Theorem caratheodorylem2 46970

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 46947	. . . . . . . . . 10 ⊢ (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂)
5	1, 4	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ⊆ dom 𝑂)
6	5	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑆 ⊆ dom 𝑂)
7		caratheodorylem2.e	. . . . . . . . 9 ⊢ (𝜑 → 𝐸:ℕ⟶𝑆)
8	7	ffvelcdmda 7025	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸‘𝑛) ∈ 𝑆)
9	6, 8	sseldd 3916	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸‘𝑛) ∈ dom 𝑂)
10		elssuni 4869	. . . . . . 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 3956	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸‘𝑛) ⊆ 𝑋)
13	12	ralrimiva 3131	. . . 4 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐸‘𝑛) ⊆ 𝑋)
14		iunss 4974	. . . 4 ⊢ (∪ 𝑛 ∈ ℕ (𝐸‘𝑛) ⊆ 𝑋 ↔ ∀𝑛 ∈ ℕ (𝐸‘𝑛) ⊆ 𝑋)
15	13, 14	sylibr 235	. . 3 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐸‘𝑛) ⊆ 𝑋)
16	1, 2, 15	omexrcl 46950	. 2 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) ∈ ℝ*)
17		nnex 12171	. . . 4 ⊢ ℕ ∈ V
18	17	a1i 11	. . 3 ⊢ (𝜑 → ℕ ∈ V)
19	1	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑂 ∈ OutMeas)
20	19, 2, 12	omecl 46946	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞))
21		eqid 2739	. . . 4 ⊢ (𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛)))
22	20, 21	fmptd 7055	. . 3 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛))):ℕ⟶(0[,]+∞))
23	18, 22	sge0xrcl 46828	. 2 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ*)
24		nfv 1921	. . 3 ⊢ Ⅎ𝑛𝜑
25		nfcv 2901	. . 3 ⊢ Ⅎ𝑛𝐸
26		nnuz 12818	. . 3 ⊢ ℕ = (ℤ≥‘1)
27	1, 2, 3	caragensspw 46952	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝒫 𝑋)
28	7, 27	fssd 6672	. . 3 ⊢ (𝜑 → 𝐸:ℕ⟶𝒫 𝑋)
29	24, 25, 1, 2, 26, 28	omeiunle 46960	. 2 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛)))))
30		elpwinss 45497	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ℕ ∩ Fin) → 𝑥 ⊆ ℕ)
31	30	resmptd 5992	. . . . . . 7 ⊢ (𝑥 ∈ (𝒫 ℕ ∩ Fin) → ((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑥) = (𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛))))
32	31	fveq2d 6831	. . . . . 6 ⊢ (𝑥 ∈ (𝒫 ℕ ∩ Fin) → (Σ^‘((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑥)) = (Σ^‘(𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))))
33	32	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → (Σ^‘((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑥)) = (Σ^‘(𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))))
34		1zzd 12549	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 1 ∈ ℤ)
35	30	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 𝑥 ⊆ ℕ)
36		elinel2 4131	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ℕ ∩ Fin) → 𝑥 ∈ Fin)
37	36	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → 𝑥 ∈ Fin)
38	34, 26, 35, 37	uzfissfz 45771	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → ∃𝑘 ∈ ℕ 𝑥 ⊆ (1...𝑘))
39		vex 3435	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
40	39	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ⊆ (1...𝑘)) → 𝑥 ∈ V)
41	1	ad2antrr 732	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ 𝑥) → 𝑂 ∈ OutMeas)
42	28	ad2antrr 732	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ 𝑥) → 𝐸:ℕ⟶𝒫 𝑋)
43		fz1ssnn 13500	. . . . . . . . . . . . . . . . . 18 ⊢ (1...𝑘) ⊆ ℕ
44		ssel2 3910	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ⊆ (1...𝑘) ∧ 𝑛 ∈ 𝑥) → 𝑛 ∈ (1...𝑘))
45	43, 44	sselid 3913	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ⊆ (1...𝑘) ∧ 𝑛 ∈ 𝑥) → 𝑛 ∈ ℕ)
46	45	adantll 720	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ 𝑥) → 𝑛 ∈ ℕ)
47	42, 46	ffvelcdmd 7026	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ 𝑥) → (𝐸‘𝑛) ∈ 𝒫 𝑋)
48		elpwi 4536	. . . . . . . . . . . . . . 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 46946	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ 𝑥) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞))
51		eqid 2739	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))
52	50, 51	fmptd 7055	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ⊆ (1...𝑘)) → (𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛))):𝑥⟶(0[,]+∞))
53	40, 52	sge0xrcl 46828	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ*)
54	53	3adant2 1137	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ*)
55		ovex 7389	. . . . . . . . . . . . 13 ⊢ (1...𝑘) ∈ V
56	55	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (1...𝑘) ∈ V)
57		elfznn 13498	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...𝑘) → 𝑛 ∈ ℕ)
58	57, 20	sylan2 599	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑘)) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞))
59		eqid 2739	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛)))
60	58, 59	fmptd 7055	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛))):(1...𝑘)⟶(0[,]+∞))
61	56, 60	sge0xrcl 46828	. . . . . . . . . . 11 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ*)
62	61	3ad2ant1 1139	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ*)
63	16	3ad2ant1 1139	. . . . . . . . . 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 1198	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ (1...𝑘)) → 𝜑)
66	57	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 ∈ ℕ)
67	65, 66, 20	syl2anc 590	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) ∧ 𝑛 ∈ (1...𝑘)) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞))
68		simp3 1144	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → 𝑥 ⊆ (1...𝑘))
69	64, 67, 68	sge0lessmpt 46842	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))) ≤ (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛)))))
70	1	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑂 ∈ OutMeas)
71	7	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐸:ℕ⟶𝑆)
