Theorem voliunlem2 25450

Description: Lemma for voliun 25453. (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 6660	. . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ dom vol)
3		mblss 25430	. . . . . 6 ⊢ (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ)
4		velpw 4556	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 ℝ ↔ 𝑥 ⊆ ℝ)
5	3, 4	sylibr 234	. . . . 5 ⊢ (𝑥 ∈ dom vol → 𝑥 ∈ 𝒫 ℝ)
6	5	ssriv 3939	. . . 4 ⊢ dom vol ⊆ 𝒫 ℝ
7	2, 6	sstrdi 3948	. . 3 ⊢ (𝜑 → ran 𝐹 ⊆ 𝒫 ℝ)
8		sspwuni 5049	. . 3 ⊢ (ran 𝐹 ⊆ 𝒫 ℝ ↔ ∪ ran 𝐹 ⊆ ℝ)
9	7, 8	sylib 218	. 2 ⊢ (𝜑 → ∪ ran 𝐹 ⊆ ℝ)
10		elpwi 4558	. . . 4 ⊢ (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
11		inundif 4430	. . . . . . . 8 ⊢ ((𝑥 ∩ ∪ ran 𝐹) ∪ (𝑥 ∖ ∪ ran 𝐹)) = 𝑥
12	11	fveq2i 6825	. . . . . . 7 ⊢ (vol*‘((𝑥 ∩ ∪ ran 𝐹) ∪ (𝑥 ∖ ∪ ran 𝐹))) = (vol*‘𝑥)
13		inss1 4188	. . . . . . . . 9 ⊢ (𝑥 ∩ ∪ ran 𝐹) ⊆ 𝑥
14		simp2 1137	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝑥 ⊆ ℝ)
15	13, 14	sstrid 3947	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ∩ ∪ ran 𝐹) ⊆ ℝ)
16		ovolsscl 25385	. . . . . . . . . 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 4087	. . . . . . . . 9 ⊢ (𝑥 ∖ ∪ ran 𝐹) ⊆ 𝑥
20	19, 14	sstrid 3947	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ∖ ∪ ran 𝐹) ⊆ ℝ)
21		ovolsscl 25385	. . . . . . . . . 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 25398	. . . . . . . 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 5128	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) ≤ ((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))))
27	18	rexrd 11165	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ ∪ ran 𝐹)) ∈ ℝ*)
28		nnuz 12778	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ≥‘1)
29		1zzd 12506	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 1 ∈ ℤ)
30		fveq2 6822	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
31	30	ineq2d 4171	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑘 → (𝑥 ∩ (𝐹‘𝑛)) = (𝑥 ∩ (𝐹‘𝑘)))
32	31	fveq2d 6826	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑘 → (vol*‘(𝑥 ∩ (𝐹‘𝑛))) = (vol*‘(𝑥 ∩ (𝐹‘𝑘))))
33		voliunlem.6	. . . . . . . . . . . . . . 15 ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝑥 ∩ (𝐹‘𝑛))))
34		fvex 6835	. . . . . . . . . . . . . . 15 ⊢ (vol*‘(𝑥 ∩ (𝐹‘𝑘))) ∈ V
35	32, 33, 34	fvmpt 6930	. . . . . . . . . . . . . 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 4188	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∩ (𝐹‘𝑘)) ⊆ 𝑥
38		ovolsscl 25385	. . . . . . . . . . . . . . . 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 2828	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (𝐻‘𝑘) ∈ ℝ)
43	28, 29, 42	serfre 13938	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → seq1( + , 𝐻):ℕ⟶ℝ)
44	43	frnd 6660	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ran seq1( + , 𝐻) ⊆ ℝ)
45		ressxr 11159	. . . . . . . . . 10 ⊢ ℝ ⊆ ℝ*
46	44, 45	sstrdi 3948	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ran seq1( + , 𝐻) ⊆ ℝ*)
47		supxrcl 13217	. . . . . . . . 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 11548	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ∈ ℝ)
51	50	rexrd 11165	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ∈ ℝ*)
52		iunin2 5020	. . . . . . . . . . 11 ⊢ ∪ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹‘𝑛)) = (𝑥 ∩ ∪ 𝑛 ∈ ℕ (𝐹‘𝑛))
53		ffn 6652	. . . . . . . . . . . . . 14 ⊢ (𝐹:ℕ⟶dom vol → 𝐹 Fn ℕ)
54		fniunfv 7183	. . . . . . . . . . . . . 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 4171	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ∩ ∪ 𝑛 ∈ ℕ (𝐹‘𝑛)) = (𝑥 ∩ ∪ ran 𝐹))
58	52, 57	eqtrid 2776	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ∪ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹‘𝑛)) = (𝑥 ∩ ∪ ran 𝐹))
59	58	fveq2d 6826	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘∪ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹‘𝑛))) = (vol*‘(𝑥 ∩ ∪ ran 𝐹)))
60		eqid 2729	. . . . . . . . . 10 ⊢ seq1( + , 𝐻) = seq1( + , 𝐻)
61		inss1 4188	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ (𝐹‘𝑛)) ⊆ 𝑥
62	61, 14	sstrid 3947	. . . . . . . . . . 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 25385	. . . . . . . . . . . . 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 25404	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘∪ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹‘𝑛))) ≤ sup(ran seq1( + , 𝐻), ℝ*, < ))
69	59, 68	eqbrtrrd 5116	. . . . . . . 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 25449	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ≤ (vol*‘𝑥))
74	43	ffvelcdmda 7018	. . . . . . . . . . . . 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 11596	. . . . . . . . . . . . 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 3121	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ∀𝑘 ∈ ℕ (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹))))
81		ffn 6652	. . . . . . . . . . 11 ⊢ (seq1( + , 𝐻):ℕ⟶ℝ → seq1( + , 𝐻) Fn ℕ)
82		breq1 5095	. . . . . . . . . . . 12 ⊢ (𝑧 = (seq1( + , 𝐻)‘𝑘) → (𝑧 ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ↔ (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹)))))
83	82	ralrn 7022	. . . . . . . . . . 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 13228	. . . . . . . . . 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 13064	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ ∪ ran 𝐹)) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹))))
90		leaddsub 11596	. . . . . . . 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 11144	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ∈ ℝ)
94	49, 93	letri3d 11258	. . . . . 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 3121	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹)))))
99		ismbl 25425	. 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 3044   ∖ cdif 3900   ∪ cun 3901   ∩ cin 3902   ⊆ wss 3903  𝒫 cpw 4551  ∪ cuni 4858  ∪ ciun 4941  Disj wdisj 5059   class class class wbr 5092   ↦ cmpt 5173  dom cdm 5619  ran crn 5620   Fn wfn 6477  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349  supcsup 9330  ℝcr 11008  1c1 11010   + caddc 11012  ℝ^*cxr 11148   < clt 11149   ≤ cle 11150   − cmin 11347  ℕcn 12128  seqcseq 13908  vol*covol 25361  volcvol 25362

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 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-inf2 9537  ax-cc 10329  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-disj 5060  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-er 8625  df-map 8755  df-pm 8756  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-sup 9332  df-inf 9333  df-oi 9402  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-3 12192  df-n0 12385  df-z 12472  df-uz 12736  df-q 12850  df-rp 12894  df-ioo 13252  df-ico 13254  df-icc 13255  df-fz 13411  df-fzo 13558  df-fl 13696  df-seq 13909  df-exp 13969  df-hash 14238  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-clim 15395  df-rlim 15396  df-sum 15594  df-ovol 25363  df-vol 25364

This theorem is referenced by:  voliunlem3  25451  iunmbl  25452