Theorem voliunlem2 25615

Description: Lemma for voliun 25618. (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 6702	. . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ dom vol)
3		mblss 25595	. . . . . 6 ⊢ (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ)
4		velpw 4562	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 ℝ ↔ 𝑥 ⊆ ℝ)
5	3, 4	sylibr 236	. . . . 5 ⊢ (𝑥 ∈ dom vol → 𝑥 ∈ 𝒫 ℝ)
6	5	ssriv 3942	. . . 4 ⊢ dom vol ⊆ 𝒫 ℝ
7	2, 6	sstrdi 3950	. . 3 ⊢ (𝜑 → ran 𝐹 ⊆ 𝒫 ℝ)
8		sspwuni 5059	. . 3 ⊢ (ran 𝐹 ⊆ 𝒫 ℝ ↔ ∪ ran 𝐹 ⊆ ℝ)
9	7, 8	sylib 220	. 2 ⊢ (𝜑 → ∪ ran 𝐹 ⊆ ℝ)
10		elpwi 4564	. . . 4 ⊢ (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
11		inundif 4435	. . . . . . . 8 ⊢ ((𝑥 ∩ ∪ ran 𝐹) ∪ (𝑥 ∖ ∪ ran 𝐹)) = 𝑥
12	11	fveq2i 6872	. . . . . . 7 ⊢ (vol*‘((𝑥 ∩ ∪ ran 𝐹) ∪ (𝑥 ∖ ∪ ran 𝐹))) = (vol*‘𝑥)
13		inss1 4190	. . . . . . . . 9 ⊢ (𝑥 ∩ ∪ ran 𝐹) ⊆ 𝑥
14		simp2 1151	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝑥 ⊆ ℝ)
15	13, 14	sstrid 3949	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ∩ ∪ ran 𝐹) ⊆ ℝ)
16		ovolsscl 25550	. . . . . . . . . 10 ⊢ (((𝑥 ∩ ∪ ran 𝐹) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ ∪ ran 𝐹)) ∈ ℝ)
17	13, 16	mp3an1 1471	. . . . . . . . 9 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ ∪ ran 𝐹)) ∈ ℝ)
18	17	3adant1 1144	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ ∪ ran 𝐹)) ∈ ℝ)
19		difss 4091	. . . . . . . . 9 ⊢ (𝑥 ∖ ∪ ran 𝐹) ⊆ 𝑥
20	19, 14	sstrid 3949	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ∖ ∪ ran 𝐹) ⊆ ℝ)
21		ovolsscl 25550	. . . . . . . . . 10 ⊢ (((𝑥 ∖ ∪ ran 𝐹) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ ∪ ran 𝐹)) ∈ ℝ)
22	19, 21	mp3an1 1471	. . . . . . . . 9 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ ∪ ran 𝐹)) ∈ ℝ)
23	22	3adant1 1144	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ ∪ ran 𝐹)) ∈ ℝ)
24		ovolun 25563	. . . . . . . 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 849	. . . . . . 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 5138	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) ≤ ((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))))
27	18	rexrd 11234	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ ∪ ran 𝐹)) ∈ ℝ*)
28		nnuz 12880	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ≥‘1)
29		1zzd 12604	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 1 ∈ ℤ)
30		fveq2 6869	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
31	30	ineq2d 4174	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑘 → (𝑥 ∩ (𝐹‘𝑛)) = (𝑥 ∩ (𝐹‘𝑘)))
32	31	fveq2d 6873	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑘 → (vol*‘(𝑥 ∩ (𝐹‘𝑛))) = (vol*‘(𝑥 ∩ (𝐹‘𝑘))))
33		voliunlem.6	. . . . . . . . . . . . . . 15 ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝑥 ∩ (𝐹‘𝑛))))
34		fvex 6882	. . . . . . . . . . . . . . 15 ⊢ (vol*‘(𝑥 ∩ (𝐹‘𝑘))) ∈ V
35	32, 33, 34	fvmpt 6977	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → (𝐻‘𝑘) = (vol*‘(𝑥 ∩ (𝐹‘𝑘))))
36	35	adantl 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (𝐻‘𝑘) = (vol*‘(𝑥 ∩ (𝐹‘𝑘))))
37		inss1 4190	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∩ (𝐹‘𝑘)) ⊆ 𝑥
38		ovolsscl 25550	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∩ (𝐹‘𝑘)) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹‘𝑘))) ∈ ℝ)
39	37, 38	mp3an1 1471	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹‘𝑘))) ∈ ℝ)
40	39	3adant1 1144	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹‘𝑘))) ∈ ℝ)
41	40	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (vol*‘(𝑥 ∩ (𝐹‘𝑘))) ∈ ℝ)
