Theorem voliunlem3 25781

Description: Lemma for voliun 25783. (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 25780	. . 3 ⊢ (𝜑 → ∪ ran 𝐹 ∈ dom vol)
5		mblvol 25759	. . 3 ⊢ (∪ ran 𝐹 ∈ dom vol → (vol‘∪ ran 𝐹) = (vol*‘∪ ran 𝐹))
6	4, 5	syl 18	. 2 ⊢ (𝜑 → (vol‘∪ ran 𝐹) = (vol*‘∪ ran 𝐹))
7	1	frnd 6715	. . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ dom vol)
8		mblss 25760	. . . . . . . 8 ⊢ (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ)
9		reex 11218	. . . . . . . . 9 ⊢ ℝ ∈ V
10	9	elpw2 5303	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 ℝ ↔ 𝑥 ⊆ ℝ)
11	8, 10	sylibr 237	. . . . . . 7 ⊢ (𝑥 ∈ dom vol → 𝑥 ∈ 𝒫 ℝ)
12	11	ssriv 3938	. . . . . 6 ⊢ dom vol ⊆ 𝒫 ℝ
13	7, 12	sstrdi 3946	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ 𝒫 ℝ)
14		sspwuni 5064	. . . . 5 ⊢ (ran 𝐹 ⊆ 𝒫 ℝ ↔ ∪ ran 𝐹 ⊆ ℝ)
15	13, 14	sylib 221	. . . 4 ⊢ (𝜑 → ∪ ran 𝐹 ⊆ ℝ)
16		ovolcl 25707	. . . 4 ⊢ (∪ ran 𝐹 ⊆ ℝ → (vol*‘∪ ran 𝐹) ∈ ℝ*)
17	15, 16	syl 18	. . 3 ⊢ (𝜑 → (vol*‘∪ ran 𝐹) ∈ ℝ*)
18		nnuz 12929	. . . . . . . 8 ⊢ ℕ = (ℤ≥‘1)
19		1zzd 12652	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℤ)
20		2fveq3 6887	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → (vol‘(𝐹‘𝑛)) = (vol‘(𝐹‘𝑘)))
21		voliunlem3.2	. . . . . . . . . . 11 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol‘(𝐹‘𝑛)))
22		fvex 6895	. . . . . . . . . . 11 ⊢ (vol‘(𝐹‘𝑘)) ∈ V
23	20, 21, 22	fvmpt 6990	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (𝐺‘𝑘) = (vol‘(𝐹‘𝑘)))
24	23	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) = (vol‘(𝐹‘𝑘)))
25		voliunlem3.4	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑖 ∈ ℕ (vol‘(𝐹‘𝑖)) ∈ ℝ)
26		2fveq3 6887	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑘 → (vol‘(𝐹‘𝑖)) = (vol‘(𝐹‘𝑘)))
27	26	eleq1d 2847	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑘 → ((vol‘(𝐹‘𝑖)) ∈ ℝ ↔ (vol‘(𝐹‘𝑘)) ∈ ℝ))
28	27	rspccva 3578	. . . . . . . . . 10 ⊢ ((∀𝑖 ∈ ℕ (vol‘(𝐹‘𝑖)) ∈ ℝ ∧ 𝑘 ∈ ℕ) → (vol‘(𝐹‘𝑘)) ∈ ℝ)
29	25, 28	sylan 592	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol‘(𝐹‘𝑘)) ∈ ℝ)
30	24, 29	eqeltrd 2862	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ)
31	18, 19, 30	serfre 14097	. . . . . . 7 ⊢ (𝜑 → seq1( + , 𝐺):ℕ⟶ℝ)
32		voliunlem3.1	. . . . . . . 8 ⊢ 𝑆 = seq1( + , 𝐺)
33	32	feq1i 6697	. . . . . . 7 ⊢ (𝑆:ℕ⟶ℝ ↔ seq1( + , 𝐺):ℕ⟶ℝ)
34	31, 33	sylibr 237	. . . . . 6 ⊢ (𝜑 → 𝑆:ℕ⟶ℝ)
35	34	frnd 6715	. . . . 5 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ)
36		ressxr 11280	. . . . 5 ⊢ ℝ ⊆ ℝ*
37	35, 36	sstrdi 3946	. . . 4 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ*)
38		supxrcl 13369	. . . 4 ⊢ (ran 𝑆 ⊆ ℝ* → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
