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Theorem voliunlem2 25075

Description: Lemma for voliun 25078. (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 6725	. . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ dom vol)
3		mblss 25055	. . . . . 6 ⊢ (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ)
4		velpw 4607	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 ℝ ↔ 𝑥 ⊆ ℝ)
5	3, 4	sylibr 233	. . . . 5 ⊢ (𝑥 ∈ dom vol → 𝑥 ∈ 𝒫 ℝ)
6	5	ssriv 3986	. . . 4 ⊢ dom vol ⊆ 𝒫 ℝ
7	2, 6	sstrdi 3994	. . 3 ⊢ (𝜑 → ran 𝐹 ⊆ 𝒫 ℝ)
8		sspwuni 5103	. . 3 ⊢ (ran 𝐹 ⊆ 𝒫 ℝ ↔ ∪ ran 𝐹 ⊆ ℝ)
9	7, 8	sylib 217	. 2 ⊢ (𝜑 → ∪ ran 𝐹 ⊆ ℝ)
10		elpwi 4609	. . . 4 ⊢ (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
11		inundif 4478	. . . . . . . 8 ⊢ ((𝑥 ∩ ∪ ran 𝐹) ∪ (𝑥 ∖ ∪ ran 𝐹)) = 𝑥
12	11	fveq2i 6894	. . . . . . 7 ⊢ (vol‘((𝑥 ∩ ∪ ran 𝐹) ∪ (𝑥 ∖ ∪ ran 𝐹))) = (vol‘𝑥)
13		inss1 4228	. . . . . . . . 9 ⊢ (𝑥 ∩ ∪ ran 𝐹) ⊆ 𝑥
14		simp2 1137	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝑥 ⊆ ℝ)
15	13, 14	sstrid 3993	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ∩ ∪ ran 𝐹) ⊆ ℝ)
16		ovolsscl 25010	. . . . . . . . . 10 ⊢ (((𝑥 ∩ ∪ ran 𝐹) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) → (vol‘(𝑥 ∩ ∪ ran 𝐹)) ∈ ℝ)
17	13, 16	mp3an1 1448	. . . . . . . . 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 4131	. . . . . . . . 9 ⊢ (𝑥 ∖ ∪ ran 𝐹) ⊆ 𝑥
20	19, 14	sstrid 3993	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ∖ ∪ ran 𝐹) ⊆ ℝ)
21		ovolsscl 25010	. . . . . . . . . 10 ⊢ (((𝑥 ∖ ∪ ran 𝐹) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) → (vol‘(𝑥 ∖ ∪ ran 𝐹)) ∈ ℝ)
22	19, 21	mp3an1 1448	. . . . . . . . 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 25023	. . . . . . . 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 837	. . . . . . 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 5184	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) → (vol‘𝑥) ≤ ((vol‘(𝑥 ∩ ∪ ran 𝐹)) + (vol‘(𝑥 ∖ ∪ ran 𝐹))))
27	18	rexrd 11266	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) → (vol‘(𝑥 ∩ ∪ ran 𝐹)) ∈ ℝ^*)
28		nnuz 12867	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ_≥‘1)
29		1zzd 12595	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 1 ∈ ℤ)
30		fveq2 6891	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
31	30	ineq2d 4212	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑘 → (𝑥 ∩ (𝐹‘𝑛)) = (𝑥 ∩ (𝐹‘𝑘)))
32	31	fveq2d 6895	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑘 → (vol‘(𝑥 ∩ (𝐹‘𝑛))) = (vol‘(𝑥 ∩ (𝐹‘𝑘))))
33		voliunlem.6	. . . . . . . . . . . . . . 15 ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝑥 ∩ (𝐹‘𝑛))))
34		fvex 6904	. . . . . . . . . . . . . . 15 ⊢ (vol*‘(𝑥 ∩ (𝐹‘𝑘))) ∈ V
35	32, 33, 34	fvmpt 6998	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → (𝐻‘𝑘) = (vol*‘(𝑥 ∩ (𝐹‘𝑘))))
36	35	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (𝐻‘𝑘) = (vol‘(𝑥 ∩ (𝐹‘𝑘))))
37		inss1 4228	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∩ (𝐹‘𝑘)) ⊆ 𝑥
38		ovolsscl 25010	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∩ (𝐹‘𝑘)) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) → (vol‘(𝑥 ∩ (𝐹‘𝑘))) ∈ ℝ)
39	37, 38	mp3an1 1448	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (vol‘(𝑥 ∩ (𝐹‘𝑘))) ∈ ℝ)
42	36, 41	eqeltrd 2833	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (𝐻‘𝑘) ∈ ℝ)
43	28, 29, 42	serfre 13999	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → seq1( + , 𝐻):ℕ⟶ℝ)
44	43	frnd 6725	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ran seq1( + , 𝐻) ⊆ ℝ)
45		ressxr 11260	. . . . . . . . . 10 ⊢ ℝ ⊆ ℝ^*
46	44, 45	sstrdi 3994	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) → ran seq1( + , 𝐻) ⊆ ℝ^)
47		supxrcl 13296	. . . . . . . . 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 11644	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) → ((vol‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ∈ ℝ)
51	50	rexrd 11266	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) → ((vol‘𝑥) − (vol‘(𝑥 ∖ ∪ ran 𝐹))) ∈ ℝ^)
52		iunin2 5074	. . . . . . . . . . 11 ⊢ ∪ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹‘𝑛)) = (𝑥 ∩ ∪ 𝑛 ∈ ℕ (𝐹‘𝑛))
53		ffn 6717	. . . . . . . . . . . . . 14 ⊢ (𝐹:ℕ⟶dom vol → 𝐹 Fn ℕ)
54		fniunfv 7248	. . . . . . . . . . . . . 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 4212	. . . . . . . . . . 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 6895	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) → (vol‘∪ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹‘𝑛))) = (vol*‘(𝑥 ∩ ∪ ran 𝐹)))