72		caratheodorylem2.5	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → Disj 𝑛 ∈ ℕ (𝐸‘𝑛))
73	72	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Disj 𝑛 ∈ ℕ (𝐸‘𝑛))
74		caratheodorylem2.g	. . . . . . . . . . . . . . 15 ⊢ 𝐺 = (𝑘 ∈ ℕ ↦ ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛))
75		nfiu1 4957	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛)
76		nfcv 2901	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘∪ 𝑚 ∈ (1...𝑛)(𝐸‘𝑚)
77		fveq2 6827	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑚 → (𝐸‘𝑛) = (𝐸‘𝑚))
78	77	cbviunv 4968	. . . . . . . . . . . . . . . . . 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 7364	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑛 → (1...𝑘) = (1...𝑛))
81	80	iuneq1d 4949	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑛 → ∪ 𝑚 ∈ (1...𝑘)(𝐸‘𝑚) = ∪ 𝑚 ∈ (1...𝑛)(𝐸‘𝑚))
82	79, 81	eqtrd 2774	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑛 → ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛) = ∪ 𝑚 ∈ (1...𝑛)(𝐸‘𝑚))
83	75, 76, 82	cbvmpt 5174	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ ↦ ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛)) = (𝑛 ∈ ℕ ↦ ∪ 𝑚 ∈ (1...𝑛)(𝐸‘𝑚))
84	74, 83	eqtri 2762	. . . . . . . . . . . . . 14 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ∪ 𝑚 ∈ (1...𝑛)(𝐸‘𝑚))
85		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ)
86	85, 26	eleqtrdi 2849	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ (ℤ≥‘1))
87	86	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ≥‘1))
88	70, 3, 26, 71, 73, 84, 87	caratheodorylem1 46969	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑂‘(𝐺‘𝑘)) = (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛)))))
89	88	eqcomd 2745	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛)))) = (𝑂‘(𝐺‘𝑘)))
90	15	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∪ 𝑛 ∈ ℕ (𝐸‘𝑛) ⊆ 𝑋)
91		fvex 6840	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸‘𝑛) ∈ V
92	55, 91	iunex 7910	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛) ∈ V
93	74	fvmpt2 6947	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℕ ∧ ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛) ∈ V) → (𝐺‘𝑘) = ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛))
94	85, 92, 93	sylancl 592	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → (𝐺‘𝑘) = ∪ 𝑛 ∈ (1...𝑘)(𝐸‘𝑛))
95	43	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → (1...𝑘) ⊆ ℕ)
96		iunss1 4936	. . . . . . . . . . . . . . . 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 3949	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → (𝐺‘𝑘) ⊆ ∪ 𝑛 ∈ ℕ (𝐸‘𝑛))
99	98	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ⊆ ∪ 𝑛 ∈ ℕ (𝐸‘𝑛))
100	70, 2, 90, 99	omessle 46941	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑂‘(𝐺‘𝑘)) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
101	89, 100	eqbrtrd 5094	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (Σ^‘(𝑛 ∈ (1...𝑘) ↦ (𝑂‘(𝐸‘𝑛)))) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
102	101	3adant3 1138	. . . . . . . . . 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 13104	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑥 ⊆ (1...𝑘)) → (Σ^‘(𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
104	103	3exp 1125	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ ℕ → (𝑥 ⊆ (1...𝑘) → (Σ^‘(𝑛 ∈ 𝑥 ↦ (𝑂‘(𝐸‘𝑛)))) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))))
105	104	adantr 481	. . . . . . 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 3138	. . . . . 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 5094	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 ℕ ∩ Fin)) → (Σ^‘((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑥)) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
109	108	ralrimiva 3131	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (𝒫 ℕ ∩ Fin)(Σ^‘((𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛))) ↾ 𝑥)) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
110	18, 22, 16	sge0lefi 46841	. . 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 258	. 2 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛)))) ≤ (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)))
112	16, 23, 29, 111	xrletrid 13097	1 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ ℕ (𝐸‘𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸‘𝑛)))))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ∃wrex 3063  Vcvv 3431   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4529  ∪ cuni 4838  ∪ ciun 4921  Disj wdisj 5039   class class class wbr 5072   ↦ cmpt 5153  dom cdm 5618   ↾ cres 5620  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356  Fincfn 8883  0cc0 11029  1c1 11030  +∞cpnf 11167  ℝ^*cxr 11169   ≤ cle 11171  ℕcn 12165  ℤ_≥cuz 12779  [,]cicc 13292  ...cfz 13452  Σ^{^}csumge0 46805  OutMeascome 46932  CaraGenccaragen 46934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-inf2 9553  ax-ac2 10376  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-disj 5040  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-se 5572  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-isom 6494  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-oadd 8399  df-omul 8400  df-er 8633  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9345  df-oi 9415  df-card 9854  df-acn 9857  df-ac 10029  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12166  df-2 12235  df-3 12236  df-n0 12429  df-z 12516  df-uz 12780  df-rp 12934  df-xadd 13055  df-ico 13295  df-icc 13296  df-fz 13453  df-fzo 13600  df-seq 13955  df-exp 14015  df-hash 14284  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-clim 15441  df-sum 15640  df-sumge0 46806  df-ome 46933  df-caragen 46935

This theorem is referenced by:  caratheodory  46971