Theorem voliunlem2 25459

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))))

Assertion

Ref	Expression
voliunlem2	⊢ (𝜑 → ∪ ran 𝐹 ∈ dom vol)

Distinct variable groups:   𝑖,𝑛,𝑥,𝐹   𝜑,𝑛,𝑥

Allowed substitution hints:   𝜑(𝑖)   𝐻(𝑥,𝑖,𝑛)

Proof of Theorem voliunlem2

Dummy variables 𝑘 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		voliunlem.3	. . . . 5 ⊢ (𝜑 → 𝐹:ℕ⟶dom vol)
2	1	frnd 6699	. . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ dom vol)
3		mblss 25439	. . . . . 6 ⊢ (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ)
4		velpw 4571	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 ℝ ↔ 𝑥 ⊆ ℝ)
5	3, 4	sylibr 234	. . . . 5 ⊢ (𝑥 ∈ dom vol → 𝑥 ∈ 𝒫 ℝ)
6	5	ssriv 3953	. . . 4 ⊢ dom vol ⊆ 𝒫 ℝ
7	2, 6	sstrdi 3962	. . 3 ⊢ (𝜑 → ran 𝐹 ⊆ 𝒫 ℝ)
8		sspwuni 5067	. . 3 ⊢ (ran 𝐹 ⊆ 𝒫 ℝ ↔ ∪ ran 𝐹 ⊆ ℝ)
9	7, 8	sylib 218	. 2 ⊢ (𝜑 → ∪ ran 𝐹 ⊆ ℝ)
10		elpwi 4573	. . . 4 ⊢ (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
11		inundif 4445	. . . . . . . 8 ⊢ ((𝑥 ∩ ∪ ran 𝐹) ∪ (𝑥 ∖ ∪ ran 𝐹)) = 𝑥
12	11	fveq2i 6864	. . . . . . 7 ⊢ (vol*‘((𝑥 ∩ ∪ ran 𝐹) ∪ (𝑥 ∖ ∪ ran 𝐹))) = (vol*‘𝑥)
13		inss1 4203	. . . . . . . . 9 ⊢ (𝑥 ∩ ∪ ran 𝐹) ⊆ 𝑥
14		simp2 1137	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝑥 ⊆ ℝ)
15	13, 14	sstrid 3961	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ∩ ∪ ran 𝐹) ⊆ ℝ)
16		ovolsscl 25394	. . . . . . . . . 10 ⊢ (((𝑥 ∩ ∪ ran 𝐹) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ ∪ ran 𝐹)) ∈ ℝ)
17	13, 16	mp3an1 1450	. . . . . . . . 9 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ ∪ ran 𝐹)) ∈ ℝ)
18	17	3adant1 1130	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ ∪ ran 𝐹)) ∈ ℝ)
19		difss 4102	. . . . . . . . 9 ⊢ (𝑥 ∖ ∪ ran 𝐹) ⊆ 𝑥
20	19, 14	sstrid 3961	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ∖ ∪ ran 𝐹) ⊆ ℝ)
21		ovolsscl 25394	. . . . . . . . . 10 ⊢ (((𝑥 ∖ ∪ ran 𝐹) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ ∪ ran 𝐹)) ∈ ℝ)
22	19, 21	mp3an1 1450	. . . . . . . . 9 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ ∪ ran 𝐹)) ∈ ℝ)
23	22	3adant1 1130	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ ∪ ran 𝐹)) ∈ ℝ)
24		ovolun 25407	. . . . . . . 8 ⊢ ((((𝑥 ∩ ∪ ran 𝐹) ⊆ ℝ ∧ (vol*‘(𝑥 ∩ ∪ ran 𝐹)) ∈ ℝ) ∧ ((𝑥 ∖ ∪ ran 𝐹) ⊆ ℝ ∧ (vol*‘(𝑥 ∖ ∪ ran 𝐹)) ∈ ℝ)) → (vol*‘((𝑥 ∩ ∪ ran 𝐹) ∪ (𝑥 ∖ ∪ ran 𝐹))) ≤ ((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))))
25	15, 18, 20, 23, 24	syl22anc 838	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘((𝑥 ∩ ∪ ran 𝐹) ∪ (𝑥 ∖ ∪ ran 𝐹))) ≤ ((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))))
26	12, 25	eqbrtrrid 5146	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) ≤ ((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))))
27	18	rexrd 11231	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ ∪ ran 𝐹)) ∈ ℝ*)
28		nnuz 12843	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ≥‘1)
29		1zzd 12571	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 1 ∈ ℤ)
30		fveq2 6861	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
31	30	ineq2d 4186	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑘 → (𝑥 ∩ (𝐹‘𝑛)) = (𝑥 ∩ (𝐹‘𝑘)))
32	31	fveq2d 6865	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑘 → (vol*‘(𝑥 ∩ (𝐹‘𝑛))) = (vol*‘(𝑥 ∩ (𝐹‘𝑘))))
33		voliunlem.6	. . . . . . . . . . . . . . 15 ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝑥 ∩ (𝐹‘𝑛))))
34		fvex 6874	. . . . . . . . . . . . . . 15 ⊢ (vol*‘(𝑥 ∩ (𝐹‘𝑘))) ∈ V
35	32, 33, 34	fvmpt 6971	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → (𝐻‘𝑘) = (vol*‘(𝑥 ∩ (𝐹‘𝑘))))
36	35	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (𝐻‘𝑘) = (vol*‘(𝑥 ∩ (𝐹‘𝑘))))
37		inss1 4203	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∩ (𝐹‘𝑘)) ⊆ 𝑥
38		ovolsscl 25394	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∩ (𝐹‘𝑘)) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹‘𝑘))) ∈ ℝ)
39	37, 38	mp3an1 1450	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹‘𝑘))) ∈ ℝ)
