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Theorem voliunlem2 23226
Description: Lemma for voliun 23229. (Contributed by Mario Carneiro, 20-Mar-2014.)
Hypotheses
Ref Expression
voliunlem.3 (𝜑𝐹:ℕ⟶dom vol)
voliunlem.5 (𝜑Disj 𝑖 ∈ ℕ (𝐹𝑖))
voliunlem.6 𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝑥 ∩ (𝐹𝑛))))
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.
StepHypRef Expression
1 voliunlem.3 . . . . 5 (𝜑𝐹:ℕ⟶dom vol)
2 frn 6010 . . . . 5 (𝐹:ℕ⟶dom vol → ran 𝐹 ⊆ dom vol)
31, 2syl 17 . . . 4 (𝜑 → ran 𝐹 ⊆ dom vol)
4 mblss 23206 . . . . . 6 (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ)
5 selpw 4137 . . . . . 6 (𝑥 ∈ 𝒫 ℝ ↔ 𝑥 ⊆ ℝ)
64, 5sylibr 224 . . . . 5 (𝑥 ∈ dom vol → 𝑥 ∈ 𝒫 ℝ)
76ssriv 3587 . . . 4 dom vol ⊆ 𝒫 ℝ
83, 7syl6ss 3595 . . 3 (𝜑 → ran 𝐹 ⊆ 𝒫 ℝ)
9 sspwuni 4577 . . 3 (ran 𝐹 ⊆ 𝒫 ℝ ↔ ran 𝐹 ⊆ ℝ)
108, 9sylib 208 . 2 (𝜑 ran 𝐹 ⊆ ℝ)
11 elpwi 4140 . . . 4 (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
12 inundif 4018 . . . . . . . 8 ((𝑥 ran 𝐹) ∪ (𝑥 ran 𝐹)) = 𝑥
1312fveq2i 6151 . . . . . . 7 (vol*‘((𝑥 ran 𝐹) ∪ (𝑥 ran 𝐹))) = (vol*‘𝑥)
14 inss1 3811 . . . . . . . . 9 (𝑥 ran 𝐹) ⊆ 𝑥
15 simp2 1060 . . . . . . . . 9 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝑥 ⊆ ℝ)
1614, 15syl5ss 3594 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ran 𝐹) ⊆ ℝ)
17 ovolsscl 23161 . . . . . . . . . 10 (((𝑥 ran 𝐹) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
1814, 17mp3an1 1408 . . . . . . . . 9 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
19183adant1 1077 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
20 difss 3715 . . . . . . . . 9 (𝑥 ran 𝐹) ⊆ 𝑥
2120, 15syl5ss 3594 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ran 𝐹) ⊆ ℝ)
22 ovolsscl 23161 . . . . . . . . . 10 (((𝑥 ran 𝐹) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
2320, 22mp3an1 1408 . . . . . . . . 9 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
24233adant1 1077 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
25 ovolun 23174 . . . . . . . 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))))
2616, 19, 21, 24, 25syl22anc 1324 . . . . . . 7 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘((𝑥 ran 𝐹) ∪ (𝑥 ran 𝐹))) ≤ ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹))))
2713, 26syl5eqbrr 4649 . . . . . 6 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) ≤ ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹))))
2819rexrd 10033 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ*)
29 nnuz 11667 . . . . . . . . . . . 12 ℕ = (ℤ‘1)
30 1zzd 11352 . . . . . . . . . . . 12 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 1 ∈ ℤ)
31 fveq2 6148 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
3231ineq2d 3792 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (𝑥 ∩ (𝐹𝑛)) = (𝑥 ∩ (𝐹𝑘)))
3332fveq2d 6152 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → (vol*‘(𝑥 ∩ (𝐹𝑛))) = (vol*‘(𝑥 ∩ (𝐹𝑘))))
34 voliunlem.6 . . . . . . . . . . . . . . 15 𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝑥 ∩ (𝐹𝑛))))
35 fvex 6158 . . . . . . . . . . . . . . 15 (vol*‘(𝑥 ∩ (𝐹𝑘))) ∈ V
3633, 34, 35fvmpt 6239 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → (𝐻𝑘) = (vol*‘(𝑥 ∩ (𝐹𝑘))))
3736adantl 482 . . . . . . . . . . . . 13 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (𝐻𝑘) = (vol*‘(𝑥 ∩ (𝐹𝑘))))
38 inss1 3811 . . . . . . . . . . . . . . . 16 (𝑥 ∩ (𝐹𝑘)) ⊆ 𝑥
39 ovolsscl 23161 . . . . . . . . . . . . . . . 16 (((𝑥 ∩ (𝐹𝑘)) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹𝑘))) ∈ ℝ)
4038, 39mp3an1 1408 . . . . . . . . . . . . . . 15 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹𝑘))) ∈ ℝ)
41403adant1 1077 . . . . . . . . . . . . . 14 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹𝑘))) ∈ ℝ)
