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Theorem voliunlem3 25460

Description: Lemma for voliun 25462. (Contributed by Mario Carneiro, 20-Mar-2014.)

**Hypotheses**
Ref	Expression
voliunlem.3	⊢ (𝜑 → 𝐹:ℕ⟶dom vol)
voliunlem.5	⊢ (𝜑 → Disj 𝑖 ∈ ℕ (𝐹‘𝑖))
voliunlem.6	⊢ 𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝑥 ∩ (𝐹‘𝑛))))
voliunlem3.1	⊢ 𝑆 = seq1( + , 𝐺)
voliunlem3.2	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol‘(𝐹‘𝑛)))
voliunlem3.4	⊢ (𝜑 → ∀𝑖 ∈ ℕ (vol‘(𝐹‘𝑖)) ∈ ℝ)

**Assertion**
Ref	Expression
voliunlem3	⊢ (𝜑 → (vol‘∪ ran 𝐹) = sup(ran 𝑆, ℝ^*, < ))

Distinct variable groups: 𝑖,𝑛,𝑥,𝐹 𝑥,𝑆 𝜑,𝑛,𝑥 Allowed substitution hints: 𝜑(𝑖) 𝑆(𝑖,𝑛) 𝐺(𝑥,𝑖,𝑛) 𝐻(𝑥,𝑖,𝑛)

Proof of Theorem voliunlem3 Dummy variables 𝑘 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		voliunlem.3	. . . 4 ⊢ (𝜑 → 𝐹:ℕ⟶dom vol)
2		voliunlem.5	. . . 4 ⊢ (𝜑 → Disj 𝑖 ∈ ℕ (𝐹‘𝑖))
3		voliunlem.6	. . . 4 ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝑥 ∩ (𝐹‘𝑛))))
4	1, 2, 3	voliunlem2 25459	. . 3 ⊢ (𝜑 → ∪ ran 𝐹 ∈ dom vol)
5		mblvol 25438	. . 3 ⊢ (∪ ran 𝐹 ∈ dom vol → (vol‘∪ ran 𝐹) = (vol*‘∪ ran 𝐹))
6	4, 5	syl 17	. 2 ⊢ (𝜑 → (vol‘∪ ran 𝐹) = (vol*‘∪ ran 𝐹))
7	1	frnd 6699	. . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ dom vol)
8		mblss 25439	. . . . . . . 8 ⊢ (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ)
9		reex 11166	. . . . . . . . 9 ⊢ ℝ ∈ V
10	9	elpw2 5292	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 ℝ ↔ 𝑥 ⊆ ℝ)
11	8, 10	sylibr 234	. . . . . . 7 ⊢ (𝑥 ∈ dom vol → 𝑥 ∈ 𝒫 ℝ)
12	11	ssriv 3953	. . . . . 6 ⊢ dom vol ⊆ 𝒫 ℝ
13	7, 12	sstrdi 3962	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ 𝒫 ℝ)
14		sspwuni 5067	. . . . 5 ⊢ (ran 𝐹 ⊆ 𝒫 ℝ ↔ ∪ ran 𝐹 ⊆ ℝ)
15	13, 14	sylib 218	. . . 4 ⊢ (𝜑 → ∪ ran 𝐹 ⊆ ℝ)
16		ovolcl 25386	. . . 4 ⊢ (∪ ran 𝐹 ⊆ ℝ → (vol‘∪ ran 𝐹) ∈ ℝ^)
17	15, 16	syl 17	. . 3 ⊢ (𝜑 → (vol‘∪ ran 𝐹) ∈ ℝ^)
18		nnuz 12843	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
19		1zzd 12571	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℤ)
20		2fveq3 6866	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → (vol‘(𝐹‘𝑛)) = (vol‘(𝐹‘𝑘)))
21		voliunlem3.2	. . . . . . . . . . 11 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol‘(𝐹‘𝑛)))
22		fvex 6874	. . . . . . . . . . 11 ⊢ (vol‘(𝐹‘𝑘)) ∈ V
23	20, 21, 22	fvmpt 6971	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (𝐺‘𝑘) = (vol‘(𝐹‘𝑘)))
24	23	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) = (vol‘(𝐹‘𝑘)))
