Theorem voliunlem2 25534

Description: Lemma for voliun 25537. (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 6674	. . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ dom vol)
3		mblss 25514	. . . . . 6 ⊢ (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ)
4		velpw 4547	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 ℝ ↔ 𝑥 ⊆ ℝ)
5	3, 4	sylibr 234	. . . . 5 ⊢ (𝑥 ∈ dom vol → 𝑥 ∈ 𝒫 ℝ)
6	5	ssriv 3926	. . . 4 ⊢ dom vol ⊆ 𝒫 ℝ
7	2, 6	sstrdi 3935	. . 3 ⊢ (𝜑 → ran 𝐹 ⊆ 𝒫 ℝ)
8		sspwuni 5043	. . 3 ⊢ (ran 𝐹 ⊆ 𝒫 ℝ ↔ ∪ ran 𝐹 ⊆ ℝ)
9	7, 8	sylib 218	. 2 ⊢ (𝜑 → ∪ ran 𝐹 ⊆ ℝ)
10		elpwi 4549	. . . 4 ⊢ (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
11		inundif 4420	. . . . . . . 8 ⊢ ((𝑥 ∩ ∪ ran 𝐹) ∪ (𝑥 ∖ ∪ ran 𝐹)) = 𝑥
12	11	fveq2i 6841	. . . . . . 7 ⊢ (vol*‘((𝑥 ∩ ∪ ran 𝐹) ∪ (𝑥 ∖ ∪ ran 𝐹))) = (vol*‘𝑥)
13		inss1 4178	. . . . . . . . 9 ⊢ (𝑥 ∩ ∪ ran 𝐹) ⊆ 𝑥
14		simp2 1138	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝑥 ⊆ ℝ)
15	13, 14	sstrid 3934	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ∩ ∪ ran 𝐹) ⊆ ℝ)
16		ovolsscl 25469	. . . . . . . . . 10 ⊢ (((𝑥 ∩ ∪ ran 𝐹) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ ∪ ran 𝐹)) ∈ ℝ)
17	13, 16	mp3an1 1451	. . . . . . . . 9 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ ∪ ran 𝐹)) ∈ ℝ)
18	17	3adant1 1131	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ ∪ ran 𝐹)) ∈ ℝ)
19		difss 4077	. . . . . . . . 9 ⊢ (𝑥 ∖ ∪ ran 𝐹) ⊆ 𝑥
20	19, 14	sstrid 3934	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ∖ ∪ ran 𝐹) ⊆ ℝ)
21		ovolsscl 25469	. . . . . . . . . 10 ⊢ (((𝑥 ∖ ∪ ran 𝐹) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ ∪ ran 𝐹)) ∈ ℝ)
22	19, 21	mp3an1 1451	. . . . . . . . 9 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ ∪ ran 𝐹)) ∈ ℝ)
23	22	3adant1 1131	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ ∪ ran 𝐹)) ∈ ℝ)
24		ovolun 25482	. . . . . . . 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 839	. . . . . . 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 5122	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) ≤ ((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))))
27	18	rexrd 11192	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ ∪ ran 𝐹)) ∈ ℝ*)
28		nnuz 12824	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ≥‘1)
29		1zzd 12555	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 1 ∈ ℤ)
30		fveq2 6838	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
31	30	ineq2d 4161	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑘 → (𝑥 ∩ (𝐹‘𝑛)) = (𝑥 ∩ (𝐹‘𝑘)))
32	31	fveq2d 6842	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑘 → (vol*‘(𝑥 ∩ (𝐹‘𝑛))) = (vol*‘(𝑥 ∩ (𝐹‘𝑘))))
33		voliunlem.6	. . . . . . . . . . . . . . 15 ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝑥 ∩ (𝐹‘𝑛))))
34		fvex 6851	. . . . . . . . . . . . . . 15 ⊢ (vol*‘(𝑥 ∩ (𝐹‘𝑘))) ∈ V
35	32, 33, 34	fvmpt 6945	. . . . . . . . . . . . . 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 4178	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∩ (𝐹‘𝑘)) ⊆ 𝑥
38		ovolsscl 25469	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∩ (𝐹‘𝑘)) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹‘𝑘))) ∈ ℝ)
39	37, 38	mp3an1 1451	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹‘𝑘))) ∈ ℝ)
40	39	3adant1 1131	. . . . . . . . . . . . . 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 2837	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (𝐻‘𝑘) ∈ ℝ)
43	28, 29, 42	serfre 13990	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → seq1( + , 𝐻):ℕ⟶ℝ)
44	43	frnd 6674	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ran seq1( + , 𝐻) ⊆ ℝ)
45		ressxr 11186	. . . . . . . . . 10 ⊢ ℝ ⊆ ℝ*
46	44, 45	sstrdi 3935	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ran seq1( + , 𝐻) ⊆ ℝ*)
47		supxrcl 13264	. . . . . . . . 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 1139	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) ∈ ℝ)
50	49, 23	resubcld 11575	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ∈ ℝ)
51	50	rexrd 11192	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ∈ ℝ*)
52		iunin2 5014	. . . . . . . . . . 11 ⊢ ∪ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹‘𝑛)) = (𝑥 ∩ ∪ 𝑛 ∈ ℕ (𝐹‘𝑛))
53		ffn 6666	. . . . . . . . . . . . . 14 ⊢ (𝐹:ℕ⟶dom vol → 𝐹 Fn ℕ)
54		fniunfv 7199	. . . . . . . . . . . . . 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 1134	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) = ∪ ran 𝐹)
57	56	ineq2d 4161	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ∩ ∪ 𝑛 ∈ ℕ (𝐹‘𝑛)) = (𝑥 ∩ ∪ ran 𝐹))
58	52, 57	eqtrid 2784	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ∪ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹‘𝑛)) = (𝑥 ∩ ∪ ran 𝐹))
59	58	fveq2d 6842	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘∪ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹‘𝑛))) = (vol*‘(𝑥 ∩ ∪ ran 𝐹)))
60		eqid 2737	. . . . . . . . . 10 ⊢ seq1( + , 𝐻) = seq1( + , 𝐻)
61		inss1 4178	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ (𝐹‘𝑛)) ⊆ 𝑥
62	61, 14	sstrid 3934	. . . . . . . . . . 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 25469	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∩ (𝐹‘𝑛)) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹‘𝑛))) ∈ ℝ)
65	61, 64	mp3an1 1451	. . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹‘𝑛))) ∈ ℝ)
66	65	3adant1 1131	. . . . . . . . . . 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 25488	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘∪ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹‘𝑛))) ≤ sup(ran seq1( + , 𝐻), ℝ*, < ))
69	59, 68	eqbrtrrd 5110	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ ∪ ran 𝐹)) ≤ sup(ran seq1( + , 𝐻), ℝ*, < ))
70	1	3ad2ant1 1134	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝐹:ℕ⟶dom vol)
71		voliunlem.5	. . . . . . . . . . . . . 14 ⊢ (𝜑 → Disj 𝑖 ∈ ℕ (𝐹‘𝑖))
72	71	3ad2ant1 1134	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → Disj 𝑖 ∈ ℕ (𝐹‘𝑖))
73	70, 72, 33, 14, 49	voliunlem1 25533	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ≤ (vol*‘𝑥))
74	43	ffvelcdmda 7034	. . . . . . . . . . . . 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 1195	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (vol*‘𝑥) ∈ ℝ)
77		leaddsub 11623	. . . . . . . . . . . . 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 1374	. . . . . . . . . . . 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 3130	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ∀𝑘 ∈ ℕ (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹))))
81		ffn 6666	. . . . . . . . . . 11 ⊢ (seq1( + , 𝐻):ℕ⟶ℝ → seq1( + , 𝐻) Fn ℕ)
82		breq1 5089	. . . . . . . . . . . 12 ⊢ (𝑧 = (seq1( + , 𝐻)‘𝑘) → (𝑧 ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ↔ (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹)))))
83	82	ralrn 7038	. . . . . . . . . . 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 13275	. . . . . . . . . 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 585	. . . . . . . . 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 13110	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ ∪ ran 𝐹)) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹))))
90		leaddsub 11623	. . . . . . . 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 1374	. . . . . . 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 11171	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ∈ ℝ)
94	49, 93	letri3d 11285	. . . . . 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 714	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))))
96	95	3expia 1122	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ) → ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹)))))
97	10, 96	sylan2 594	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 ℝ) → ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹)))))
98	97	ralrimiva 3130	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹)))))
99		ismbl 25509	. 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 584	1 ⊢ (𝜑 → ∪ ran 𝐹 ∈ dom vol)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ∖ cdif 3887   ∪ cun 3888   ∩ cin 3889   ⊆ wss 3890  𝒫 cpw 4542  ∪ cuni 4851  ∪ ciun 4934  Disj wdisj 5053   class class class wbr 5086   ↦ cmpt 5167  dom cdm 5628  ran crn 5629   Fn wfn 6491  ⟶wf 6492  ‘cfv 6496  (class class class)co 7364  supcsup 9350  ℝcr 11034  1c1 11036   + caddc 11038  ℝ^*cxr 11175   < clt 11176   ≤ cle 11177   − cmin 11374  ℕcn 12171  seqcseq 13960  vol*covol 25445  volcvol 25446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5306  ax-pr 5374  ax-un 7686  ax-inf2 9559  ax-cc 10354  ax-cnex 11091  ax-resscn 11092  ax-1cn 11093  ax-icn 11094  ax-addcl 11095  ax-addrcl 11096  ax-mulcl 11097  ax-mulrcl 11098  ax-mulcom 11099  ax-addass 11100  ax-mulass 11101  ax-distr 11102  ax-i2m1 11103  ax-1ne0 11104  ax-1rid 11105  ax-rnegex 11106  ax-rrecex 11107  ax-cnre 11108  ax-pre-lttri 11109  ax-pre-lttrn 11110  ax-pre-ltadd 11111  ax-pre-mulgt0 11112  ax-pre-sup 11113