4241adantr 481 . . . . . . . . . . . . 13 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (vol*‘(𝑥 ∩ (𝐹𝑘))) ∈ ℝ)
4337, 42eqeltrd 2698 . . . . . . . . . . . 12 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (𝐻𝑘) ∈ ℝ)
4429, 30, 43serfre 12770 . . . . . . . . . . 11 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → seq1( + , 𝐻):ℕ⟶ℝ)
45 frn 6010 . . . . . . . . . . 11 (seq1( + , 𝐻):ℕ⟶ℝ → ran seq1( + , 𝐻) ⊆ ℝ)
4644, 45syl 17 . . . . . . . . . 10 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ran seq1( + , 𝐻) ⊆ ℝ)
47 ressxr 10027 . . . . . . . . . 10 ℝ ⊆ ℝ*
4846, 47syl6ss 3595 . . . . . . . . 9 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ran seq1( + , 𝐻) ⊆ ℝ*)
49 supxrcl 12088 . . . . . . . . 9 (ran seq1( + , 𝐻) ⊆ ℝ* → sup(ran seq1( + , 𝐻), ℝ*, < ) ∈ ℝ*)
5048, 49syl 17 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → sup(ran seq1( + , 𝐻), ℝ*, < ) ∈ ℝ*)
51 simp3 1061 . . . . . . . . . 10 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) ∈ ℝ)
5251, 24resubcld 10402 . . . . . . . . 9 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹))) ∈ ℝ)
5352rexrd 10033 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹))) ∈ ℝ*)
54 iunin2 4550 . . . . . . . . . . 11 𝑛 ∈ ℕ (𝑥 ∩ (𝐹𝑛)) = (𝑥 𝑛 ∈ ℕ (𝐹𝑛))
55 ffn 6002 . . . . . . . . . . . . . 14 (𝐹:ℕ⟶dom vol → 𝐹 Fn ℕ)
56 fniunfv 6459 . . . . . . . . . . . . . 14 (𝐹 Fn ℕ → 𝑛 ∈ ℕ (𝐹𝑛) = ran 𝐹)
571, 55, 563syl 18 . . . . . . . . . . . . 13 (𝜑 𝑛 ∈ ℕ (𝐹𝑛) = ran 𝐹)
58573ad2ant1 1080 . . . . . . . . . . . 12 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝑛 ∈ ℕ (𝐹𝑛) = ran 𝐹)
5958ineq2d 3792 . . . . . . . . . . 11 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 𝑛 ∈ ℕ (𝐹𝑛)) = (𝑥 ran 𝐹))
6054, 59syl5eq 2667 . . . . . . . . . 10 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝑛 ∈ ℕ (𝑥 ∩ (𝐹𝑛)) = (𝑥 ran 𝐹))
6160fveq2d 6152 . . . . . . . . 9 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹𝑛))) = (vol*‘(𝑥 ran 𝐹)))
62 eqid 2621 . . . . . . . . . 10 seq1( + , 𝐻) = seq1( + , 𝐻)
63 inss1 3811 . . . . . . . . . . . 12 (𝑥 ∩ (𝐹𝑛)) ⊆ 𝑥
6463, 15syl5ss 3594 . . . . . . . . . . 11 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ∩ (𝐹𝑛)) ⊆ ℝ)
6564adantr 481 . . . . . . . . . 10 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (𝑥 ∩ (𝐹𝑛)) ⊆ ℝ)
66 ovolsscl 23161 . . . . . . . . . . . . 13 (((𝑥 ∩ (𝐹𝑛)) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹𝑛))) ∈ ℝ)
6763, 66mp3an1 1408 . . . . . . . . . . . 12 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹𝑛))) ∈ ℝ)
68673adant1 1077 . . . . . . . . . . 11 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹𝑛))) ∈ ℝ)
6968adantr 481 . . . . . . . . . 10 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol*‘(𝑥 ∩ (𝐹𝑛))) ∈ ℝ)
7062, 34, 65, 69ovoliun 23180 . . . . . . . . 9 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹𝑛))) ≤ sup(ran seq1( + , 𝐻), ℝ*, < ))
7161, 70eqbrtrrd 4637 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ≤ sup(ran seq1( + , 𝐻), ℝ*, < ))
7213ad2ant1 1080 . . . . . . . . . . . . 13 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝐹:ℕ⟶dom vol)
73 voliunlem.5 . . . . . . . . . . . . . 14 (𝜑Disj 𝑖 ∈ ℕ (𝐹𝑖))
74733ad2ant1 1080 . . . . . . . . . . . . 13 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → Disj 𝑖 ∈ ℕ (𝐹𝑖))
7572, 74, 34, 15, 51voliunlem1 23225 . . . . . . . . . . . 12 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ran 𝐹))) ≤ (vol*‘𝑥))
7644ffvelrnda 6315 . . . . . . . . . . . . 13 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘𝑘) ∈ ℝ)
7724adantr 481 . . . . . . . . . . . . 13 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
78 simpl3 1064 . . . . . . . . . . . . 13 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (vol*‘𝑥) ∈ ℝ)
79 leaddsub 10448 . . . . . . . . . . . . 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)))))