25		voliunlem3.4	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑖 ∈ ℕ (vol‘(𝐹‘𝑖)) ∈ ℝ)
26		2fveq3 6866	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑘 → (vol‘(𝐹‘𝑖)) = (vol‘(𝐹‘𝑘)))
27	26	eleq1d 2814	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑘 → ((vol‘(𝐹‘𝑖)) ∈ ℝ ↔ (vol‘(𝐹‘𝑘)) ∈ ℝ))
28	27	rspccva 3590	. . . . . . . . . 10 ⊢ ((∀𝑖 ∈ ℕ (vol‘(𝐹‘𝑖)) ∈ ℝ ∧ 𝑘 ∈ ℕ) → (vol‘(𝐹‘𝑘)) ∈ ℝ)
29	25, 28	sylan 580	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol‘(𝐹‘𝑘)) ∈ ℝ)
30	24, 29	eqeltrd 2829	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ)
31	18, 19, 30	serfre 14003	. . . . . . 7 ⊢ (𝜑 → seq1( + , 𝐺):ℕ⟶ℝ)
32		voliunlem3.1	. . . . . . . 8 ⊢ 𝑆 = seq1( + , 𝐺)
33	32	feq1i 6682	. . . . . . 7 ⊢ (𝑆:ℕ⟶ℝ ↔ seq1( + , 𝐺):ℕ⟶ℝ)
34	31, 33	sylibr 234	. . . . . 6 ⊢ (𝜑 → 𝑆:ℕ⟶ℝ)
35	34	frnd 6699	. . . . 5 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ)
36		ressxr 11225	. . . . 5 ⊢ ℝ ⊆ ℝ^*
37	35, 36	sstrdi 3962	. . . 4 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ^*)
38		supxrcl 13282	. . . 4 ⊢ (ran 𝑆 ⊆ ℝ^* → sup(ran 𝑆, ℝ^, < ) ∈ ℝ^)
39	37, 38	syl 17	. . 3 ⊢ (𝜑 → sup(ran 𝑆, ℝ^, < ) ∈ ℝ^)
40		eqid 2730	. . . . 5 ⊢ seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝐹‘𝑛)))) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝐹‘𝑛))))
41		eqid 2730	. . . . 5 ⊢ (𝑛 ∈ ℕ ↦ (vol‘(𝐹‘𝑛))) = (𝑛 ∈ ℕ ↦ (vol‘(𝐹‘𝑛)))
42	1	ffvelcdmda 7059	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ dom vol)
43		mblss 25439	. . . . . 6 ⊢ ((𝐹‘𝑛) ∈ dom vol → (𝐹‘𝑛) ⊆ ℝ)
44	42, 43	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ⊆ ℝ)
45		mblvol 25438	. . . . . . 7 ⊢ ((𝐹‘𝑛) ∈ dom vol → (vol‘(𝐹‘𝑛)) = (vol*‘(𝐹‘𝑛)))
46	42, 45	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘(𝐹‘𝑛)) = (vol*‘(𝐹‘𝑛)))
47		2fveq3 6866	. . . . . . . . 9 ⊢ (𝑖 = 𝑛 → (vol‘(𝐹‘𝑖)) = (vol‘(𝐹‘𝑛)))
48	47	eleq1d 2814	. . . . . . . 8 ⊢ (𝑖 = 𝑛 → ((vol‘(𝐹‘𝑖)) ∈ ℝ ↔ (vol‘(𝐹‘𝑛)) ∈ ℝ))
49	48	rspccva 3590	. . . . . . 7 ⊢ ((∀𝑖 ∈ ℕ (vol‘(𝐹‘𝑖)) ∈ ℝ ∧ 𝑛 ∈ ℕ) → (vol‘(𝐹‘𝑛)) ∈ ℝ)
50	25, 49	sylan 580	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘(𝐹‘𝑛)) ∈ ℝ)
51	46, 50	eqeltrrd 2830	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol*‘(𝐹‘𝑛)) ∈ ℝ)
52	40, 41, 44, 51	ovoliun 25413	. . . 4 ⊢ (𝜑 → (vol‘∪ 𝑛 ∈ ℕ (𝐹‘𝑛)) ≤ sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝐹‘𝑛)))), ℝ^*, < ))
53	1	ffnd 6692	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn ℕ)