42	36, 41	eqeltrd 2864	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (𝐻‘𝑘) ∈ ℝ)
43	28, 29, 42	serfre 14046	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → seq1( + , 𝐻):ℕ⟶ℝ)
44	43	frnd 6702	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ran seq1( + , 𝐻) ⊆ ℝ)
45		ressxr 11228	. . . . . . . . . 10 ⊢ ℝ ⊆ ℝ*
46	44, 45	sstrdi 3950	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ran seq1( + , 𝐻) ⊆ ℝ*)
47		supxrcl 13320	. . . . . . . . 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 1152	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) ∈ ℝ)
50	49, 23	resubcld 11617	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ∈ ℝ)
51	50	rexrd 11234	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ∈ ℝ*)
52		iunin2 5030	. . . . . . . . . . 11 ⊢ ∪ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹‘𝑛)) = (𝑥 ∩ ∪ 𝑛 ∈ ℕ (𝐹‘𝑛))
53		ffn 6693	. . . . . . . . . . . . . 14 ⊢ (𝐹:ℕ⟶dom vol → 𝐹 Fn ℕ)
54		fniunfv 7233	. . . . . . . . . . . . . 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 1147	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) = ∪ ran 𝐹)
57	56	ineq2d 4174	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ∩ ∪ 𝑛 ∈ ℕ (𝐹‘𝑛)) = (𝑥 ∩ ∪ ran 𝐹))
58	52, 57	eqtrid 2811	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ∪ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹‘𝑛)) = (𝑥 ∩ ∪ ran 𝐹))
59	58	fveq2d 6873	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘∪ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹‘𝑛))) = (vol*‘(𝑥 ∩ ∪ ran 𝐹)))
60		eqid 2764	. . . . . . . . . 10 ⊢ seq1( + , 𝐻) = seq1( + , 𝐻)
61		inss1 4190	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ (𝐹‘𝑛)) ⊆ 𝑥
62	61, 14	sstrid 3949	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ∩ (𝐹‘𝑛)) ⊆ ℝ)
63	62	adantr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (𝑥 ∩ (𝐹‘𝑛)) ⊆ ℝ)
64		ovolsscl 25550	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∩ (𝐹‘𝑛)) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹‘𝑛))) ∈ ℝ)
65	61, 64	mp3an1 1471	. . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹‘𝑛))) ∈ ℝ)
66	65	3adant1 1144	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹‘𝑛))) ∈ ℝ)
67	66	adantr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol*‘(𝑥 ∩ (𝐹‘𝑛))) ∈ ℝ)
68	60, 33, 63, 67	ovoliun 25569	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘∪ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹‘𝑛))) ≤ sup(ran seq1( + , 𝐻), ℝ*, < ))
69	59, 68	eqbrtrrd 5126	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ ∪ ran 𝐹)) ≤ sup(ran seq1( + , 𝐻), ℝ*, < ))
70	1	3ad2ant1 1147	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝐹:ℕ⟶dom vol)
71		voliunlem.5	. . . . . . . . . . . . . 14 ⊢ (𝜑 → Disj 𝑖 ∈ ℕ (𝐹‘𝑖))
72	71	3ad2ant1 1147	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → Disj 𝑖 ∈ ℕ (𝐹‘𝑖))
73	70, 72, 33, 14, 49	voliunlem1 25614	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ≤ (vol*‘𝑥))
74	43	ffvelcdmda 7067	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘𝑘) ∈ ℝ)
75	23	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (vol*‘(𝑥 ∖ ∪ ran 𝐹)) ∈ ℝ)
76		simpl3 1208	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (vol*‘𝑥) ∈ ℝ)
77		leaddsub 11665	. . . . . . . . . . . . 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 1392	. . . . . . . . . . . 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 234	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹))))