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-disj 5054  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5523  df-eprel 5528  df-po 5536  df-so 5537  df-fr 5581  df-se 5582  df-we 5583  df-xp 5634  df-rel 5635  df-cnv 5636  df-co 5637  df-dm 5638  df-rn 5639  df-res 5640  df-ima 5641  df-pred 6263  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7321  df-ov 7367  df-oprab 7368  df-mpo 7369  df-om 7815  df-1st 7939  df-2nd 7940  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-er 8640  df-map 8772  df-pm 8773  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-sup 9352  df-inf 9353  df-oi 9422  df-card 9860  df-pnf 11178  df-mnf 11179  df-xr 11180  df-ltxr 11181  df-le 11182  df-sub 11376  df-neg 11377  df-div 11805  df-nn 12172  df-2 12241  df-3 12242  df-n0 12435  df-z 12522  df-uz 12786  df-q 12896  df-rp 12940  df-ioo 13299  df-ico 13301  df-icc 13302  df-fz 13459  df-fzo 13606  df-fl 13748  df-seq 13961  df-exp 14021  df-hash 14290  df-cj 15058  df-re 15059  df-im 15060  df-sqrt 15194  df-abs 15195  df-clim 15447  df-rlim 15448  df-sum 15646  df-ovol 25447  df-vol 25448

This theorem is referenced by:  voliunlem3  25535  iunmbl  25536