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Theorem voliunlem3 25820

Description: Lemma for voliun 25822. (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 25819	. . 3 ⊢ (𝜑 → ∪ ran 𝐹 ∈ dom vol)
5		mblvol 25798	. . 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 6707	. . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ dom vol)
8		mblss 25799	. . . . . . . 8 ⊢ (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ)
9		reex 11248	. . . . . . . . 9 ⊢ ℝ ∈ V
10	9	elpw2 5296	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 ℝ ↔ 𝑥 ⊆ ℝ)
11	8, 10	sylibr 237	. . . . . . 7 ⊢ (𝑥 ∈ dom vol → 𝑥 ∈ 𝒫 ℝ)
12	11	ssriv 3935	. . . . . 6 ⊢ dom vol ⊆ 𝒫 ℝ
13	7, 12	sstrdi 3943	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ 𝒫 ℝ)
14		sspwuni 5060	. . . . 5 ⊢ (ran 𝐹 ⊆ 𝒫 ℝ ↔ ∪ ran 𝐹 ⊆ ℝ)
15	13, 14	sylib 221	. . . 4 ⊢ (𝜑 → ∪ ran 𝐹 ⊆ ℝ)
16		ovolcl 25746	. . . 4 ⊢ (∪ ran 𝐹 ⊆ ℝ → (vol‘∪ ran 𝐹) ∈ ℝ^)
17	15, 16	syl 18	. . 3 ⊢ (𝜑 → (vol‘∪ ran 𝐹) ∈ ℝ^)
18		nnuz 12959	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
19		1zzd 12682	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℤ)
20		2fveq3 6879	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → (vol‘(𝐹‘𝑛)) = (vol‘(𝐹‘𝑘)))
21		voliunlem3.2	. . . . . . . . . . 11 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol‘(𝐹‘𝑛)))
22		fvex 6887	. . . . . . . . . . 11 ⊢ (vol‘(𝐹‘𝑘)) ∈ V
23	20, 21, 22	fvmpt 6982	. . . . . . . . . 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 6879	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑘 → (vol‘(𝐹‘𝑖)) = (vol‘(𝐹‘𝑘)))
27	26	eleq1d 2845	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑘 → ((vol‘(𝐹‘𝑖)) ∈ ℝ ↔ (vol‘(𝐹‘𝑘)) ∈ ℝ))
28	27	rspccva 3575	. . . . . . . . . 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 2860	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ)
31	18, 19, 30	serfre 14128	. . . . . . 7 ⊢ (𝜑 → seq1( + , 𝐺):ℕ⟶ℝ)
32		voliunlem3.1	. . . . . . . 8 ⊢ 𝑆 = seq1( + , 𝐺)
33	32	feq1i 6689	. . . . . . 7 ⊢ (𝑆:ℕ⟶ℝ ↔ seq1( + , 𝐺):ℕ⟶ℝ)
34	31, 33	sylibr 237	. . . . . 6 ⊢ (𝜑 → 𝑆:ℕ⟶ℝ)
35	34	frnd 6707	. . . . 5 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ)
36		ressxr 11310	. . . . 5 ⊢ ℝ ⊆ ℝ^*
37	35, 36	sstrdi 3943	. . . 4 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ^*)
38		supxrcl 13400	. . . 4 ⊢ (ran 𝑆 ⊆ ℝ^* → sup(ran 𝑆, ℝ^, < ) ∈ ℝ^)
39	37, 38	syl 18	. . 3 ⊢ (𝜑 → sup(ran 𝑆, ℝ^, < ) ∈ ℝ^)
40		eqid 2760	. . . . 5 ⊢ seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝐹‘𝑛)))) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝐹‘𝑛))))
41		eqid 2760	. . . . 5 ⊢ (𝑛 ∈ ℕ ↦ (vol‘(𝐹‘𝑛))) = (𝑛 ∈ ℕ ↦ (vol‘(𝐹‘𝑛)))
42	1	ffvelcdmda 7073	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ dom vol)
43		mblss 25799	. . . . . 6 ⊢ ((𝐹‘𝑛) ∈ dom vol → (𝐹‘𝑛) ⊆ ℝ)
44	42, 43	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ⊆ ℝ)
45		mblvol 25798	. . . . . . 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 6879	. . . . . . . . 9 ⊢ (𝑖 = 𝑛 → (vol‘(𝐹‘𝑖)) = (vol‘(𝐹‘𝑛)))
48	47	eleq1d 2845	. . . . . . . 8 ⊢ (𝑖 = 𝑛 → ((vol‘(𝐹‘𝑖)) ∈ ℝ ↔ (vol‘(𝐹‘𝑛)) ∈ ℝ))
49	48	rspccva 3575	. . . . . . 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 2861	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol*‘(𝐹‘𝑛)) ∈ ℝ)
52	40, 41, 44, 51	ovoliun 25773	. . . 4 ⊢ (𝜑 → (vol‘∪ 𝑛 ∈ ℕ (𝐹‘𝑛)) ≤ sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝐹‘𝑛)))), ℝ^*, < ))
53	1	ffnd 6699	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn ℕ)
54		fniunfv 7240	. . . . . 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 6878	. . . 4 ⊢ (𝜑 → (vol‘∪ 𝑛 ∈ ℕ (𝐹‘𝑛)) = (vol‘∪ ran 𝐹))
57	46	mpteq2dva 5198	. . . . . . . . 9 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (vol‘(𝐹‘𝑛))) = (𝑛 ∈ ℕ ↦ (vol*‘(𝐹‘𝑛))))
58	21, 57	eqtrid 2807	. . . . . . . 8 ⊢ (𝜑 → 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘(𝐹‘𝑛))))
59	58	seqeq3d 14106	. . . . . . 7 ⊢ (𝜑 → seq1( + , 𝐺) = seq1( + , (𝑛 ∈ ℕ ↦ (vol*‘(𝐹‘𝑛)))))
60	32, 59	eqtr2id 2808	. . . . . 6 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ (vol*‘(𝐹‘𝑛)))) = 𝑆)
61	60	rneqd 5917	. . . . 5 ⊢ (𝜑 → ran seq1( + , (𝑛 ∈ ℕ ↦ (vol*‘(𝐹‘𝑛)))) = ran 𝑆)
62	61	supeq1d 9416	. . . 4 ⊢ (𝜑 → sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝐹‘𝑛)))), ℝ^, < ) = sup(ran 𝑆, ℝ^*, < ))
63	52, 56, 62	3brtr3d 5136	. . 3 ⊢ (𝜑 → (vol‘∪ ran 𝐹) ≤ sup(ran 𝑆, ℝ^, < ))
64		ovolge0 25749	. . . . . . . . . 10 ⊢ (∪ ran 𝐹 ⊆ ℝ → 0 ≤ (vol*‘∪ ran 𝐹))
65	15, 64	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (vol*‘∪ ran 𝐹))
66		mnflt0 13209	. . . . . . . . . 10 ⊢ -∞ < 0