39	37, 38	syl 18	. . 3 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
40		eqid 2762	. . . . 5 ⊢ seq1( + , (𝑛 ∈ ℕ ↦ (vol*‘(𝐹‘𝑛)))) = seq1( + , (𝑛 ∈ ℕ ↦ (vol*‘(𝐹‘𝑛))))
41		eqid 2762	. . . . 5 ⊢ (𝑛 ∈ ℕ ↦ (vol*‘(𝐹‘𝑛))) = (𝑛 ∈ ℕ ↦ (vol*‘(𝐹‘𝑛)))
42	1	ffvelcdmda 7080	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ dom vol)
43		mblss 25760	. . . . . 6 ⊢ ((𝐹‘𝑛) ∈ dom vol → (𝐹‘𝑛) ⊆ ℝ)
44	42, 43	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ⊆ ℝ)
45		mblvol 25759	. . . . . . 7 ⊢ ((𝐹‘𝑛) ∈ dom vol → (vol‘(𝐹‘𝑛)) = (vol*‘(𝐹‘𝑛)))
46	42, 45	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘(𝐹‘𝑛)) = (vol*‘(𝐹‘𝑛)))
47		2fveq3 6887	. . . . . . . . 9 ⊢ (𝑖 = 𝑛 → (vol‘(𝐹‘𝑖)) = (vol‘(𝐹‘𝑛)))
48	47	eleq1d 2847	. . . . . . . 8 ⊢ (𝑖 = 𝑛 → ((vol‘(𝐹‘𝑖)) ∈ ℝ ↔ (vol‘(𝐹‘𝑛)) ∈ ℝ))
49	48	rspccva 3578	. . . . . . 7 ⊢ ((∀𝑖 ∈ ℕ (vol‘(𝐹‘𝑖)) ∈ ℝ ∧ 𝑛 ∈ ℕ) → (vol‘(𝐹‘𝑛)) ∈ ℝ)
50	25, 49	sylan 592	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘(𝐹‘𝑛)) ∈ ℝ)
51	46, 50	eqeltrrd 2863	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol*‘(𝐹‘𝑛)) ∈ ℝ)
52	40, 41, 44, 51	ovoliun 25734	. . . 4 ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ (𝐹‘𝑛)) ≤ sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol*‘(𝐹‘𝑛)))), ℝ*, < ))
53	1	ffnd 6707	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn ℕ)
54		fniunfv 7247	. . . . . 6 ⊢ (𝐹 Fn ℕ → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) = ∪ ran 𝐹)
55	53, 54	syl 18	. . . . 5 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) = ∪ ran 𝐹)
56	55	fveq2d 6886	. . . 4 ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ (𝐹‘𝑛)) = (vol*‘∪ ran 𝐹))
57	46	mpteq2dva 5202	. . . . . . . . 9 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (vol‘(𝐹‘𝑛))) = (𝑛 ∈ ℕ ↦ (vol*‘(𝐹‘𝑛))))
58	21, 57	eqtrid 2809	. . . . . . . 8 ⊢ (𝜑 → 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘(𝐹‘𝑛))))
59	58	seqeq3d 14075	. . . . . . 7 ⊢ (𝜑 → seq1( + , 𝐺) = seq1( + , (𝑛 ∈ ℕ ↦ (vol*‘(𝐹‘𝑛)))))
60	32, 59	eqtr2id 2810	. . . . . 6 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ (vol*‘(𝐹‘𝑛)))) = 𝑆)
61	60	rneqd 5926	. . . . 5 ⊢ (𝜑 → ran seq1( + , (𝑛 ∈ ℕ ↦ (vol*‘(𝐹‘𝑛)))) = ran 𝑆)
62	61	supeq1d 9419	. . . 4 ⊢ (𝜑 → sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol*‘(𝐹‘𝑛)))), ℝ*, < ) = sup(ran 𝑆, ℝ*, < ))
63	52, 56, 62	3brtr3d 5140	. . 3 ⊢ (𝜑 → (vol*‘∪ ran 𝐹) ≤ sup(ran 𝑆, ℝ*, < ))
64		ovolge0 25710	. . . . . . . . . 10 ⊢ (∪ ran 𝐹 ⊆ ℝ → 0 ≤ (vol*‘∪ ran 𝐹))
65	15, 64	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (vol*‘∪ ran 𝐹))
66		mnflt0 13178	. . . . . . . . . 10 ⊢ -∞ < 0
67		mnfxr 11293	. . . . . . . . . . 11 ⊢ -∞ ∈ ℝ*