54		fniunfv 7224	. . . . . 6 ⊢ (𝐹 Fn ℕ → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) = ∪ ran 𝐹)
55	53, 54	syl 17	. . . . 5 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) = ∪ ran 𝐹)
56	55	fveq2d 6865	. . . 4 ⊢ (𝜑 → (vol‘∪ 𝑛 ∈ ℕ (𝐹‘𝑛)) = (vol‘∪ ran 𝐹))
57	46	mpteq2dva 5203	. . . . . . . . 9 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (vol‘(𝐹‘𝑛))) = (𝑛 ∈ ℕ ↦ (vol*‘(𝐹‘𝑛))))
58	21, 57	eqtrid 2777	. . . . . . . 8 ⊢ (𝜑 → 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘(𝐹‘𝑛))))
59	58	seqeq3d 13981	. . . . . . 7 ⊢ (𝜑 → seq1( + , 𝐺) = seq1( + , (𝑛 ∈ ℕ ↦ (vol*‘(𝐹‘𝑛)))))
60	32, 59	eqtr2id 2778	. . . . . 6 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ (vol*‘(𝐹‘𝑛)))) = 𝑆)
61	60	rneqd 5905	. . . . 5 ⊢ (𝜑 → ran seq1( + , (𝑛 ∈ ℕ ↦ (vol*‘(𝐹‘𝑛)))) = ran 𝑆)
62	61	supeq1d 9404	. . . 4 ⊢ (𝜑 → sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝐹‘𝑛)))), ℝ^, < ) = sup(ran 𝑆, ℝ^*, < ))
63	52, 56, 62	3brtr3d 5141	. . 3 ⊢ (𝜑 → (vol‘∪ ran 𝐹) ≤ sup(ran 𝑆, ℝ^, < ))
64		ovolge0 25389	. . . . . . . . . 10 ⊢ (∪ ran 𝐹 ⊆ ℝ → 0 ≤ (vol*‘∪ ran 𝐹))
65	15, 64	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (vol*‘∪ ran 𝐹))
66		mnflt0 13092	. . . . . . . . . 10 ⊢ -∞ < 0
67		mnfxr 11238	. . . . . . . . . . 11 ⊢ -∞ ∈ ℝ^*
68		0xr 11228	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ^*
69		xrltletr 13124	. . . . . . . . . . 11 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ (vol‘∪ ran 𝐹) ∈ ℝ^) → ((-∞ < 0 ∧ 0 ≤ (vol‘∪ ran 𝐹)) → -∞ < (vol‘∪ ran 𝐹)))
70	67, 68, 69	mp3an12 1453	. . . . . . . . . 10 ⊢ ((vol‘∪ ran 𝐹) ∈ ℝ^ → ((-∞ < 0 ∧ 0 ≤ (vol‘∪ ran 𝐹)) → -∞ < (vol‘∪ ran 𝐹)))
71	66, 70	mpani 696	. . . . . . . . 9 ⊢ ((vol‘∪ ran 𝐹) ∈ ℝ^ → (0 ≤ (vol‘∪ ran 𝐹) → -∞ < (vol‘∪ ran 𝐹)))
72	17, 65, 71	sylc 65	. . . . . . . 8 ⊢ (𝜑 → -∞ < (vol*‘∪ ran 𝐹))
73		xrrebnd 13135	. . . . . . . . . 10 ⊢ ((vol‘∪ ran 𝐹) ∈ ℝ^ → ((vol‘∪ ran 𝐹) ∈ ℝ ↔ (-∞ < (vol‘∪ ran 𝐹) ∧ (vol*‘∪ ran 𝐹) < +∞)))
74	17, 73	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((vol‘∪ ran 𝐹) ∈ ℝ ↔ (-∞ < (vol‘∪ ran 𝐹) ∧ (vol*‘∪ ran 𝐹) < +∞)))
75	9	elpw2 5292	. . . . . . . . . . . 12 ⊢ (∪ ran 𝐹 ∈ 𝒫 ℝ ↔ ∪ ran 𝐹 ⊆ ℝ)
76	15, 75	sylibr 234	. . . . . . . . . . 11 ⊢ (𝜑 → ∪ ran 𝐹 ∈ 𝒫 ℝ)
77		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ 𝜑) → 𝑥 = ∪ ran 𝐹)
78	77	sseq1d 3981	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ 𝜑) → (𝑥 ⊆ ℝ ↔ ∪ ran 𝐹 ⊆ ℝ))
79	77	fveq2d 6865	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ 𝜑) → (vol‘𝑥) = (vol‘∪ ran 𝐹))