67		mnfxr 11323	. . . . . . . . . . 11 ⊢ -∞ ∈ ℝ^*
68		0xr 11313	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ^*
69		xrltletr 13241	. . . . . . . . . . 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 13253	. . . . . . . . . 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 5296	. . . . . . . . . . . 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 3962	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ 𝜑) → (𝑥 ⊆ ℝ ↔ ∪ ran 𝐹 ⊆ ℝ))
79	77	fveq2d 6878	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ 𝜑) → (vol‘𝑥) = (vol‘∪ ran 𝐹))
80	79	eleq1d 2845	. . . . . . . . . . . . . . 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 4165	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑥 = ∪ ran 𝐹 ∧ 𝜑) ∧ 𝑛 ∈ ℕ) → (𝑥 ∩ (𝐹‘𝑛)) = (∪ ran 𝐹 ∩ (𝐹‘𝑛)))
83		fnfvelrn 7069	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4899	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4169	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑥 = ∪ ran 𝐹 ∧ 𝜑) ∧ 𝑛 ∈ ℕ) → (𝑥 ∩ (𝐹‘𝑛)) = (𝐹‘𝑛))
91	90	fveq2d 6878	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑥 = ∪ ran 𝐹 ∧ 𝜑) ∧ 𝑛 ∈ ℕ) → (vol*‘(𝑥 ∩ (𝐹‘𝑛))) = (vol‘(𝐹‘𝑛)))
94	93	mpteq2dva 5198	. . . . . . . . . . . . . . . . . . . . . . . . 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 2820	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → 𝐻 = 𝐺)
97	96	seqeq3d 14106	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → seq1( + , 𝐻) = seq1( + , 𝐺))
98	97, 32	eqtr4di 2813	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → seq1( + , 𝐻) = 𝑆)
99	98	fveq1d 6876	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → (seq1( + , 𝐻)‘𝑘) = (𝑆‘𝑘))
100		difeq1 4067	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = ∪ ran 𝐹 → (𝑥 ∖ ∪ ran 𝐹) = (∪ ran 𝐹 ∖ ∪ ran 𝐹))
101		difid 4325	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∪ ran 𝐹 ∖ ∪ ran 𝐹) = ∅
102	100, 101	eqtrdi 2811	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = ∪ ran 𝐹 → (𝑥 ∖ ∪ ran 𝐹) = ∅)
103	102	fveq2d 6878	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = ∪ ran 𝐹 → (vol‘(𝑥 ∖ ∪ ran 𝐹)) = (vol‘∅))
104		ovol0 25761	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (vol*‘∅) = 0
105	103, 104	eqtrdi 2811	. . . . . . . . . . . . . . . . . . . . 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 7427	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))) = ((𝑆‘𝑘) + 0))
108	34	ffvelcdmda 7073	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ∈ ℝ)
109	108	adantl 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → (𝑆‘𝑘) ∈ ℝ)
110	109	recnd 11294	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → (𝑆‘𝑘) ∈ ℂ)
111	110	addridd 11467	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → ((𝑆‘𝑘) + 0) = (𝑆‘𝑘))
112	107, 111	eqtrd 2795	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = ∪ ran 𝐹 ∧ (𝜑 ∧ 𝑘 ∈ ℕ)) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))) = (𝑆‘𝑘))
113		fveq2 6874	. . . . . . . . . . . . . . . . . . 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 5116	. . . . . . . . . . . . . . . . 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 25818	. . . . . . . . . . . . 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 3517	. . . . . . . . . . 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 13249	. . . . . . . . 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 11312	. . . . . . . . . . 11 ⊢ ((𝑆‘𝑘) ∈ ℝ → (𝑆‘𝑘) ∈ ℝ^*)
135		pnfge 13214	. . . . . . . . . . 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 5107	. . . . . . . . . 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 3153	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ (vol*‘∪ ran 𝐹))
144	34	ffnd 6699	. . . . . 6 ⊢ (𝜑 → 𝑆 Fn ℕ)
145		breq1 5106	. . . . . . 7 ⊢ (𝑧 = (𝑆‘𝑘) → (𝑧 ≤ (vol‘∪ ran 𝐹) ↔ (𝑆‘𝑘) ≤ (vol‘∪ ran 𝐹)))
146	145	ralrn 7077	. . . . . 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 13411	. . . . 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 13239	. 2 ⊢ (𝜑 → (vol‘∪ ran 𝐹) = sup(ran 𝑆, ℝ^, < ))
153	6, 152	eqtrd 2795	1 ⊢ (𝜑 → (vol‘∪ ran 𝐹) = sup(ran 𝑆, ℝ^*, < ))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3076 ∖ cdif 3896 ∩ cin 3898 ⊆ wss 3899 ∅c0 4279 𝒫 cpw 4557 ∪ cuni 4867 ∪ ciun 4951 Disj wdisj 5070 class class class wbr 5103 ↦ cmpt 5186 dom cdm 5648 ran crn 5649 Fn wfn 6523 ⟶wf 6524 ‘cfv 6528 (class class class)co 7409 supcsup 9410 ℝcr 11156 0cc0 11157 1c1 11158 + caddc 11160 +∞cpnf 11297 -∞cmnf 11298 ℝ^*cxr 11299 < clt 11300 ≤ cle 11301 ℕcn 12290 seqcseq 14098 vol*covol 25730 volcvol 25731

