Theorem caratheodorylem2 42808

Description: Caratheodory's construction is sigma-additive. Main part of Step (e) in the proof of Theorem 113C of [Fremlin1] p. 21. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
caratheodorylem2.o	⊢ (𝜑 → 𝑂 ∈ OutMeas)
caratheodorylem2.x	⊢ 𝑋 = ∪ dom 𝑂
caratheodorylem2.s	⊢ 𝑆 = (CaraGen‘𝑂)
caratheodorylem2.e	⊢ (𝜑 → 𝐸:ℕ⟶𝑆)
caratheodorylem2.5	⊢ (𝜑 → Disj 𝑛 ∈ ℕ (𝐸‘𝑛))
caratheodorylem2.g	⊢ 𝐺 = (𝑘 ∈ ℕ ↦ ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛))

Assertion

Ref	Expression
caratheodorylem2	⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛)))))

Distinct variable groups:   𝑘,𝐸,𝑛   𝑛,𝐺   𝑘,𝑂,𝑛   𝑛,𝑋   𝜑,𝑘,𝑛

Allowed substitution hints:   𝑆(𝑘,𝑛)   𝐺(𝑘)   𝑋(𝑘)

Proof of Theorem caratheodorylem2

Dummy variables 𝑚 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		caratheodorylem2.o	. . 3 ⊢ (𝜑 → 𝑂 ∈ OutMeas)
2		caratheodorylem2.x	. . 3 ⊢ 𝑋 = ∪ dom 𝑂
3		caratheodorylem2.s	. . . . . . . . . . 11 ⊢ 𝑆 = (CaraGen‘𝑂)
4	3	caragenss 42785	. . . . . . . . . 10 ⊢ (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂)
5	1, 4	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ⊆ dom 𝑂)
6	5	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑆 ⊆ dom 𝑂)
7		caratheodorylem2.e	. . . . . . . . 9 ⊢ (𝜑 → 𝐸:ℕ⟶𝑆)
8	7	ffvelrnda 6850	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸‘𝑛) ∈ 𝑆)
9	6, 8	sseldd 3967	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸‘𝑛) ∈ dom 𝑂)
10		elssuni 4867	. . . . . . 7 ⊢ ((𝐸‘𝑛) ∈ dom 𝑂 → (𝐸‘𝑛) ⊆ ∪ dom 𝑂)
11	9, 10	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸‘𝑛) ⊆ ∪ dom 𝑂)
12	11, 2	sseqtrrdi 4017	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸‘𝑛) ⊆ 𝑋)
13	12	ralrimiva 3182	. . . 4 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐸‘𝑛) ⊆ 𝑋)
14		iunss 4968	. . . 4 ⊢ (∪ 𝑛 ∈ ℕ (𝐸‘𝑛) ⊆ 𝑋 ↔ ∀𝑛 ∈ ℕ (𝐸‘𝑛) ⊆ 𝑋)
15	13, 14	sylibr 236	. . 3 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐸‘𝑛) ⊆ 𝑋)
16	1, 2, 15	omexrcl 42788	. 2 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) ∈ ℝ*)
17		nnex 11643	. . . 4 ⊢ ℕ ∈ V
18	17	a1i 11	. . 3 ⊢ (𝜑 → ℕ ∈ V)
19	1	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑂 ∈ OutMeas)
20	19, 2, 12	omecl 42784	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞))
21		eqid 2821	. . . 4 ⊢ (𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛)))
22	20, 21	fmptd 6877	. . 3 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛))):ℕ⟶(0[,]+∞))
23	18, 22	sge0xrcl 42666	. 2 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ*)
24		nfv 1911	. . 3 ⊢ Ⅎ𝑛𝜑
25		nfcv 2977	. . 3 ⊢ Ⅎ𝑛𝐸
26		nnuz 12280	. . 3 ⊢ ℕ = (ℤ≥‘1)
27	1, 2, 3	caragensspw 42790	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝒫 𝑋)
28	7, 27	fssd 6527	. . 3 ⊢ (𝜑 → 𝐸:ℕ⟶𝒫 𝑋)
29	24, 25, 1, 2, 26, 28	omeiunle 42798	. 2 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛)))))
30		elpwinss 41309	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ℕ ∩ Fin) → 𝑥 ⊆ ℕ)
31	30	resmptd 5907	. . . . . . 7 ⊢ (𝑥 ∈ (𝒫 ℕ ∩ Fin) → ((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑥) = (𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛))))
32	31	fveq2d 6673	. . . . . 6 ⊢ (𝑥 ∈ (𝒫 ℕ ∩ Fin) → (Σ^‘((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑥)) = (Σ^‘(𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))))
33	32	adantl 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → (Σ^‘((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑥)) = (Σ^‘(𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))))
34		1zzd 12012	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 1 ∈ ℤ)
35	30	adantl 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 𝑥 ⊆ ℕ)
36		elinel2 4172	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ℕ ∩ Fin) → 𝑥 ∈ Fin)
37	36	adantl 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 𝑥 ∈ Fin)
38	34, 26, 35, 37	uzfissfz 41592	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → ∃𝑘 ∈ ℕ 𝑥 ⊆ (1...𝑘))
39		vex 3497	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
40	39	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ⊆ (1...𝑘)) → 𝑥 ∈ V)
41	1	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ 𝑥) → 𝑂 ∈ OutMeas)
42	28	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ 𝑥) → 𝐸:ℕ⟶𝒫 𝑋)
43		fz1ssnn 12937	. . . . . . . . . . . . . . . . . 18 ⊢ (1...𝑘) ⊆ ℕ
44		ssel2 3961	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ⊆ (1...𝑘) ∧ 𝑛 ∈ 𝑥) → 𝑛 ∈ (1...𝑘))
45	43, 44	sseldi 3964	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ⊆ (1...𝑘) ∧ 𝑛 ∈ 𝑥) → 𝑛 ∈ ℕ)
