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Theorem voliunlem2 23540
Description: Lemma for voliun 23543. (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 6194 . . . . 5 (𝐹:ℕ⟶dom vol → ran 𝐹 ⊆ dom vol)
31, 2syl 17 . . . 4 (𝜑 → ran 𝐹 ⊆ dom vol)
4 mblss 23520 . . . . . 6 (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ)
5 selpw 4305 . . . . . 6 (𝑥 ∈ 𝒫 ℝ ↔ 𝑥 ⊆ ℝ)
64, 5sylibr 224 . . . . 5 (𝑥 ∈ dom vol → 𝑥 ∈ 𝒫 ℝ)
76ssriv 3757 . . . 4 dom vol ⊆ 𝒫 ℝ
83, 7syl6ss 3765 . . 3 (𝜑 → ran 𝐹 ⊆ 𝒫 ℝ)
9 sspwuni 4746 . . 3 (ran 𝐹 ⊆ 𝒫 ℝ ↔ ran 𝐹 ⊆ ℝ)
108, 9sylib 208 . 2 (𝜑 ran 𝐹 ⊆ ℝ)
11 elpwi 4308 . . . 4 (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
12 inundif 4189 . . . . . . . 8 ((𝑥 ran 𝐹) ∪ (𝑥 ran 𝐹)) = 𝑥
1312fveq2i 6336 . . . . . . 7 (vol*‘((𝑥 ran 𝐹) ∪ (𝑥 ran 𝐹))) = (vol*‘𝑥)
14 inss1 3982 . . . . . . . . 9 (𝑥 ran 𝐹) ⊆ 𝑥
15 simp2 1131 . . . . . . . . 9 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝑥 ⊆ ℝ)
1614, 15syl5ss 3764 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ran 𝐹) ⊆ ℝ)
17 ovolsscl 23475 . . . . . . . . . 10 (((𝑥 ran 𝐹) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
1814, 17mp3an1 1559 . . . . . . . . 9 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
19183adant1 1124 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
20 difss 3889 . . . . . . . . 9 (𝑥 ran 𝐹) ⊆ 𝑥
2120, 15syl5ss 3764 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ran 𝐹) ⊆ ℝ)
22 ovolsscl 23475 . . . . . . . . . 10 (((𝑥 ran 𝐹) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
2320, 22mp3an1 1559 . . . . . . . . 9 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
24233adant1 1124 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
25 ovolun 23488 . . . . . . . 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 1477 . . . . . . 7 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘((𝑥 ran 𝐹) ∪ (𝑥 ran 𝐹))) ≤ ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹))))
2713, 26syl5eqbrr 4823 . . . . . 6 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) ≤ ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹))))
2819rexrd 10292 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ*)
29 nnuz 11926 . . . . . . . . . . . 12 ℕ = (ℤ‘1)
30 1zzd 11611 . . . . . . . . . . . 12 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 1 ∈ ℤ)
31 fveq2 6333 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
3231ineq2d 3966 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (𝑥 ∩ (𝐹𝑛)) = (𝑥 ∩ (𝐹𝑘)))
3332fveq2d 6337 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → (vol*‘(𝑥 ∩ (𝐹𝑛))) = (vol*‘(𝑥 ∩ (𝐹𝑘))))
34 voliunlem.6 . . . . . . . . . . . . . . 15 𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝑥 ∩ (𝐹𝑛))))
35 fvex 6343 . . . . . . . . . . . . . . 15 (vol*‘(𝑥 ∩ (𝐹𝑘))) ∈ V
3633, 34, 35fvmpt 6425 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → (𝐻𝑘) = (vol*‘(𝑥 ∩ (𝐹𝑘))))
3736adantl 467 . . . . . . . . . . . . 13 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (𝐻𝑘) = (vol*‘(𝑥 ∩ (𝐹𝑘))))
38 inss1 3982 . . . . . . . . . . . . . . . 16 (𝑥 ∩ (𝐹𝑘)) ⊆ 𝑥
39 ovolsscl 23475 . . . . . . . . . . . . . . . 16 (((𝑥 ∩ (𝐹𝑘)) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹𝑘))) ∈ ℝ)
4038, 39mp3an1 1559 . . . . . . . . . . . . . . 15 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹𝑘))) ∈ ℝ)
41403adant1 1124 . . . . . . . . . . . . . 14 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹𝑘))) ∈ ℝ)
4241adantr 466 . . . . . . . . . . . . 13 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (vol*‘(𝑥 ∩ (𝐹𝑘))) ∈ ℝ)
4337, 42eqeltrd 2850 . . . . . . . . . . . 12 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (𝐻𝑘) ∈ ℝ)