68		0xr 11283	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ*
69		xrltletr 13210	. . . . . . . . . . 11 ⊢ ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ* ∧ (vol*‘∪ ran 𝐹) ∈ ℝ*) → ((-∞ < 0 ∧ 0 ≤ (vol*‘∪ ran 𝐹)) → -∞ < (vol*‘∪ ran 𝐹)))
70	67, 68, 69	mp3an12 1480	. . . . . . . . . 10 ⊢ ((vol*‘∪ ran 𝐹) ∈ ℝ* → ((-∞ < 0 ∧ 0 ≤ (vol*‘∪ ran 𝐹)) → -∞ < (vol*‘∪ ran 𝐹)))
71	66, 70	mpani 709	. . . . . . . . 9 ⊢ ((vol*‘∪ ran 𝐹) ∈ ℝ* → (0 ≤ (vol*‘∪ ran 𝐹) → -∞ < (vol*‘∪ ran 𝐹)))
72	17, 65, 71	sylc 66	. . . . . . . 8 ⊢ (𝜑 → -∞ < (vol*‘∪ ran 𝐹))
73		xrrebnd 13222	. . . . . . . . . 10 ⊢ ((vol*‘∪ ran 𝐹) ∈ ℝ* → ((vol*‘∪ ran 𝐹) ∈ ℝ ↔ (-∞ < (vol*‘∪ ran 𝐹) ∧ (vol*‘∪ ran 𝐹) < +∞)))
74	17, 73	syl 18	. . . . . . . . 9 ⊢ (𝜑 → ((vol*‘∪ ran 𝐹) ∈ ℝ ↔ (-∞ < (vol*‘∪ ran 𝐹) ∧ (vol*‘∪ ran 𝐹) < +∞)))
75	9	elpw2 5303	. . . . . . . . . . . 12 ⊢ (∪ ran 𝐹 ∈ 𝒫 ℝ ↔ ∪ ran 𝐹 ⊆ ℝ)
76	15, 75	sylibr 237	. . . . . . . . . . 11 ⊢ (𝜑 → ∪ ran 𝐹 ∈ 𝒫 ℝ)
77		simpl 488	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ 𝜑) → 𝑥 = ∪ ran 𝐹)
78	77	sseq1d 3965	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ 𝜑) → (𝑥 ⊆ ℝ ↔ ∪ ran 𝐹 ⊆ ℝ))
79	77	fveq2d 6886	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ 𝜑) → (vol*‘𝑥) = (vol*‘∪ ran 𝐹))
80	79	eleq1d 2847	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ 𝜑) → ((vol*‘𝑥) ∈ ℝ ↔ (vol*‘∪ ran 𝐹) ∈ ℝ))
81		simpll 779	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑥 = ∪ ran 𝐹 ∧ 𝜑) ∧ 𝑛 ∈ ℕ) → 𝑥 = ∪ ran 𝐹)
82	81	ineq1d 4168	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑥 = ∪ ran 𝐹 ∧ 𝜑) ∧ 𝑛 ∈ ℕ) → (𝑥 ∩ (𝐹‘𝑛)) = (∪ ran 𝐹 ∩ (𝐹‘𝑛)))
83		fnfvelrn 7076	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝐹 Fn ℕ ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ran 𝐹)
84	53, 83	sylan 592	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ran 𝐹)
85		elssuni 4902	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝐹‘𝑛) ∈ ran 𝐹 → (𝐹‘𝑛) ⊆ ∪ ran 𝐹)
86	84, 85	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ⊆ ∪ ran 𝐹)
87	86	adantll 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑥 = ∪ ran 𝐹 ∧ 𝜑) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ⊆ ∪ ran 𝐹)
88		sseqin2 4172	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐹‘𝑛) ⊆ ∪ ran 𝐹 ↔ (∪ ran 𝐹 ∩ (𝐹‘𝑛)) = (𝐹‘𝑛))
89	87, 88	sylib 221	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑥 = ∪ ran 𝐹 ∧ 𝜑) ∧ 𝑛 ∈ ℕ) → (∪ ran 𝐹 ∩ (𝐹‘𝑛)) = (𝐹‘𝑛))
90	82, 89	eqtrd 2797	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑥 = ∪ ran 𝐹 ∧ 𝜑) ∧ 𝑛 ∈ ℕ) → (𝑥 ∩ (𝐹‘𝑛)) = (𝐹‘𝑛))
91	90	fveq2d 6886	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑥 = ∪ ran 𝐹 ∧ 𝜑) ∧ 𝑛 ∈ ℕ) → (vol*‘(𝑥 ∩ (𝐹‘𝑛))) = (vol*‘(𝐹‘𝑛)))