60		eqid 2732	. . . . . . . . . 10 ⊢ seq1( + , 𝐻) = seq1( + , 𝐻)
61		inss1 4228	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ (𝐹‘𝑛)) ⊆ 𝑥
62	61, 14	sstrid 3993	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ∩ (𝐹‘𝑛)) ⊆ ℝ)
63	62	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (𝑥 ∩ (𝐹‘𝑛)) ⊆ ℝ)
64		ovolsscl 25010	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∩ (𝐹‘𝑛)) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) → (vol‘(𝑥 ∩ (𝐹‘𝑛))) ∈ ℝ)
65	61, 64	mp3an1 1448	. . . . . . . . . . . 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 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝑥 ∩ (𝐹‘𝑛))) ∈ ℝ)
68	60, 33, 63, 67	ovoliun 25029	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) → (vol‘∪ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹‘𝑛))) ≤ sup(ran seq1( + , 𝐻), ℝ^*, < ))
69	59, 68	eqbrtrrd 5172	. . . . . . . 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 25074	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (vol‘(𝑥 ∖ ∪ ran 𝐹))) ≤ (vol*‘𝑥))
74	43	ffvelcdmda 7086	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘𝑘) ∈ ℝ)
75	23	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (vol‘(𝑥 ∖ ∪ ran 𝐹)) ∈ ℝ)
76		simpl3 1193	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (vol‘𝑥) ∈ ℝ)
77		leaddsub 11692	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 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 231	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘𝑘) ≤ ((vol‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹))))
80	79	ralrimiva 3146	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) → ∀𝑘 ∈ ℕ (seq1( + , 𝐻)‘𝑘) ≤ ((vol‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹))))
81		ffn 6717	. . . . . . . . . . 11 ⊢ (seq1( + , 𝐻):ℕ⟶ℝ → seq1( + , 𝐻) Fn ℕ)
82		breq1 5151	. . . . . . . . . . . 12 ⊢ (𝑧 = (seq1( + , 𝐻)‘𝑘) → (𝑧 ≤ ((vol‘𝑥) − (vol‘(𝑥 ∖ ∪ ran 𝐹))) ↔ (seq1( + , 𝐻)‘𝑘) ≤ ((vol‘𝑥) − (vol‘(𝑥 ∖ ∪ ran 𝐹)))))
83	82	ralrn 7089	. . . . . . . . . . 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 256	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) → ∀𝑧 ∈ ran seq1( + , 𝐻)𝑧 ≤ ((vol‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹))))
86		supxrleub 13307	. . . . . . . . . 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 256	. . . . . . . 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 13143	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) → (vol‘(𝑥 ∩ ∪ ran 𝐹)) ≤ ((vol‘𝑥) − (vol‘(𝑥 ∖ ∪ ran 𝐹))))
90		leaddsub 11692	. . . . . . . 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 1371	. . . . . . 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 256	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) → ((vol‘(𝑥 ∩ ∪ ran 𝐹)) + (vol‘(𝑥 ∖ ∪ ran 𝐹))) ≤ (vol‘𝑥))
93	18, 23	readdcld 11245	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol‘𝑥) ∈ ℝ) → ((vol‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ∈ ℝ)
94	49, 93	letri3d 11358	. . . . . 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 711	. . . . 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 3146	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝒫 ℝ((vol‘𝑥) ∈ ℝ → (vol‘𝑥) = ((vol‘(𝑥 ∩ ∪ ran 𝐹)) + (vol‘(𝑥 ∖ ∪ ran 𝐹)))))
99		ismbl 25050	. 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)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∖ cdif 3945 ∪ cun 3946 ∩ cin 3947 ⊆ wss 3948 𝒫 cpw 4602 ∪ cuni 4908 ∪ ciun 4997 Disj wdisj 5113 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5676 ran crn 5677 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7411 supcsup 9437 ℝcr 11111 1c1 11113 + caddc 11115 ℝ^*cxr 11249 < clt 11250 ≤ cle 11251 − cmin 11446 ℕcn 12214 seqcseq 13968 vol*covol 24986 volcvol 24987

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-cc 10432 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-n0 12475 df-z 12561 df-uz 12825 df-q 12935 df-rp 12977 df-ioo 13330 df-ico 13332 df-icc 13333 df-fz 13487 df-fzo 13630 df-fl 13759 df-seq 13969 df-exp 14030 df-hash 14293 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-clim 15434 df-rlim 15435 df-sum 15635 df-ovol 24988 df-vol 24989

This theorem is referenced by: voliunlem3 25076 iunmbl 25077

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