8076, 77, 78, 79syl3anc 1323 . . . . . . . . . . . 12 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ran 𝐹))) ≤ (vol*‘𝑥) ↔ (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹)))))
8175, 80mpbid 222 . . . . . . . . . . 11 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹))))
8281ralrimiva 2960 . . . . . . . . . 10 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ∀𝑘 ∈ ℕ (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹))))
83 ffn 6002 . . . . . . . . . . 11 (seq1( + , 𝐻):ℕ⟶ℝ → seq1( + , 𝐻) Fn ℕ)
84 breq1 4616 . . . . . . . . . . . 12 (𝑧 = (seq1( + , 𝐻)‘𝑘) → (𝑧 ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹))) ↔ (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹)))))
8584ralrn 6318 . . . . . . . . . . 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)))))
8644, 83, 853syl 18 . . . . . . . . . 10 ((𝜑𝑥 ⊆ 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)))))
8782, 86mpbird 247 . . . . . . . . 9 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ∀𝑧 ∈ ran seq1( + , 𝐻)𝑧 ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹))))
88 supxrleub 12099 . . . . . . . . . 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)))))
8948, 53, 88syl2anc 692 . . . . . . . . 9 ((𝜑𝑥 ⊆ 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)))))
9087, 89mpbird 247 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → sup(ran seq1( + , 𝐻), ℝ*, < ) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹))))
9128, 50, 53, 71, 90xrletrd 11937 . . . . . . 7 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹))))
92 leaddsub 10448 . . . . . . . 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)))))
9319, 24, 51, 92syl3anc 1323 . . . . . . 7 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹))) ≤ (vol*‘𝑥) ↔ (vol*‘(𝑥 ran 𝐹)) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹)))))
9491, 93mpbird 247 . . . . . 6 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹))) ≤ (vol*‘𝑥))
9519, 24readdcld 10013 . . . . . . 7 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹))) ∈ ℝ)
9651, 95letri3d 10123 . . . . . 6 ((𝜑𝑥 ⊆ 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))))
9727, 94, 96mpbir2and 956 . . . . 5 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹))))
98973expia 1264 . . . 4 ((𝜑𝑥 ⊆ ℝ) → ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹)))))
9911, 98sylan2 491 . . 3 ((𝜑𝑥 ∈ 𝒫 ℝ) → ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹)))))
10099ralrimiva 2960 . 2 (𝜑 → ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹)))))
101 ismbl 23201 . 2 ( ran 𝐹 ∈ dom vol ↔ ( ran 𝐹 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹))))))
10210, 100, 101sylanbrc 697 1 (𝜑 ran 𝐹 ∈ dom vol)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  cdif 3552  cun 3553  cin 3554  wss 3555  𝒫 cpw 4130   cuni 4402   ciun 4485  Disj wdisj 4583   class class class wbr 4613  cmpt 4673  dom cdm 5074  ran crn 5075   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  supcsup 8290  cr 9879  1c1 9881   + caddc 9883  *cxr 10017   < clt 10018  cle 10019  cmin 10210  cn 10964  seqcseq 12741  vol*covol 23138  volcvol 23139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cc 9201  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-disj 4584  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-sup 8292  df-inf 8293  df-oi 8359  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-z 11322  df-uz 11632  df-q 11733  df-rp 11777  df-ioo 12121  df-ico 12123  df-icc 12124  df-fz 12269  df-fzo 12407  df-fl 12533  df-seq 12742  df-exp 12801  df-hash 13058  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-clim 14153  df-rlim 14154  df-sum 14351  df-ovol 23140  df-vol 23141
This theorem is referenced by:  voliunlem3  23227  iunmbl  23228
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