92	46	adantll 727	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑥 = ∪ ran 𝐹 ∧ 𝜑) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐹‘𝑛)) = (vol*‘(𝐹‘𝑛)))
93	91, 92	eqtr4d 2800	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑥 = ∪ ran 𝐹 ∧ 𝜑) ∧ 𝑛 ∈ ℕ) → (vol*‘(𝑥 ∩ (𝐹‘𝑛))) = (vol‘(𝐹‘𝑛)))
94	93	mpteq2dva 5202	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ 𝜑) → (𝑛 ∈ ℕ ↦ (vol*‘(𝑥 ∩ (𝐹‘𝑛)))) = (𝑛 ∈ ℕ ↦ (vol‘(𝐹‘𝑛))))
95	94	adantrr 730	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → (𝑛 ∈ ℕ ↦ (vol*‘(𝑥 ∩ (𝐹‘𝑛)))) = (𝑛 ∈ ℕ ↦ (vol‘(𝐹‘𝑛))))
96	95, 3, 21	3eqtr4g 2822	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → 𝐻 = 𝐺)
97	96	seqeq3d 14075	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → seq1( + , 𝐻) = seq1( + , 𝐺))
98	97, 32	eqtr4di 2815	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → seq1( + , 𝐻) = 𝑆)
99	98	fveq1d 6884	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → (seq1( + , 𝐻)‘𝑘) = (𝑆‘𝑘))
100		difeq1 4070	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = ∪ ran 𝐹 → (𝑥 ∖ ∪ ran 𝐹) = (∪ ran 𝐹 ∖ ∪ ran 𝐹))
101		difid 4328	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∪ ran 𝐹 ∖ ∪ ran 𝐹) = ∅
102	100, 101	eqtrdi 2813	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = ∪ ran 𝐹 → (𝑥 ∖ ∪ ran 𝐹) = ∅)
103	102	fveq2d 6886	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = ∪ ran 𝐹 → (vol*‘(𝑥 ∖ ∪ ran 𝐹)) = (vol*‘∅))
104		ovol0 25722	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (vol*‘∅) = 0
105	103, 104	eqtrdi 2813	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = ∪ ran 𝐹 → (vol*‘(𝑥 ∖ ∪ ran 𝐹)) = 0)
106	105	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → (vol*‘(𝑥 ∖ ∪ ran 𝐹)) = 0)
107	99, 106	oveq12d 7434	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))) = ((𝑆‘𝑘) + 0))
108	34	ffvelcdmda 7080	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ∈ ℝ)
109	108	adantl 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → (𝑆‘𝑘) ∈ ℝ)
110	109	recnd 11264	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → (𝑆‘𝑘) ∈ ℂ)
111	110	addridd 11437	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → ((𝑆‘𝑘) + 0) = (𝑆‘𝑘))
112	107, 111	eqtrd 2797	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))) = (𝑆‘𝑘))
113		fveq2 6882	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ∪ ran 𝐹 → (vol*‘𝑥) = (vol*‘∪ ran 𝐹))
114	113	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → (vol*‘𝑥) = (vol*‘∪ ran 𝐹))
115	112, 114	breq12d 5120	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → (((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ≤ (vol*‘𝑥) ↔ (𝑆‘𝑘) ≤ (vol*‘∪ ran 𝐹)))
