Theorem caratheodorylem2 46955

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 46932	. . . . . . . . . 10 ⊢ (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂)
5	1, 4	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ⊆ dom 𝑂)
6	5	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑆 ⊆ dom 𝑂)
7		caratheodorylem2.e	. . . . . . . . 9 ⊢ (𝜑 → 𝐸:ℕ⟶𝑆)
8	7	ffvelcdmda 7036	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸‘𝑛) ∈ 𝑆)
9	6, 8	sseldd 3922	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸‘𝑛) ∈ dom 𝑂)
10		elssuni 4881	. . . . . . 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 3963	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸‘𝑛) ⊆ 𝑋)
13	12	ralrimiva 3129	. . . 4 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐸‘𝑛) ⊆ 𝑋)
14		iunss 4987	. . . 4 ⊢ (∪ 𝑛 ∈ ℕ (𝐸‘𝑛) ⊆ 𝑋 ↔ ∀𝑛 ∈ ℕ (𝐸‘𝑛) ⊆ 𝑋)
15	13, 14	sylibr 234	. . 3 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐸‘𝑛) ⊆ 𝑋)
16	1, 2, 15	omexrcl 46935	. 2 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) ∈ ℝ*)
17		nnex 12180	. . . 4 ⊢ ℕ ∈ V
18	17	a1i 11	. . 3 ⊢ (𝜑 → ℕ ∈ V)
19	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑂 ∈ OutMeas)
20	19, 2, 12	omecl 46931	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞))
21		eqid 2736	. . . 4 ⊢ (𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛)))
22	20, 21	fmptd 7066	. . 3 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛))):ℕ⟶(0[,]+∞))
23	18, 22	sge0xrcl 46813	. 2 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ*)
24		nfv 1916	. . 3 ⊢ Ⅎ𝑛𝜑
25		nfcv 2898	. . 3 ⊢ Ⅎ𝑛𝐸
26		nnuz 12827	. . 3 ⊢ ℕ = (ℤ≥‘1)
27	1, 2, 3	caragensspw 46937	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝒫 𝑋)
28	7, 27	fssd 6685	. . 3 ⊢ (𝜑 → 𝐸:ℕ⟶𝒫 𝑋)
29	24, 25, 1, 2, 26, 28	omeiunle 46945	. 2 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛)))))
30		elpwinss 45480	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ℕ ∩ Fin) → 𝑥 ⊆ ℕ)
31	30	resmptd 6005	. . . . . . 7 ⊢ (𝑥 ∈ (𝒫 ℕ ∩ Fin) → ((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑥) = (𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛))))
32	31	fveq2d 6844	. . . . . 6 ⊢ (𝑥 ∈ (𝒫 ℕ ∩ Fin) → (Σ^‘((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑥)) = (Σ^‘(𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))))
33	32	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → (Σ^‘((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑥)) = (Σ^‘(𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))))
34		1zzd 12558	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 1 ∈ ℤ)
35	30	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 𝑥 ⊆ ℕ)
36		elinel2 4142	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ℕ ∩ Fin) → 𝑥 ∈ Fin)
37	36	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 𝑥 ∈ Fin)
38	34, 26, 35, 37	uzfissfz 45756	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → ∃𝑘 ∈ ℕ 𝑥 ⊆ (1...𝑘))
39		vex 3433	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
40	39	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ⊆ (1...𝑘)) → 𝑥 ∈ V)
41	1	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ 𝑥) → 𝑂 ∈ OutMeas)
42	28	ad2antrr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ 𝑥) → 𝐸:ℕ⟶𝒫 𝑋)
43		fz1ssnn 13509	. . . . . . . . . . . . . . . . . 18 ⊢ (1...𝑘) ⊆ ℕ
44		ssel2 3916	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ⊆ (1...𝑘) ∧ 𝑛 ∈ 𝑥) → 𝑛 ∈ (1...𝑘))
45	43, 44	sselid 3919	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ⊆ (1...𝑘) ∧ 𝑛 ∈ 𝑥) → 𝑛 ∈ ℕ)
46	45	adantll 715	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ 𝑥) → 𝑛 ∈ ℕ)
47	42, 46	ffvelcdmd 7037	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ 𝑥) → (𝐸‘𝑛) ∈ 𝒫 𝑋)
48		elpwi 4548	. . . . . . . . . . . . . . 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 46931	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ 𝑥) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞))
51		eqid 2736	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))
52	50, 51	fmptd 7066	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ⊆ (1...𝑘)) → (𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛))):𝑥⟶(0[,]+∞))
53	40, 52	sge0xrcl 46813	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ*)
54	53	3adant2 1132	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ*)
55		ovex 7400	. . . . . . . . . . . . 13 ⊢ (1...𝑘) ∈ V
56	55	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (1...𝑘) ∈ V)
57		elfznn 13507	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...𝑘) → 𝑛 ∈ ℕ)
58	57, 20	sylan2 594	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑘)) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞))
59		eqid 2736	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛)))
60	58, 59	fmptd 7066	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛))):(1...𝑘)⟶(0[,]+∞))
61	56, 60	sge0xrcl 46813	. . . . . . . . . . 11 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ*)
62	61	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ*)
63	16	3ad2ant1 1134	. . . . . . . . . 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 1193	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ (1...𝑘)) → 𝜑)
66	57	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 ∈ ℕ)
67	65, 66, 20	syl2anc 585	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ (1...𝑘)) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞))
68		simp3 1139	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → 𝑥 ⊆ (1...𝑘))
69	64, 67, 68	sge0lessmpt 46827	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))) ≤ (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛)))))
70	1	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑂 ∈ OutMeas)