40	39	3adant1 1130	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹‘𝑘))) ∈ ℝ)
41	40	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (vol*‘(𝑥 ∩ (𝐹‘𝑘))) ∈ ℝ)
42	36, 41	eqeltrd 2829	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (𝐻‘𝑘) ∈ ℝ)
43	28, 29, 42	serfre 14003	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → seq1( + , 𝐻):ℕ⟶ℝ)
44	43	frnd 6699	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ran seq1( + , 𝐻) ⊆ ℝ)
45		ressxr 11225	. . . . . . . . . 10 ⊢ ℝ ⊆ ℝ*
46	44, 45	sstrdi 3962	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ran seq1( + , 𝐻) ⊆ ℝ*)
47		supxrcl 13282	. . . . . . . . 9 ⊢ (ran seq1( + , 𝐻) ⊆ ℝ* → sup(ran seq1( + , 𝐻), ℝ*, < ) ∈ ℝ*)
48	46, 47	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → sup(ran seq1( + , 𝐻), ℝ*, < ) ∈ ℝ*)
49		simp3 1138	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) ∈ ℝ)
50	49, 23	resubcld 11613	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ∈ ℝ)
51	50	rexrd 11231	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ∈ ℝ*)
52		iunin2 5038	. . . . . . . . . . 11 ⊢ ∪ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹‘𝑛)) = (𝑥 ∩ ∪ 𝑛 ∈ ℕ (𝐹‘𝑛))
53		ffn 6691	. . . . . . . . . . . . . 14 ⊢ (𝐹:ℕ⟶dom vol → 𝐹 Fn ℕ)
54		fniunfv 7224	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn ℕ → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) = ∪ ran 𝐹)
55	1, 53, 54	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) = ∪ ran 𝐹)
56	55	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) = ∪ ran 𝐹)
57	56	ineq2d 4186	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ∩ ∪ 𝑛 ∈ ℕ (𝐹‘𝑛)) = (𝑥 ∩ ∪ ran 𝐹))
58	52, 57	eqtrid 2777	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ∪ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹‘𝑛)) = (𝑥 ∩ ∪ ran 𝐹))
59	58	fveq2d 6865	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘∪ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹‘𝑛))) = (vol*‘(𝑥 ∩ ∪ ran 𝐹)))
60		eqid 2730	. . . . . . . . . 10 ⊢ seq1( + , 𝐻) = seq1( + , 𝐻)
61		inss1 4203	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ (𝐹‘𝑛)) ⊆ 𝑥
62	61, 14	sstrid 3961	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ∩ (𝐹‘𝑛)) ⊆ ℝ)
63	62	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (𝑥 ∩ (𝐹‘𝑛)) ⊆ ℝ)
64		ovolsscl 25394	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∩ (𝐹‘𝑛)) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹‘𝑛))) ∈ ℝ)
65	61, 64	mp3an1 1450	. . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹‘𝑛))) ∈ ℝ)
66	65	3adant1 1130	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹‘𝑛))) ∈ ℝ)
67	66	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol*‘(𝑥 ∩ (𝐹‘𝑛))) ∈ ℝ)
68	60, 33, 63, 67	ovoliun 25413	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘∪ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹‘𝑛))) ≤ sup(ran seq1( + , 𝐻), ℝ*, < ))
69	59, 68	eqbrtrrd 5134	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ ∪ ran 𝐹)) ≤ sup(ran seq1( + , 𝐻), ℝ*, < ))
70	1	3ad2ant1 1133	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝐹:ℕ⟶dom vol)
71		voliunlem.5	. . . . . . . . . . . . . 14 ⊢ (𝜑 → Disj 𝑖 ∈ ℕ (𝐹‘𝑖))
72	71	3ad2ant1 1133	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → Disj 𝑖 ∈ ℕ (𝐹‘𝑖))
73	70, 72, 33, 14, 49	voliunlem1 25458	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ≤ (vol*‘𝑥))
74	43	ffvelcdmda 7059	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘𝑘) ∈ ℝ)
75	23	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (vol*‘(𝑥 ∖ ∪ ran 𝐹)) ∈ ℝ)
76		simpl3 1194	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (vol*‘𝑥) ∈ ℝ)
77		leaddsub 11661	. . . . . . . . . . . . 13 ⊢ (((seq1( + , 𝐻)‘𝑘) ∈ ℝ ∧ (vol*‘(𝑥 ∖ ∪ ran 𝐹)) ∈ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ≤ (vol*‘𝑥) ↔ (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹)))))
78	74, 75, 76, 77	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ≤ (vol*‘𝑥) ↔ (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹)))))
79	73, 78	mpbid 232	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹))))