46	45	adantll 712	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ 𝑥) → 𝑛 ∈ ℕ)
47	42, 46	ffvelrnd 6851	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ 𝑥) → (𝐸‘𝑛) ∈ 𝒫 𝑋)
48		elpwi 4547	. . . . . . . . . . . . . . 15 ⊢ ((𝐸‘𝑛) ∈ 𝒫 𝑋 → (𝐸‘𝑛) ⊆ 𝑋)
49	47, 48	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ 𝑥) → (𝐸‘𝑛) ⊆ 𝑋)
50	41, 2, 49	omecl 42784	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ 𝑥) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞))
51		eqid 2821	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))
52	50, 51	fmptd 6877	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ⊆ (1...𝑘)) → (𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛))):𝑥⟶(0[,]+∞))
53	40, 52	sge0xrcl 42666	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ*)
54	53	3adant2 1127	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ*)
55		ovex 7188	. . . . . . . . . . . . 13 ⊢ (1...𝑘) ∈ V
56	55	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (1...𝑘) ∈ V)
57		elfznn 12935	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...𝑘) → 𝑛 ∈ ℕ)
58	57, 20	sylan2 594	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑘)) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞))
59		eqid 2821	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛)))
60	58, 59	fmptd 6877	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛))):(1...𝑘)⟶(0[,]+∞))
61	56, 60	sge0xrcl 42666	. . . . . . . . . . 11 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ*)
62	61	3ad2ant1 1129	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ*)
63	16	3ad2ant1 1129	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) ∈ ℝ*)
64	55	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (1...𝑘) ∈ V)
65		simpl1 1187	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ (1...𝑘)) → 𝜑)
66	57	adantl 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 ∈ ℕ)
67	65, 66, 20	syl2anc 586	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ (1...𝑘)) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞))
68		simp3 1134	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → 𝑥 ⊆ (1...𝑘))
69	64, 67, 68	sge0lessmpt 42680	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))) ≤ (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛)))))
70	1	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑂 ∈ OutMeas)
71	7	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐸:ℕ⟶𝑆)
72		caratheodorylem2.5	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → Disj 𝑛 ∈ ℕ (𝐸‘𝑛))
73	72	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Disj 𝑛 ∈ ℕ (𝐸‘𝑛))
74		caratheodorylem2.g	. . . . . . . . . . . . . . 15 ⊢ 𝐺 = (𝑘 ∈ ℕ ↦ ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛))
75		nfiu1 4952	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛)
76		nfcv 2977	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘∪ 𝑚 ∈ (1...𝑛)(𝐸‘𝑚)
77		fveq2 6669	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑚 → (𝐸‘𝑛) = (𝐸‘𝑚))
78	77	cbviunv 4964	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛) = ∪ 𝑚 ∈ (1...𝑘)(𝐸‘𝑚)
79	78	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑛 → ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛) = ∪ 𝑚 ∈ (1...𝑘)(𝐸‘𝑚))
80		oveq2 7163	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑛 → (1...𝑘) = (1...𝑛))
81	80	iuneq1d 4945	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑛 → ∪ 𝑚 ∈ (1...𝑘)(𝐸‘𝑚) = ∪ 𝑚 ∈ (1...𝑛)(𝐸‘𝑚))
82	79, 81	eqtrd 2856	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑛 → ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛) = ∪ 𝑚 ∈ (1...𝑛)(𝐸‘𝑚))
83	75, 76, 82	cbvmpt 5166	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ ↦ ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛)) = (𝑛 ∈ ℕ ↦ ∪ 𝑚 ∈ (1...𝑛)(𝐸‘𝑚))
84	74, 83	eqtri 2844	. . . . . . . . . . . . . 14 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ∪ 𝑚 ∈ (1...𝑛)(𝐸‘𝑚))
85		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ)
86	85, 26	eleqtrdi 2923	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ (ℤ≥‘1))
87	86	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ≥‘1))
88	70, 3, 26, 71, 73, 84, 87	caratheodorylem1 42807	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑂‘(𝐺‘𝑘)) = (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛)))))
89	88	eqcomd 2827	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛)))) = (𝑂‘(𝐺‘𝑘)))
90	15	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∪ 𝑛 ∈ ℕ (𝐸‘𝑛) ⊆ 𝑋)