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 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7735 ax-inf2 9620 ax-cc 10470 ax-cnex 11213 ax-resscn 11214 ax-1cn 11215 ax-icn 11216 ax-addcl 11217 ax-addrcl 11218 ax-mulcl 11219 ax-mulrcl 11220 ax-mulcom 11221 ax-addass 11222 ax-mulass 11223 ax-distr 11224 ax-i2m1 11225 ax-1ne0 11226 ax-1rid 11227 ax-rnegex 11228 ax-rrecex 11229 ax-cnre 11230 ax-pre-lttri 11231 ax-pre-lttrn 11232 ax-pre-ltadd 11233 ax-pre-mulgt0 11234 ax-pre-sup 11235

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 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-se 5602 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-pred 6294 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6530 df-fn 6531 df-f 6532 df-f1 6533 df-fo 6534 df-f1o 6535 df-fv 6536 df-isom 6537 df-riota 7366 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7677 df-om 7862 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8455 df-2o 8456 df-er 8696 df-map 8828 df-pm 8829 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-sup 9412 df-inf 9413 df-oi 9482 df-dju 9939 df-card 9977 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11500 df-neg 11501 df-div 11929 df-nn 12291 df-2 12360 df-3 12361 df-n0 12562 df-z 12649 df-uz 12921 df-q 13031 df-rp 13076 df-xadd 13197 df-ioo 13435 df-ico 13437 df-icc 13438 df-fz 13595 df-fzo 13743 df-fl 13886 df-seq 14099 df-exp 14159 df-hash 14428 df-cj 15219 df-re 15220 df-im 15221 df-sqrt 15355 df-abs 15356 df-clim 15608 df-rlim 15609 df-sum 15807 df-xmet 21618 df-met 21619 df-ovol 25732 df-vol 25733

This theorem is used by: voliun 25822

W3C validator