80	79	ralrimiva 3126	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ∀𝑘 ∈ ℕ (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹))))
81		ffn 6691	. . . . . . . . . . 11 ⊢ (seq1( + , 𝐻):ℕ⟶ℝ → seq1( + , 𝐻) Fn ℕ)
82		breq1 5113	. . . . . . . . . . . 12 ⊢ (𝑧 = (seq1( + , 𝐻)‘𝑘) → (𝑧 ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ↔ (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹)))))
83	82	ralrn 7063	. . . . . . . . . . 11 ⊢ (seq1( + , 𝐻) Fn ℕ → (∀𝑧 ∈ ran seq1( + , 𝐻)𝑧 ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ↔ ∀𝑘 ∈ ℕ (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹)))))
84	43, 81, 83	3syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (∀𝑧 ∈ ran seq1( + , 𝐻)𝑧 ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ↔ ∀𝑘 ∈ ℕ (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹)))))
85	80, 84	mpbird 257	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ∀𝑧 ∈ ran seq1( + , 𝐻)𝑧 ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹))))
86		supxrleub 13293	. . . . . . . . . 10 ⊢ ((ran seq1( + , 𝐻) ⊆ ℝ* ∧ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ∈ ℝ*) → (sup(ran seq1( + , 𝐻), ℝ*, < ) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ↔ ∀𝑧 ∈ ran seq1( + , 𝐻)𝑧 ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹)))))
87	46, 51, 86	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (sup(ran seq1( + , 𝐻), ℝ*, < ) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ↔ ∀𝑧 ∈ ran seq1( + , 𝐻)𝑧 ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹)))))
88	85, 87	mpbird 257	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → sup(ran seq1( + , 𝐻), ℝ*, < ) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹))))
89	27, 48, 51, 69, 88	xrletrd 13129	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ ∪ ran 𝐹)) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹))))
90		leaddsub 11661	. . . . . . . 8 ⊢ (((vol*‘(𝑥 ∩ ∪ ran 𝐹)) ∈ ℝ ∧ (vol*‘(𝑥 ∖ ∪ ran 𝐹)) ∈ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ≤ (vol*‘𝑥) ↔ (vol*‘(𝑥 ∩ ∪ ran 𝐹)) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹)))))
91	18, 23, 49, 90	syl3anc 1373	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ≤ (vol*‘𝑥) ↔ (vol*‘(𝑥 ∩ ∪ ran 𝐹)) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹)))))
92	89, 91	mpbird 257	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ≤ (vol*‘𝑥))
93	18, 23	readdcld 11210	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ∈ ℝ)
94	49, 93	letri3d 11323	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘𝑥) = ((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ↔ ((vol*‘𝑥) ≤ ((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ∧ ((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ≤ (vol*‘𝑥))))
95	26, 92, 94	mpbir2and 713	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))))
96	95	3expia 1121	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ) → ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹)))))
97	10, 96	sylan2 593	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 ℝ) → ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹)))))
98	97	ralrimiva 3126	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹)))))
99		ismbl 25434	. 2 ⊢ (∪ ran 𝐹 ∈ dom vol ↔ (∪ ran 𝐹 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))))))
100	9, 98, 99	sylanbrc 583	1 ⊢ (𝜑 → ∪ ran 𝐹 ∈ dom vol)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3045   ∖ cdif 3914   ∪ cun 3915   ∩ cin 3916   ⊆ wss 3917  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  1c1 11076   + caddc 11078  ℝ^*cxr 11214   < clt 11215   ≤ cle 11216   − cmin 11412  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-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  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-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-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-ovol 25372  df-vol 25373

This theorem is referenced by:  voliunlem3  25460  iunmbl  25461