80	79	eleq1d 2814	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ 𝜑) → ((vol‘𝑥) ∈ ℝ ↔ (vol‘∪ ran 𝐹) ∈ ℝ))
81		simpll 766	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑥 = ∪ ran 𝐹 ∧ 𝜑) ∧ 𝑛 ∈ ℕ) → 𝑥 = ∪ ran 𝐹)
82	81	ineq1d 4185	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑥 = ∪ ran 𝐹 ∧ 𝜑) ∧ 𝑛 ∈ ℕ) → (𝑥 ∩ (𝐹‘𝑛)) = (∪ ran 𝐹 ∩ (𝐹‘𝑛)))
83		fnfvelrn 7055	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝐹 Fn ℕ ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ran 𝐹)
84	53, 83	sylan 580	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ran 𝐹)
85		elssuni 4904	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝐹‘𝑛) ∈ ran 𝐹 → (𝐹‘𝑛) ⊆ ∪ ran 𝐹)
86	84, 85	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ⊆ ∪ ran 𝐹)
87	86	adantll 714	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑥 = ∪ ran 𝐹 ∧ 𝜑) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ⊆ ∪ ran 𝐹)
88		sseqin2 4189	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐹‘𝑛) ⊆ ∪ ran 𝐹 ↔ (∪ ran 𝐹 ∩ (𝐹‘𝑛)) = (𝐹‘𝑛))
89	87, 88	sylib 218	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑥 = ∪ ran 𝐹 ∧ 𝜑) ∧ 𝑛 ∈ ℕ) → (∪ ran 𝐹 ∩ (𝐹‘𝑛)) = (𝐹‘𝑛))
90	82, 89	eqtrd 2765	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑥 = ∪ ran 𝐹 ∧ 𝜑) ∧ 𝑛 ∈ ℕ) → (𝑥 ∩ (𝐹‘𝑛)) = (𝐹‘𝑛))
91	90	fveq2d 6865	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑥 = ∪ ran 𝐹 ∧ 𝜑) ∧ 𝑛 ∈ ℕ) → (vol‘(𝑥 ∩ (𝐹‘𝑛))) = (vol‘(𝐹‘𝑛)))
92	46	adantll 714	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑥 = ∪ ran 𝐹 ∧ 𝜑) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐹‘𝑛)) = (vol*‘(𝐹‘𝑛)))
93	91, 92	eqtr4d 2768	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑥 = ∪ ran 𝐹 ∧ 𝜑) ∧ 𝑛 ∈ ℕ) → (vol*‘(𝑥 ∩ (𝐹‘𝑛))) = (vol‘(𝐹‘𝑛)))
94	93	mpteq2dva 5203	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ 𝜑) → (𝑛 ∈ ℕ ↦ (vol*‘(𝑥 ∩ (𝐹‘𝑛)))) = (𝑛 ∈ ℕ ↦ (vol‘(𝐹‘𝑛))))
95	94	adantrr 717	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → (𝑛 ∈ ℕ ↦ (vol*‘(𝑥 ∩ (𝐹‘𝑛)))) = (𝑛 ∈ ℕ ↦ (vol‘(𝐹‘𝑛))))
96	95, 3, 21	3eqtr4g 2790	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → 𝐻 = 𝐺)
97	96	seqeq3d 13981	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → seq1( + , 𝐻) = seq1( + , 𝐺))
98	97, 32	eqtr4di 2783	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → seq1( + , 𝐻) = 𝑆)
99	98	fveq1d 6863	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → (seq1( + , 𝐻)‘𝑘) = (𝑆‘𝑘))
