝐹‘(𝑋‘𝑧))))
183	180, 182	eqtr4d 2767	. . . . . 6 ⊢ (𝜑 → ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧))) = (((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) / 𝐿))
184	183	oveq2d 7385	. . . . 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 5128	. . . 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 2767	. . . . . 6 ⊢ (𝜑 → (((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) / 𝐿) = ((1 − (𝑆 / 𝐿)) · (𝐹‘(𝑋‘𝑧))))
187	186	oveq2d 7385	. . . . 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 12094	. . . . . . . 8 ⊢ (𝜑 → 0 < (𝑆 / 𝐿))
190		lemul2 12011	. . . . . . . 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 1376	. . . . . . 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 232	. . . . . 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 11768	. . . . 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 5124	. . . 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 11307	. . 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 6844	. . 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 11178	. . . . 5 ⊢ (𝜑 → (ℂ_fld Σ_g ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵)) ∈ ℂ)
198	135	recnd 11178	. . . . 5 ⊢ (𝜑 → ((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))) ∈ ℂ)
199	197, 198, 57, 66	divdird 11972	. . . 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 3941	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑇‘𝑥) ∈ ℝ)
201	42	ffvelcdmda 7038	. . . . . . . . . . 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 11180	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥))) ∈ ℝ)
204	203	recnd 11178	. . . . . . . 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 6844	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝐹‘(𝑋‘𝑥)) = (𝐹‘(𝑋‘𝑧)))
207	82, 206	oveq12d 7387	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥))) = ((𝑇‘𝑧) · (𝐹‘(𝑋‘𝑧))))
208	68, 69, 71, 8, 205, 35, 46, 198, 207	gsumunsn 19866	. . . . . 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 6911	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ 𝐷 ↦ (𝐹‘𝑦)))
210		fveq2 6840	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑋‘𝑥) → (𝐹‘𝑦) = (𝐹‘(𝑋‘𝑥)))
211	77, 87, 209, 210	fmptco 7083	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∘ 𝑋) = (𝑥 ∈ 𝐴 ↦ (𝐹‘(𝑋‘𝑥))))
212	5, 73, 202, 86, 211	offval2 7653	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ∘_f · (𝐹 ∘ 𝑋)) = (𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥)))))
213	212	reseq1d 5938	. . . . . . . 8 ⊢ (𝜑 → ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ (𝐵 ∪ {𝑧})) = ((𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥)))) ↾ (𝐵 ∪ {𝑧})))
214	6	resmptd 6000	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥)))) ↾ (𝐵 ∪ {𝑧})) = (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥)))))
215	213, 214	eqtrd 2764	. . . . . . 7 ⊢ (𝜑 → ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ (𝐵 ∪ {𝑧})) = (𝑥 ∈ (𝐵 ∪ {𝑧}) ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥)))))
216	215	oveq2d 7385	. . . . . 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 5938	. . . . . . . . 9 ⊢ (𝜑 → ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵) = ((𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥)))) ↾ 𝐵))
218	7	resmptd 6000	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥)))) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥)))))
219	217, 218	eqtrd 2764	. . . . . . . 8 ⊢ (𝜑 → ((𝑇 ∘_f · (𝐹 ∘ 𝑋)) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ ((𝑇‘𝑥) · (𝐹‘(𝑋‘𝑥)))))
220	219	oveq2d 7385	. . . . . . 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 7384	. . . . . 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 2774	. . . . 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 7384	. . . 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 11963	. . . . 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 7387	. . . 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 2774	. . 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 5134	. 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 511	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 Vcvv 3444 ∪ cun 3909 ⊆ wss 3911 {csn 4585 class class class wbr 5102 ↦ cmpt 5183 ↾ cres 5633 ∘ ccom 5635 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 ∘_f cof 7631 Fincfn 8895 ℂcc 11042 ℝcr 11043 0cc0 11044 1c1 11045 + caddc 11047 · cmul 11049 +∞cpnf 11181 < clt 11184 ≤ cle 11185 − cmin 11381 / cdiv 11811 ℝ⁺crp 12927 [,)cico 13284 [,]cicc 13285 Σ_g cgsu 17379 SubGrpcsubg 19028 CMndccmn 19686 Abelcabl 19687 Ringcrg 20118 SubRingcsubrg 20454 DivRingcdr 20614 ℂ_fldccnfld 21240 ℝ_fldcrefld 21489

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-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-addf 11123

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-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-tpos 8182 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-oi 9439 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-rp 12928 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-seq 13943 df-hash 14272 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-0g 17380 df-gsum 17381 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-grp 18844 df-minusg 18845 df-mulg 18976 df-subg 19031 df-cntz 19225 df-cmn 19688 df-abl 19689 df-mgp 20026 df-rng 20038 df-ur 20067 df-ring 20120 df-cring 20121 df-oppr 20222 df-dvdsr 20242 df-unit 20243 df-invr 20273 df-dvr 20286 df-subrng 20431 df-subrg 20455 df-drng 20616 df-cnfld 21241 df-refld 21490

This theorem is referenced by: jensen 26875

W3C validator