116	115	expr 462	. . . . . . . . . . . . . . . 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 276	. . . . . . . . . . . . . . 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 347	. . . . . . . . . . . . . 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 347	. . . . . . . . . . . . 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 816	. . . . . . . . . . . 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 1151	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝐹:ℕ⟶dom vol)
122	2	3ad2ant1 1151	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → Disj 𝑖 ∈ ℕ (𝐹‘𝑖))
123		simp2 1155	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝑥 ⊆ ℝ)
124		simp3 1156	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) ∈ ℝ)
125	121, 122, 3, 123, 124	voliunlem1 25779	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ≤ (vol*‘𝑥))
126	125	3exp1 1371	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ⊆ ℝ → ((vol*‘𝑥) ∈ ℝ → (𝑘 ∈ ℕ → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ≤ (vol*‘𝑥)))))
127	120, 126	vtoclg 3520	. . . . . . . . . . 11 ⊢ (∪ ran 𝐹 ∈ 𝒫 ℝ → (𝜑 → (∪ ran 𝐹 ⊆ ℝ → ((vol*‘∪ ran 𝐹) ∈ ℝ → (𝑘 ∈ ℕ → (𝑆‘𝑘) ≤ (vol*‘∪ ran 𝐹))))))
128	76, 127	mpcom 39	. . . . . . . . . 10 ⊢ (𝜑 → (∪ ran 𝐹 ⊆ ℝ → ((vol*‘∪ ran 𝐹) ∈ ℝ → (𝑘 ∈ ℕ → (𝑆‘𝑘) ≤ (vol*‘∪ ran 𝐹)))))
129	15, 128	mpd 16	. . . . . . . . 9 ⊢ (𝜑 → ((vol*‘∪ ran 𝐹) ∈ ℝ → (𝑘 ∈ ℕ → (𝑆‘𝑘) ≤ (vol*‘∪ ran 𝐹))))
130	74, 129	sylbird 263	. . . . . . . 8 ⊢ (𝜑 → ((-∞ < (vol*‘∪ ran 𝐹) ∧ (vol*‘∪ ran 𝐹) < +∞) → (𝑘 ∈ ℕ → (𝑆‘𝑘) ≤ (vol*‘∪ ran 𝐹))))
131	72, 130	mpand 708	. . . . . . 7 ⊢ (𝜑 → ((vol*‘∪ ran 𝐹) < +∞ → (𝑘 ∈ ℕ → (𝑆‘𝑘) ≤ (vol*‘∪ ran 𝐹))))
132		nltpnft 13218	. . . . . . . . 9 ⊢ ((vol*‘∪ ran 𝐹) ∈ ℝ* → ((vol*‘∪ ran 𝐹) = +∞ ↔ ¬ (vol*‘∪ ran 𝐹) < +∞))
133	17, 132	syl 18	. . . . . . . 8 ⊢ (𝜑 → ((vol*‘∪ ran 𝐹) = +∞ ↔ ¬ (vol*‘∪ ran 𝐹) < +∞))
134		rexr 11282	. . . . . . . . . . 11 ⊢ ((𝑆‘𝑘) ∈ ℝ → (𝑆‘𝑘) ∈ ℝ*)
135		pnfge 13183	. . . . . . . . . . 11 ⊢ ((𝑆‘𝑘) ∈ ℝ* → (𝑆‘𝑘) ≤ +∞)
136	108, 134, 135	3syl 19	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ≤ +∞)
137	136	ex 418	. . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ ℕ → (𝑆‘𝑘) ≤ +∞))
138		breq2 5111	. . . . . . . . . 10 ⊢ ((vol*‘∪ ran 𝐹) = +∞ → ((𝑆‘𝑘) ≤ (vol*‘∪ ran 𝐹) ↔ (𝑆‘𝑘) ≤ +∞))
139	138	imbi2d 343	. . . . . . . . 9 ⊢ ((vol*‘∪ ran 𝐹) = +∞ → ((𝑘 ∈ ℕ → (𝑆‘𝑘) ≤ (vol*‘∪ ran 𝐹)) ↔ (𝑘 ∈ ℕ → (𝑆‘𝑘) ≤ +∞)))
140	137, 139	syl5ibrcom 250	. . . . . . . 8 ⊢ (𝜑 → ((vol*‘∪ ran 𝐹) = +∞ → (𝑘 ∈ ℕ → (𝑆‘𝑘) ≤ (vol*‘∪ ran 𝐹))))
141	133, 140	sylbird 263	. . . . . . 7 ⊢ (𝜑 → (¬ (vol*‘∪ ran 𝐹) < +∞ → (𝑘 ∈ ℕ → (𝑆‘𝑘) ≤ (vol*‘∪ ran 𝐹))))