100		difeq1 4085	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = ∪ ran 𝐹 → (𝑥 ∖ ∪ ran 𝐹) = (∪ ran 𝐹 ∖ ∪ ran 𝐹))
101		difid 4342	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∪ ran 𝐹 ∖ ∪ ran 𝐹) = ∅
102	100, 101	eqtrdi 2781	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = ∪ ran 𝐹 → (𝑥 ∖ ∪ ran 𝐹) = ∅)
103	102	fveq2d 6865	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = ∪ ran 𝐹 → (vol‘(𝑥 ∖ ∪ ran 𝐹)) = (vol‘∅))
104		ovol0 25401	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (vol*‘∅) = 0
105	103, 104	eqtrdi 2781	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = ∪ ran 𝐹 → (vol*‘(𝑥 ∖ ∪ ran 𝐹)) = 0)
106	105	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → (vol*‘(𝑥 ∖ ∪ ran 𝐹)) = 0)
107	99, 106	oveq12d 7408	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))) = ((𝑆‘𝑘) + 0))
108	34	ffvelcdmda 7059	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ∈ ℝ)
109	108	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → (𝑆‘𝑘) ∈ ℝ)
110	109	recnd 11209	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → (𝑆‘𝑘) ∈ ℂ)
111	110	addridd 11381	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → ((𝑆‘𝑘) + 0) = (𝑆‘𝑘))
112	107, 111	eqtrd 2765	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))) = (𝑆‘𝑘))
113		fveq2 6861	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ∪ ran 𝐹 → (vol‘𝑥) = (vol‘∪ ran 𝐹))
114	113	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → (vol‘𝑥) = (vol‘∪ ran 𝐹))
115	112, 114	breq12d 5123	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → (((seq1( + , 𝐻)‘𝑘) + (vol‘(𝑥 ∖ ∪ ran 𝐹))) ≤ (vol‘𝑥) ↔ (𝑆‘𝑘) ≤ (vol*‘∪ ran 𝐹)))
116	115	expr 456	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ 𝜑) → (𝑘 ∈ ℕ → (((seq1( + , 𝐻)‘𝑘) + (vol‘(𝑥 ∖ ∪ ran 𝐹))) ≤ (vol‘𝑥) ↔ (𝑆‘𝑘) ≤ (vol*‘∪ ran 𝐹))))
117	116	pm5.74d 273	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ 𝜑) → ((𝑘 ∈ ℕ → ((seq1( + , 𝐻)‘𝑘) + (vol‘(𝑥 ∖ ∪ ran 𝐹))) ≤ (vol‘𝑥)) ↔ (𝑘 ∈ ℕ → (𝑆‘𝑘) ≤ (vol*‘∪ ran 𝐹))))
118	80, 117	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ 𝜑) → (((vol‘𝑥) ∈ ℝ → (𝑘 ∈ ℕ → ((seq1( + , 𝐻)‘𝑘) + (vol‘(𝑥 ∖ ∪ ran 𝐹))) ≤ (vol‘𝑥))) ↔ ((vol‘∪ ran 𝐹) ∈ ℝ → (𝑘 ∈ ℕ → (𝑆‘𝑘) ≤ (vol*‘∪ ran 𝐹)))))
119	78, 118	imbi12d 344	. . . . . . . . . . . . 13 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ 𝜑) → ((𝑥 ⊆ ℝ → ((vol‘𝑥) ∈ ℝ → (𝑘 ∈ ℕ → ((seq1( + , 𝐻)‘𝑘) + (vol‘(𝑥 ∖ ∪ ran 𝐹))) ≤ (vol‘𝑥)))) ↔ (∪ ran 𝐹 ⊆ ℝ → ((vol‘∪ ran 𝐹) ∈ ℝ → (𝑘 ∈ ℕ → (𝑆‘𝑘) ≤ (vol*‘∪ ran 𝐹))))))