4429, 30, 43serfre 13038 . . . . . . . . . . 11 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → seq1( + , 𝐻):ℕ⟶ℝ)
45 frn 6194 . . . . . . . . . . 11 (seq1( + , 𝐻):ℕ⟶ℝ → ran seq1( + , 𝐻) ⊆ ℝ)
4644, 45syl 17 . . . . . . . . . 10 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ran seq1( + , 𝐻) ⊆ ℝ)
47 ressxr 10286 . . . . . . . . . 10 ℝ ⊆ ℝ*
4846, 47syl6ss 3765 . . . . . . . . 9 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ran seq1( + , 𝐻) ⊆ ℝ*)
49 supxrcl 12351 . . . . . . . . 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 1132 . . . . . . . . . 10 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) ∈ ℝ)
5251, 24resubcld 10661 . . . . . . . . 9 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹))) ∈ ℝ)
5352rexrd 10292 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹))) ∈ ℝ*)
54 iunin2 4719 . . . . . . . . . . 11 𝑛 ∈ ℕ (𝑥 ∩ (𝐹𝑛)) = (𝑥 𝑛 ∈ ℕ (𝐹𝑛))
55 ffn 6186 . . . . . . . . . . . . . 14 (𝐹:ℕ⟶dom vol → 𝐹 Fn ℕ)
56 fniunfv 6649 . . . . . . . . . . . . . 14 (𝐹 Fn ℕ → 𝑛 ∈ ℕ (𝐹𝑛) = ran 𝐹)
571, 55, 563syl 18 . . . . . . . . . . . . 13 (𝜑 𝑛 ∈ ℕ (𝐹𝑛) = ran 𝐹)
58573ad2ant1 1127 . . . . . . . . . . . 12 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝑛 ∈ ℕ (𝐹𝑛) = ran 𝐹)
5958ineq2d 3966 . . . . . . . . . . 11 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 𝑛 ∈ ℕ (𝐹𝑛)) = (𝑥 ran 𝐹))
6054, 59syl5eq 2817 . . . . . . . . . 10 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝑛 ∈ ℕ (𝑥 ∩ (𝐹𝑛)) = (𝑥 ran 𝐹))
6160fveq2d 6337 . . . . . . . . 9 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹𝑛))) = (vol*‘(𝑥 ran 𝐹)))
62 eqid 2771 . . . . . . . . . 10 seq1( + , 𝐻) = seq1( + , 𝐻)
63 inss1 3982 . . . . . . . . . . . 12 (𝑥 ∩ (𝐹𝑛)) ⊆ 𝑥
6463, 15syl5ss 3764 . . . . . . . . . . 11 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ∩ (𝐹𝑛)) ⊆ ℝ)
6564adantr 466 . . . . . . . . . 10 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (𝑥 ∩ (𝐹𝑛)) ⊆ ℝ)
66 ovolsscl 23475 . . . . . . . . . . . . 13 (((𝑥 ∩ (𝐹𝑛)) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹𝑛))) ∈ ℝ)
6763, 66mp3an1 1559 . . . . . . . . . . . 12 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹𝑛))) ∈ ℝ)
68673adant1 1124 . . . . . . . . . . 11 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹𝑛))) ∈ ℝ)
6968adantr 466 . . . . . . . . . 10 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol*‘(𝑥 ∩ (𝐹𝑛))) ∈ ℝ)
7062, 34, 65, 69ovoliun 23494 . . . . . . . . 9 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹𝑛))) ≤ sup(ran seq1( + , 𝐻), ℝ*, < ))
7161, 70eqbrtrrd 4811 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ≤ sup(ran seq1( + , 𝐻), ℝ*, < ))
7213ad2ant1 1127 . . . . . . . . . . . . 13 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝐹:ℕ⟶dom vol)
73 voliunlem.5 . . . . . . . . . . . . . 14 (𝜑Disj 𝑖 ∈ ℕ (𝐹𝑖))
74733ad2ant1 1127 . . . . . . . . . . . . 13 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → Disj 𝑖 ∈ ℕ (𝐹𝑖))
7572, 74, 34, 15, 51voliunlem1 23539 . . . . . . . . . . . 12 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ran 𝐹))) ≤ (vol*‘𝑥))
7644ffvelrnda 6503 . . . . . . . . . . . . 13 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘𝑘) ∈ ℝ)
7724adantr 466 . . . . . . . . . . . . 13 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
78 simpl3 1231 . . . . . . . . . . . . 13 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (vol*‘𝑥) ∈ ℝ)
79 leaddsub 10707 . . . . . . . . . . . . 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 1476 . . . . . . . . . . . 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 3115 . . . . . . . . . 10 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ∀𝑘 ∈ ℕ (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹))))