142	131, 141	pm2.61d 181	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ ℕ → (𝑆‘𝑘) ≤ (vol*‘∪ ran 𝐹)))
143	142	ralrimiv 3155	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ (vol*‘∪ ran 𝐹))
144	34	ffnd 6707	. . . . . 6 ⊢ (𝜑 → 𝑆 Fn ℕ)
145		breq1 5110	. . . . . . 7 ⊢ (𝑧 = (𝑆‘𝑘) → (𝑧 ≤ (vol*‘∪ ran 𝐹) ↔ (𝑆‘𝑘) ≤ (vol*‘∪ ran 𝐹)))
146	145	ralrn 7084	. . . . . 6 ⊢ (𝑆 Fn ℕ → (∀𝑧 ∈ ran 𝑆 𝑧 ≤ (vol*‘∪ ran 𝐹) ↔ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ (vol*‘∪ ran 𝐹)))
147	144, 146	syl 18	. . . . 5 ⊢ (𝜑 → (∀𝑧 ∈ ran 𝑆 𝑧 ≤ (vol*‘∪ ran 𝐹) ↔ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ (vol*‘∪ ran 𝐹)))
148	143, 147	mpbird 260	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ ran 𝑆 𝑧 ≤ (vol*‘∪ ran 𝐹))
149		supxrleub 13380	. . . . 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 596	. . . 4 ⊢ (𝜑 → (sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘∪ ran 𝐹) ↔ ∀𝑧 ∈ ran 𝑆 𝑧 ≤ (vol*‘∪ ran 𝐹)))
151	148, 150	mpbird 260	. . 3 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘∪ ran 𝐹))
152	17, 39, 63, 151	xrletrid 13208	. 2 ⊢ (𝜑 → (vol*‘∪ ran 𝐹) = sup(ran 𝑆, ℝ*, < ))
153	6, 152	eqtrd 2797	1 ⊢ (𝜑 → (vol‘∪ ran 𝐹) = sup(ran 𝑆, ℝ*, < ))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3078   ∖ cdif 3899   ∩ cin 3901   ⊆ wss 3902  ∅c0 4282  𝒫 cpw 4560  ∪ cuni 4870  ∪ ciun 4954  Disj wdisj 5074   class class class wbr 5107   ↦ cmpt 5190  dom cdm 5659  ran crn 5660   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7416  supcsup 9413  ℝcr 11126  0cc0 11127  1c1 11128   + caddc 11130  +∞cpnf 11267  -∞cmnf 11268  ℝ^*cxr 11269   < clt 11270   ≤ cle 11271  ℕcn 12260  seqcseq 14067  vol*covol 25691  volcvol 25692

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 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-inf2 9623  ax-cc 10440  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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-disj 5075  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7681  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-2o 8459  df-er 8699  df-map 8831  df-pm 8832  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-sup 9415  df-inf 9416  df-oi 9485  df-dju 9909  df-card 9947  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-q 13001  df-rp 13045  df-xadd 13166  df-ioo 13404  df-ico 13406  df-icc 13407  df-fz 13564  df-fzo 13712  df-fl 13855  df-seq 14068  df-exp 14128  df-hash 14397  df-cj 15188  df-re 15189  df-im 15190  df-sqrt 15324  df-abs 15325  df-clim 15577  df-rlim 15578  df-sum 15776  df-xmet 21579  df-met 21580  df-ovol 25693  df-vol 25694

This theorem is used by:  voliun  25783