120	119	pm5.74da 803	. . . . . . . . . . . 12 ⊢ (𝑥 = ∪ ran 𝐹 → ((𝜑 → (𝑥 ⊆ ℝ → ((vol‘𝑥) ∈ ℝ → (𝑘 ∈ ℕ → ((seq1( + , 𝐻)‘𝑘) + (vol‘(𝑥 ∖ ∪ ran 𝐹))) ≤ (vol‘𝑥))))) ↔ (𝜑 → (∪ ran 𝐹 ⊆ ℝ → ((vol‘∪ ran 𝐹) ∈ ℝ → (𝑘 ∈ ℕ → (𝑆‘𝑘) ≤ (vol*‘∪ ran 𝐹)))))))
121	1	3ad2ant1 1133	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝐹:ℕ⟶dom vol)
122	2	3ad2ant1 1133	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → Disj 𝑖 ∈ ℕ (𝐹‘𝑖))
123		simp2 1137	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝑥 ⊆ ℝ)
124		simp3 1138	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) → (vol‘𝑥) ∈ ℝ)
125	121, 122, 3, 123, 124	voliunlem1 25458	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (vol‘(𝑥 ∖ ∪ ran 𝐹))) ≤ (vol*‘𝑥))
126	125	3exp1 1353	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ⊆ ℝ → ((vol‘𝑥) ∈ ℝ → (𝑘 ∈ ℕ → ((seq1( + , 𝐻)‘𝑘) + (vol‘(𝑥 ∖ ∪ ran 𝐹))) ≤ (vol*‘𝑥)))))
127	120, 126	vtoclg 3523	. . . . . . . . . . 11 ⊢ (∪ ran 𝐹 ∈ 𝒫 ℝ → (𝜑 → (∪ ran 𝐹 ⊆ ℝ → ((vol‘∪ ran 𝐹) ∈ ℝ → (𝑘 ∈ ℕ → (𝑆‘𝑘) ≤ (vol‘∪ ran 𝐹))))))
128	76, 127	mpcom 38	. . . . . . . . . 10 ⊢ (𝜑 → (∪ ran 𝐹 ⊆ ℝ → ((vol‘∪ ran 𝐹) ∈ ℝ → (𝑘 ∈ ℕ → (𝑆‘𝑘) ≤ (vol‘∪ ran 𝐹)))))
129	15, 128	mpd 15	. . . . . . . . 9 ⊢ (𝜑 → ((vol‘∪ ran 𝐹) ∈ ℝ → (𝑘 ∈ ℕ → (𝑆‘𝑘) ≤ (vol‘∪ ran 𝐹))))
130	74, 129	sylbird 260	. . . . . . . 8 ⊢ (𝜑 → ((-∞ < (vol‘∪ ran 𝐹) ∧ (vol‘∪ ran 𝐹) < +∞) → (𝑘 ∈ ℕ → (𝑆‘𝑘) ≤ (vol*‘∪ ran 𝐹))))
131	72, 130	mpand 695	. . . . . . 7 ⊢ (𝜑 → ((vol‘∪ ran 𝐹) < +∞ → (𝑘 ∈ ℕ → (𝑆‘𝑘) ≤ (vol‘∪ ran 𝐹))))
132		nltpnft 13131	. . . . . . . . 9 ⊢ ((vol‘∪ ran 𝐹) ∈ ℝ^ → ((vol‘∪ ran 𝐹) = +∞ ↔ ¬ (vol‘∪ ran 𝐹) < +∞))
133	17, 132	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((vol‘∪ ran 𝐹) = +∞ ↔ ¬ (vol‘∪ ran 𝐹) < +∞))
134		rexr 11227	. . . . . . . . . . 11 ⊢ ((𝑆‘𝑘) ∈ ℝ → (𝑆‘𝑘) ∈ ℝ^*)
135		pnfge 13097	. . . . . . . . . . 11 ⊢ ((𝑆‘𝑘) ∈ ℝ^* → (𝑆‘𝑘) ≤ +∞)
136	108, 134, 135	3syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ≤ +∞)
137	136	ex 412	. . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ ℕ → (𝑆‘𝑘) ≤ +∞))
138		breq2 5114	. . . . . . . . . 10 ⊢ ((vol‘∪ ran 𝐹) = +∞ → ((𝑆‘𝑘) ≤ (vol‘∪ ran 𝐹) ↔ (𝑆‘𝑘) ≤ +∞))
139	138	imbi2d 340	. . . . . . . . 9 ⊢ ((vol‘∪ ran 𝐹) = +∞ → ((𝑘 ∈ ℕ → (𝑆‘𝑘) ≤ (vol‘∪ ran 𝐹)) ↔ (𝑘 ∈ ℕ → (𝑆‘𝑘) ≤ +∞)))
140	137, 139	syl5ibrcom 247	. . . . . . . 8 ⊢ (𝜑 → ((vol‘∪ ran 𝐹) = +∞ → (𝑘 ∈ ℕ → (𝑆‘𝑘) ≤ (vol‘∪ ran 𝐹))))
141	133, 140	sylbird 260	. . . . . . 7 ⊢ (𝜑 → (¬ (vol‘∪ ran 𝐹) < +∞ → (𝑘 ∈ ℕ → (𝑆‘𝑘) ≤ (vol‘∪ ran 𝐹))))
