Theorem caratheodorylem2 47358

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 47335	. . . . . . . . . 10 ⊢ (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂)
5	1, 4	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ⊆ dom 𝑂)
6	5	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑆 ⊆ dom 𝑂)
7		caratheodorylem2.e	. . . . . . . . 9 ⊢ (𝜑 → 𝐸:ℕ⟶𝑆)
8	7	ffvelcdmda 7078	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸‘𝑛) ∈ 𝑆)
9	6, 8	sseldd 3932	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸‘𝑛) ∈ dom 𝑂)
10		elssuni 4899	. . . . . . 7 ⊢ ((𝐸‘𝑛) ∈ dom 𝑂 → (𝐸‘𝑛) ⊆ ∪ dom 𝑂)
11	9, 10	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸‘𝑛) ⊆ ∪ dom 𝑂)
12	11, 2	sseqtrrdi 3972	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸‘𝑛) ⊆ 𝑋)
13	12	ralrimiva 3154	. . . 4 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐸‘𝑛) ⊆ 𝑋)
14		iunss 5003	. . . 4 ⊢ (∪ 𝑛 ∈ ℕ (𝐸‘𝑛) ⊆ 𝑋 ↔ ∀𝑛 ∈ ℕ (𝐸‘𝑛) ⊆ 𝑋)
15	13, 14	sylibr 237	. . 3 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐸‘𝑛) ⊆ 𝑋)
16	1, 2, 15	omexrcl 47338	. 2 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) ∈ ℝ*)
17		nnex 12266	. . . 4 ⊢ ℕ ∈ V
18	17	a1i 11	. . 3 ⊢ (𝜑 → ℕ ∈ V)
19	1	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑂 ∈ OutMeas)
20	19, 2, 12	omecl 47334	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞))
21		eqid 2760	. . . 4 ⊢ (𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛)))
22	20, 21	fmptd 7108	. . 3 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛))):ℕ⟶(0[,]+∞))
23	18, 22	sge0xrcl 47216	. 2 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ*)
24		nfv 1947	. . 3 ⊢ Ⅎ𝑛𝜑
25		nfcv 2922	. . 3 ⊢ Ⅎ𝑛𝐸
26		nnuz 12929	. . 3 ⊢ ℕ = (ℤ≥‘1)
27	1, 2, 3	caragensspw 47340	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝒫 𝑋)
28	7, 27	fssd 6721	. . 3 ⊢ (𝜑 → 𝐸:ℕ⟶𝒫 𝑋)
29	24, 25, 1, 2, 26, 28	omeiunle 47348	. 2 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛)))))
30		elpwinss 45886	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ℕ ∩ Fin) → 𝑥 ⊆ ℕ)
31	30	resmptd 6036	. . . . . . 7 ⊢ (𝑥 ∈ (𝒫 ℕ ∩ Fin) → ((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑥) = (𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛))))
32	31	fveq2d 6883	. . . . . 6 ⊢ (𝑥 ∈ (𝒫 ℕ ∩ Fin) → (Σ^‘((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑥)) = (Σ^‘(𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))))
33	32	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → (Σ^‘((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑥)) = (Σ^‘(𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))))
34		1zzd 12652	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 1 ∈ ℤ)
35	30	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 𝑥 ⊆ ℕ)
36		elinel2 4148	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ℕ ∩ Fin) → 𝑥 ∈ Fin)
37	36	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 𝑥 ∈ Fin)
38	34, 26, 35, 37	uzfissfz 46159	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → ∃𝑘 ∈ ℕ 𝑥 ⊆ (1...𝑘))
39		vex 3454	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
40	39	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ⊆ (1...𝑘)) → 𝑥 ∈ V)
41	1	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ 𝑥) → 𝑂 ∈ OutMeas)
42	28	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ 𝑥) → 𝐸:ℕ⟶𝒫 𝑋)
43		fz1ssnn 13613	. . . . . . . . . . . . . . . . . 18 ⊢ (1...𝑘) ⊆ ℕ
44		ssel2 3926	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ⊆ (1...𝑘) ∧ 𝑛 ∈ 𝑥) → 𝑛 ∈ (1...𝑘))
45	43, 44	sselid 3929	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ⊆ (1...𝑘) ∧ 𝑛 ∈ 𝑥) → 𝑛 ∈ ℕ)
46	45	adantll 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ 𝑥) → 𝑛 ∈ ℕ)
47	42, 46	ffvelcdmd 7079	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ 𝑥) → (𝐸‘𝑛) ∈ 𝒫 𝑋)
48		elpwi 4564	. . . . . . . . . . . . . . 15 ⊢ ((𝐸‘𝑛) ∈ 𝒫 𝑋 → (𝐸‘𝑛) ⊆ 𝑋)
49	47, 48	syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ 𝑥) → (𝐸‘𝑛) ⊆ 𝑋)
50	41, 2, 49	omecl 47334	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ 𝑥) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞))
51		eqid 2760	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))
52	50, 51	fmptd 7108	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ⊆ (1...𝑘)) → (𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛))):𝑥⟶(0[,]+∞))
53	40, 52	sge0xrcl 47216	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ*)
54	53	3adant2 1149	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ*)
55		ovex 7447	. . . . . . . . . . . . 13 ⊢ (1...𝑘) ∈ V
56	55	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (1...𝑘) ∈ V)
57		elfznn 13611	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...𝑘) → 𝑛 ∈ ℕ)
58	57, 20	sylan2 605	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑘)) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞))
59		eqid 2760	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛)))
60	58, 59	fmptd 7108	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛))):(1...𝑘)⟶(0[,]+∞))
61	56, 60	sge0xrcl 47216	. . . . . . . . . . 11 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ*)
62	61	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ*)
63	16	3ad2ant1 1151	. . . . . . . . . 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 1210	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ (1...𝑘)) → 𝜑)
66	57	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 ∈ ℕ)
67	65, 66, 20	syl2anc 596	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ (1...𝑘)) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞))
68		simp3 1156	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → 𝑥 ⊆ (1...𝑘))
69	64, 67, 68	sge0lessmpt 47230	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))) ≤ (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛)))))
70	1	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑂 ∈ OutMeas)