91		fvex 6682	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸‘𝑛) ∈ V
92	55, 91	iunex 7668	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛) ∈ V
93	74	fvmpt2 6778	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℕ ∧ ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛) ∈ V) → (𝐺‘𝑘) = ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛))
94	85, 92, 93	sylancl 588	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → (𝐺‘𝑘) = ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛))
95	43	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → (1...𝑘) ⊆ ℕ)
96		iunss1 4932	. . . . . . . . . . . . . . . 16 ⊢ ((1...𝑘) ⊆ ℕ → ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛) ⊆ ∪ 𝑛 ∈ ℕ (𝐸‘𝑛))
97	95, 96	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛) ⊆ ∪ 𝑛 ∈ ℕ (𝐸‘𝑛))
98	94, 97	eqsstrd 4004	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → (𝐺‘𝑘) ⊆ ∪ 𝑛 ∈ ℕ (𝐸‘𝑛))
99	98	adantl 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ⊆ ∪ 𝑛 ∈ ℕ (𝐸‘𝑛))
100	70, 2, 90, 99	omessle 42779	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑂‘(𝐺‘𝑘)) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
101	89, 100	eqbrtrd 5087	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛)))) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
102	101	3adant3 1128	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛)))) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
103	54, 62, 63, 69, 102	xrletrd 12554	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
104	103	3exp 1115	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ ℕ → (𝑥 ⊆ (1...𝑘) → (Σ^‘(𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))))
105	104	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → (𝑘 ∈ ℕ → (𝑥 ⊆ (1...𝑘) → (Σ^‘(𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))))
106	105	rexlimdv 3283	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → (∃𝑘 ∈ ℕ 𝑥 ⊆ (1...𝑘) → (Σ^‘(𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛))))
107	38, 106	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → (Σ^‘(𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
108	33, 107	eqbrtrd 5087	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → (Σ^‘((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑥)) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
109	108	ralrimiva 3182	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (𝒫 ℕ ∩ Fin)(Σ^‘((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑥)) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
110	18, 22, 16	sge0lefi 42679	. . 3 ⊢ (𝜑 → ((Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛)))) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) ↔ ∀𝑥 ∈ (𝒫 ℕ ∩ Fin)(Σ^‘((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑥)) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛))))
111	109, 110	mpbird 259	. 2 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛)))) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
112	16, 23, 29, 111	xrletrid 12547	1 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛)))))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139  Vcvv 3494   ∩ cin 3934   ⊆ wss 3935  𝒫 cpw 4538  ∪ cuni 4837  ∪ ciun 4918  Disj wdisj 5030   class class class wbr 5065   ↦ cmpt 5145  dom cdm 5554   ↾ cres 5556  ⟶wf 6350  ‘cfv 6354  (class class class)co 7155  Fincfn 8508  0cc0 10536  1c1 10537  +∞cpnf 10671  ℝ^*cxr 10673   ≤ cle 10675  ℕcn 11637  ℤ_≥cuz 12242  [,]cicc 12740  ...cfz 12891  Σ^{^}csumge0 42643  OutMeascome 42770  CaraGenccaragen 42772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-inf2 9103  ax-ac2 9884  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613  ax-pre-sup 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-disj 5031  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-1st 7688  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-oadd 8105  df-omul 8106  df-er 8288  df-map 8407  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512  df-sup 8905  df-oi 8973  df-card 9367  df-acn 9370  df-ac 9541  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-div 11297  df-nn 11638  df-2 11699  df-3 11700  df-n0 11897  df-z 11981  df-uz 12243  df-rp 12389  df-xadd 12507  df-ico 12743  df-icc 12744  df-fz 12892  df-fzo 13033  df-seq 13369  df-exp 13429  df-hash 13690  df-cj 14457  df-re 14458  df-im 14459  df-sqrt 14593  df-abs 14594  df-clim 14844  df-sum 15042  df-sumge0 42644  df-ome 42771  df-caragen 42773

This theorem is referenced by:  caratheodory  42809