71	7	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐸:ℕ⟶𝑆)
72		caratheodorylem2.5	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → Disj 𝑛 ∈ ℕ (𝐸‘𝑛))
73	72	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Disj 𝑛 ∈ ℕ (𝐸‘𝑛))
74		caratheodorylem2.g	. . . . . . . . . . . . . . 15 ⊢ 𝐺 = (𝑘 ∈ ℕ ↦ ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛))
75		nfiu1 4969	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛)
76		nfcv 2898	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘∪ 𝑚 ∈ (1...𝑛)(𝐸‘𝑚)
77		fveq2 6840	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑚 → (𝐸‘𝑛) = (𝐸‘𝑚))
78	77	cbviunv 4981	. . . . . . . . . . . . . . . . . 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 7375	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑛 → (1...𝑘) = (1...𝑛))
81	80	iuneq1d 4961	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑛 → ∪ 𝑚 ∈ (1...𝑘)(𝐸‘𝑚) = ∪ 𝑚 ∈ (1...𝑛)(𝐸‘𝑚))
82	79, 81	eqtrd 2771	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑛 → ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛) = ∪ 𝑚 ∈ (1...𝑛)(𝐸‘𝑚))
83	75, 76, 82	cbvmpt 5187	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ ↦ ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛)) = (𝑛 ∈ ℕ ↦ ∪ 𝑚 ∈ (1...𝑛)(𝐸‘𝑚))
84	74, 83	eqtri 2759	. . . . . . . . . . . . . 14 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ∪ 𝑚 ∈ (1...𝑛)(𝐸‘𝑚))
85		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ)
86	85, 26	eleqtrdi 2846	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ (ℤ≥‘1))
87	86	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ≥‘1))
88	70, 3, 26, 71, 73, 84, 87	caratheodorylem1 46954	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑂‘(𝐺‘𝑘)) = (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛)))))
89	88	eqcomd 2742	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛)))) = (𝑂‘(𝐺‘𝑘)))
90	15	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∪ 𝑛 ∈ ℕ (𝐸‘𝑛) ⊆ 𝑋)
91		fvex 6853	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸‘𝑛) ∈ V
92	55, 91	iunex 7921	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛) ∈ V
93	74	fvmpt2 6959	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℕ ∧ ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛) ∈ V) → (𝐺‘𝑘) = ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛))
94	85, 92, 93	sylancl 587	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → (𝐺‘𝑘) = ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛))
95	43	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → (1...𝑘) ⊆ ℕ)
96		iunss1 4948	. . . . . . . . . . . . . . . 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 3956	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → (𝐺‘𝑘) ⊆ ∪ 𝑛 ∈ ℕ (𝐸‘𝑛))
99	98	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ⊆ ∪ 𝑛 ∈ ℕ (𝐸‘𝑛))
100	70, 2, 90, 99	omessle 46926	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑂‘(𝐺‘𝑘)) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
101	89, 100	eqbrtrd 5107	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛)))) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
102	101	3adant3 1133	. . . . . . . . . 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 13113	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
104	103	3exp 1120	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ ℕ → (𝑥 ⊆ (1...𝑘) → (Σ^‘(𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))))
105	104	adantr 480	. . . . . . 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 3136	. . . . . 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 5107	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → (Σ^‘((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑥)) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
109	108	ralrimiva 3129	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (𝒫 ℕ ∩ Fin)(Σ^‘((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑥)) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
110	18, 22, 16	sge0lefi 46826	. . 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 257	. 2 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛)))) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
112	16, 23, 29, 111	xrletrid 13106	1 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛)))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061  Vcvv 3429   ∩ cin 3888   ⊆ wss 3889  𝒫 cpw 4541  ∪ cuni 4850  ∪ ciun 4933  Disj wdisj 5052   class class class wbr 5085   ↦ cmpt 5166  dom cdm 5631   ↾ cres 5633  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  Fincfn 8893  0cc0 11038  1c1 11039  +∞cpnf 11176  ℝ^*cxr 11178   ≤ cle 11180  ℕcn 12174  ℤ_≥cuz 12788  [,]cicc 13301  ...cfz 13461  Σ^{^}csumge0 46790  OutMeascome 46917  CaraGenccaragen 46919

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-inf2 9562  ax-ac2 10385  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-disj 5053  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  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 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-oadd 8409  df-omul 8410  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-sup 9355  df-oi 9425  df-card 9863  df-acn 9866  df-ac 10038  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-nn 12175  df-2 12244  df-3 12245  df-n0 12438  df-z 12525  df-uz 12789  df-rp 12943  df-xadd 13064  df-ico 13304  df-icc 13305  df-fz 13462  df-fzo 13609  df-seq 13964  df-exp 14024  df-hash 14293  df-cj 15061  df-re 15062  df-im 15063  df-sqrt 15197  df-abs 15198  df-clim 15450  df-sum 15649  df-sumge0 46791  df-ome 46918  df-caragen 46920

This theorem is referenced by:  caratheodory  46956