71	7	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐸:ℕ⟶𝑆)
72		caratheodorylem2.5	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → Disj 𝑛 ∈ ℕ (𝐸‘𝑛))
73	72	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Disj 𝑛 ∈ ℕ (𝐸‘𝑛))
74		caratheodorylem2.g	. . . . . . . . . . . . . . 15 ⊢ 𝐺 = (𝑘 ∈ ℕ ↦ ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛))
75		nfiu1 4986	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛)
76		nfcv 2922	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘∪ 𝑚 ∈ (1...𝑛)(𝐸‘𝑚)
77		fveq2 6879	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑚 → (𝐸‘𝑛) = (𝐸‘𝑚))
78	77	cbviunv 4997	. . . . . . . . . . . . . . . . . 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 7422	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑛 → (1...𝑘) = (1...𝑛))
81	80	iuneq1d 4979	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑛 → ∪ 𝑚 ∈ (1...𝑘)(𝐸‘𝑚) = ∪ 𝑚 ∈ (1...𝑛)(𝐸‘𝑚))
82	79, 81	eqtrd 2795	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑛 → ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛) = ∪ 𝑚 ∈ (1...𝑛)(𝐸‘𝑚))
83	75, 76, 82	cbvmpt 5207	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ ↦ ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛)) = (𝑛 ∈ ℕ ↦ ∪ 𝑚 ∈ (1...𝑛)(𝐸‘𝑚))
84	74, 83	eqtri 2783	. . . . . . . . . . . . . 14 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ∪ 𝑚 ∈ (1...𝑛)(𝐸‘𝑚))
85		id 23	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ)
86	85, 26	eleqtrdi 2870	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ (ℤ≥‘1))
87	86	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ≥‘1))
88	70, 3, 26, 71, 73, 84, 87	caratheodorylem1 47357	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑂‘(𝐺‘𝑘)) = (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛)))))
89	88	eqcomd 2766	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛)))) = (𝑂‘(𝐺‘𝑘)))
90	15	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∪ 𝑛 ∈ ℕ (𝐸‘𝑛) ⊆ 𝑋)
91		fvex 6892	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸‘𝑛) ∈ V
92	55, 91	iunex 7966	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛) ∈ V
93	74	fvmpt2 6999	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℕ ∧ ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛) ∈ V) → (𝐺‘𝑘) = ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛))
94	85, 92, 93	sylancl 598	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → (𝐺‘𝑘) = ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛))
95	43	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → (1...𝑘) ⊆ ℕ)
96		iunss1 4966	. . . . . . . . . . . . . . . 16 ⊢ ((1...𝑘) ⊆ ℕ → ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛) ⊆ ∪ 𝑛 ∈ ℕ (𝐸‘𝑛))
97	95, 96	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛) ⊆ ∪ 𝑛 ∈ ℕ (𝐸‘𝑛))
98	94, 97	eqsstrd 3965	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → (𝐺‘𝑘) ⊆ ∪ 𝑛 ∈ ℕ (𝐸‘𝑛))
99	98	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ⊆ ∪ 𝑛 ∈ ℕ (𝐸‘𝑛))
100	70, 2, 90, 99	omessle 47329	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑂‘(𝐺‘𝑘)) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
101	89, 100	eqbrtrd 5127	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛)))) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
102	101	3adant3 1150	. . . . . . . . . 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 13216	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
104	103	3exp 1137	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ ℕ → (𝑥 ⊆ (1...𝑘) → (Σ^‘(𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))))
105	104	adantr 486	. . . . . . 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 3161	. . . . . 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 16	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → (Σ^‘(𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
108	33, 107	eqbrtrd 5127	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → (Σ^‘((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑥)) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
109	108	ralrimiva 3154	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (𝒫 ℕ ∩ Fin)(Σ^‘((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑥)) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
110	18, 22, 16	sge0lefi 47229	. . 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 260	. 2 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛)))) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
112	16, 23, 29, 111	xrletrid 13209	1 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛)))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557  ∪ cuni 4867  ∪ ciun 4951  Disj wdisj 5070   class class class wbr 5103   ↦ cmpt 5186  dom cdm 5655   ↾ cres 5657  ⟶wf 6529  ‘cfv 6533  (class class class)co 7414  Fincfn 8955  0cc0 11127  1c1 11128  +∞cpnf 11267  ℝ^*cxr 11269   ≤ cle 11271  ℕcn 12260  ℤ_≥cuz 12890  [,]cicc 13404  ...cfz 13564  Σ^{^}csumge0 47193  OutMeascome 47320  CaraGenccaragen 47322

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-inf2 9623  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204  ax-pre-sup 11205

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-disj 5071  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-1o 8458  df-oadd 8462  df-omul 8463  df-er 8699  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-sup 9415  df-oi 9485  df-card 9947  df-acn 9950  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-div 11899  df-nn 12261  df-2 12330  df-3 12331  df-n0 12532  df-z 12619  df-uz 12891  df-rp 13046  df-xadd 13167  df-ico 13407  df-icc 13408  df-fz 13565  df-fzo 13713  df-seq 14069  df-exp 14129  df-hash 14398  df-cj 15189  df-re 15190  df-im 15191  df-sqrt 15325  df-abs 15326  df-clim 15578  df-sum 15777  df-sumge0 47194  df-ome 47321  df-caragen 47323

This theorem is used by:  caratheodory  47359