142	131, 141	pm2.61d 179	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ ℕ → (𝑆‘𝑘) ≤ (vol*‘∪ ran 𝐹)))
143	142	ralrimiv 3125	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ (vol*‘∪ ran 𝐹))
144	34	ffnd 6692	. . . . . 6 ⊢ (𝜑 → 𝑆 Fn ℕ)
145		breq1 5113	. . . . . . 7 ⊢ (𝑧 = (𝑆‘𝑘) → (𝑧 ≤ (vol‘∪ ran 𝐹) ↔ (𝑆‘𝑘) ≤ (vol‘∪ ran 𝐹)))
146	145	ralrn 7063	. . . . . 6 ⊢ (𝑆 Fn ℕ → (∀𝑧 ∈ ran 𝑆 𝑧 ≤ (vol‘∪ ran 𝐹) ↔ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ (vol‘∪ ran 𝐹)))
147	144, 146	syl 17	. . . . 5 ⊢ (𝜑 → (∀𝑧 ∈ ran 𝑆 𝑧 ≤ (vol‘∪ ran 𝐹) ↔ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ (vol‘∪ ran 𝐹)))
148	143, 147	mpbird 257	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ ran 𝑆 𝑧 ≤ (vol*‘∪ ran 𝐹))
149		supxrleub 13293	. . . . 5 ⊢ ((ran 𝑆 ⊆ ℝ^* ∧ (vol‘∪ ran 𝐹) ∈ ℝ^) → (sup(ran 𝑆, ℝ^, < ) ≤ (vol‘∪ ran 𝐹) ↔ ∀𝑧 ∈ ran 𝑆 𝑧 ≤ (vol*‘∪ ran 𝐹)))
150	37, 17, 149	syl2anc 584	. . . 4 ⊢ (𝜑 → (sup(ran 𝑆, ℝ^, < ) ≤ (vol‘∪ ran 𝐹) ↔ ∀𝑧 ∈ ran 𝑆 𝑧 ≤ (vol*‘∪ ran 𝐹)))
151	148, 150	mpbird 257	. . 3 ⊢ (𝜑 → sup(ran 𝑆, ℝ^, < ) ≤ (vol‘∪ ran 𝐹))
152	17, 39, 63, 151	xrletrid 13122	. 2 ⊢ (𝜑 → (vol‘∪ ran 𝐹) = sup(ran 𝑆, ℝ^, < ))
153	6, 152	eqtrd 2765	1 ⊢ (𝜑 → (vol‘∪ ran 𝐹) = sup(ran 𝑆, ℝ^*, < ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3045 ∖ cdif 3914 ∩ cin 3916 ⊆ wss 3917 ∅c0 4299 𝒫 cpw 4566 ∪ cuni 4874 ∪ ciun 4958 Disj wdisj 5077 class class class wbr 5110 ↦ cmpt 5191 dom cdm 5641 ran crn 5642 Fn wfn 6509 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 supcsup 9398 ℝcr 11074 0cc0 11075 1c1 11076 + caddc 11078 +∞cpnf 11212 -∞cmnf 11213 ℝ^*cxr 11214 < clt 11215 ≤ cle 11216 ℕcn 12193 seqcseq 13973 vol*covol 25370 volcvol 25371

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 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cc 10395 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-disj 5078 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-pm 8805 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-sup 9400 df-inf 9401 df-oi 9470 df-dju 9861 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-n0 12450 df-z 12537 df-uz 12801 df-q 12915 df-rp 12959 df-xadd 13080 df-ioo 13317 df-ico 13319 df-icc 13320 df-fz 13476 df-fzo 13623 df-fl 13761 df-seq 13974 df-exp 14034 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15461 df-rlim 15462 df-sum 15660 df-xmet 21264 df-met 21265 df-ovol 25372 df-vol 25373

This theorem is referenced by: voliun 25462

W3C validator