80	79	ralrimiva 3156	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ∀𝑘 ∈ ℕ (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹))))
81		ffn 6693	. . . . . . . . . . 11 ⊢ (seq1( + , 𝐻):ℕ⟶ℝ → seq1( + , 𝐻) Fn ℕ)
82		breq1 5105	. . . . . . . . . . . 12 ⊢ (𝑧 = (seq1( + , 𝐻)‘𝑘) → (𝑧 ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ↔ (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹)))))
83	82	ralrn 7071	. . . . . . . . . . 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 259	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ∀𝑧 ∈ ran seq1( + , 𝐻)𝑧 ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹))))
86		supxrleub 13331	. . . . . . . . . 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 593	. . . . . . . . 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 259	. . . . . . . 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 13166	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ ∪ ran 𝐹)) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹))))
90		leaddsub 11665	. . . . . . . 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 1392	. . . . . . 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 259	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ≤ (vol*‘𝑥))
93	18, 23	readdcld 11213	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ∈ ℝ)
94	49, 93	letri3d 11327	. . . . . 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 723	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))))
96	95	3expia 1135	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ) → ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹)))))
97	10, 96	sylan2 602	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 ℝ) → ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹)))))
98	97	ralrimiva 3156	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹)))))
99		ismbl 25590	. 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 592	1 ⊢ (𝜑 → ∪ ran 𝐹 ∈ dom vol)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  ∀wral 3078   ∖ cdif 3903   ∪ cun 3904   ∩ cin 3905   ⊆ wss 3906  𝒫 cpw 4557  ∪ cuni 4867  ∪ ciun 4951  Disj wdisj 5069   class class class wbr 5102   ↦ cmpt 5183  dom cdm 5649  ran crn 5650   Fn wfn 6518  ⟶wf 6519  ‘cfv 6523  (class class class)co 7398  supcsup 9388  ℝcr 11074  1c1 11076   + caddc 11078  ℝ^*cxr 11217   < clt 11218   ≤ cle 11219   − cmin 11416  ℕcn 12212  seqcseq 14016  vol*covol 25526  volcvol 25527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-inf2 9598  ax-cc 10394  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 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  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 5070  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-se 5603  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-isom 6532  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-er 8680  df-map 8812  df-pm 8813  df-en 8930  df-dom 8931  df-sdom 8932  df-fin 8933  df-sup 9390  df-inf 9391  df-oi 9460  df-card 9899  df-pnf 11220  df-mnf 11221  df-xr 11222  df-ltxr 11223  df-le 11224  df-sub 11418  df-neg 11419  df-div 11847  df-nn 12213  df-2 12282  df-3 12283  df-n0 12484  df-z 12571  df-uz 12842  df-q 12952  df-rp 12996  df-ioo 13355  df-ico 13357  df-icc 13358  df-fz 13515  df-fzo 13662  df-fl 13804  df-seq 14017  df-exp 14077  df-hash 14346  df-cj 15128  df-re 15129  df-im 15130  df-sqrt 15264  df-abs 15265  df-clim 15517  df-rlim 15518  df-sum 15716  df-ovol 25528  df-vol 25529

This theorem is referenced by:  voliunlem3  25616  iunmbl  25617