83 ffn 6186 . . . . . . . . . . 11 (seq1( + , 𝐻):ℕ⟶ℝ → seq1( + , 𝐻) Fn ℕ)
84 breq1 4790 . . . . . . . . . . . 12 (𝑧 = (seq1( + , 𝐻)‘𝑘) → (𝑧 ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹))) ↔ (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹)))))
8584ralrn 6506 . . . . . . . . . . 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 12362 . . . . . . . . . 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 567 . . . . . . . . 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 12199 . . . . . . 7 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹))))
92 leaddsub 10707 . . . . . . . 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 1476 . . . . . . 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 10272 . . . . . . 7 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹))) ∈ ℝ)
9651, 95letri3d 10382 . . . . . 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 686 . . . . 5 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹))))
98973expia 1114 . . . 4 ((𝜑𝑥 ⊆ ℝ) → ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹)))))
9911, 98sylan2 574 . . 3 ((𝜑𝑥 ∈ 𝒫 ℝ) → ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹)))))
10099ralrimiva 3115 . 2 (𝜑 → ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹)))))
101 ismbl 23515 . 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 566 1 (𝜑 ran 𝐹 ∈ dom vol)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  cdif 3721  cun 3722  cin 3723  wss 3724  𝒫 cpw 4298   cuni 4575   ciun 4655  Disj wdisj 4755   class class class wbr 4787  cmpt 4864  dom cdm 5250  ran crn 5251   Fn wfn 6027  wf 6028  cfv 6032  (class class class)co 6794  supcsup 8503  cr 10138  1c1 10140   + caddc 10142  *cxr 10276   < clt 10277  cle 10278  cmin 10469  cn 11223  seqcseq 13009  vol*covol 23451  volcvol 23452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097  ax-inf2 8703  ax-cc 9460  ax-cnex 10195  ax-resscn 10196  ax-1cn 10197  ax-icn 10198  ax-addcl 10199  ax-addrcl 10200  ax-mulcl 10201  ax-mulrcl 10202  ax-mulcom 10203  ax-addass 10204  ax-mulass 10205  ax-distr 10206  ax-i2m1 10207  ax-1ne0 10208  ax-1rid 10209  ax-rnegex 10210  ax-rrecex 10211  ax-cnre 10212  ax-pre-lttri 10213  ax-pre-lttrn 10214  ax-pre-ltadd 10215  ax-pre-mulgt0 10216  ax-pre-sup 10217
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-disj 4756  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-se 5210  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5824  df-ord 5870  df-on 5871  df-lim 5872  df-suc 5873  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-isom 6041  df-riota 6755  df-ov 6797  df-oprab 6798  df-mpt2 6799  df-om 7214  df-1st 7316  df-2nd 7317  df-wrecs 7560  df-recs 7622  df-rdg 7660  df-1o 7714  df-oadd 7718  df-er 7897  df-map 8012  df-pm 8013  df-en 8111  df-dom 8112  df-sdom 8113  df-fin 8114  df-sup 8505  df-inf 8506  df-oi 8572  df-card 8966  df-pnf 10279  df-mnf 10280  df-xr 10281  df-ltxr 10282  df-le 10283  df-sub 10471  df-neg 10472  df-div 10888  df-nn 11224  df-2 11282  df-3 11283  df-n0 11496  df-z 11581  df-uz 11890  df-q 11993  df-rp 12037  df-ioo 12385  df-ico 12387  df-icc 12388  df-fz 12535  df-fzo 12675  df-fl 12802  df-seq 13010  df-exp 13069  df-hash 13323  df-cj 14048  df-re 14049  df-im 14050  df-sqrt 14184  df-abs 14185  df-clim 14428  df-rlim 14429  df-sum 14626  df-ovol 23453  df-vol 23454
This theorem is referenced by:  voliunlem3  23541  iunmbl  23542
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