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Theorem voliunlem2 24821
Description: Lemma for voliun 24824. (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)
21frnd 6664 . . . 4 (𝜑 → ran 𝐹 ⊆ dom vol)
3 mblss 24801 . . . . . 6 (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ)
4 velpw 4557 . . . . . 6 (𝑥 ∈ 𝒫 ℝ ↔ 𝑥 ⊆ ℝ)
53, 4sylibr 233 . . . . 5 (𝑥 ∈ dom vol → 𝑥 ∈ 𝒫 ℝ)
65ssriv 3940 . . . 4 dom vol ⊆ 𝒫 ℝ
72, 6sstrdi 3948 . . 3 (𝜑 → ran 𝐹 ⊆ 𝒫 ℝ)
8 sspwuni 5052 . . 3 (ran 𝐹 ⊆ 𝒫 ℝ ↔ ran 𝐹 ⊆ ℝ)
97, 8sylib 217 . 2 (𝜑 ran 𝐹 ⊆ ℝ)
10 elpwi 4559 . . . 4 (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
11 inundif 4430 . . . . . . . 8 ((𝑥 ran 𝐹) ∪ (𝑥 ran 𝐹)) = 𝑥
1211fveq2i 6833 . . . . . . 7 (vol*‘((𝑥 ran 𝐹) ∪ (𝑥 ran 𝐹))) = (vol*‘𝑥)
13 inss1 4180 . . . . . . . . 9 (𝑥 ran 𝐹) ⊆ 𝑥
14 simp2 1137 . . . . . . . . 9 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝑥 ⊆ ℝ)
1513, 14sstrid 3947 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ran 𝐹) ⊆ ℝ)
16 ovolsscl 24756 . . . . . . . . . 10 (((𝑥 ran 𝐹) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
1713, 16mp3an1 1448 . . . . . . . . 9 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
18173adant1 1130 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
19 difss 4083 . . . . . . . . 9 (𝑥 ran 𝐹) ⊆ 𝑥
2019, 14sstrid 3947 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ran 𝐹) ⊆ ℝ)
21 ovolsscl 24756 . . . . . . . . . 10 (((𝑥 ran 𝐹) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
2219, 21mp3an1 1448 . . . . . . . . 9 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
23223adant1 1130 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
24 ovolun 24769 . . . . . . . 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))))
2515, 18, 20, 23, 24syl22anc 837 . . . . . . 7 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘((𝑥 ran 𝐹) ∪ (𝑥 ran 𝐹))) ≤ ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹))))
2612, 25eqbrtrrid 5133 . . . . . 6 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) ≤ ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹))))
2718rexrd 11131 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ*)
28 nnuz 12727 . . . . . . . . . . . 12 ℕ = (ℤ‘1)
29 1zzd 12457 . . . . . . . . . . . 12 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 1 ∈ ℤ)
30 fveq2 6830 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
3130ineq2d 4164 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (𝑥 ∩ (𝐹𝑛)) = (𝑥 ∩ (𝐹𝑘)))
3231fveq2d 6834 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → (vol*‘(𝑥 ∩ (𝐹𝑛))) = (vol*‘(𝑥 ∩ (𝐹𝑘))))
33 voliunlem.6 . . . . . . . . . . . . . . 15 𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝑥 ∩ (𝐹𝑛))))
34 fvex 6843 . . . . . . . . . . . . . . 15 (vol*‘(𝑥 ∩ (𝐹𝑘))) ∈ V
3532, 33, 34fvmpt 6936 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → (𝐻𝑘) = (vol*‘(𝑥 ∩ (𝐹𝑘))))
3635adantl 483 . . . . . . . . . . . . 13 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (𝐻𝑘) = (vol*‘(𝑥 ∩ (𝐹𝑘))))
37 inss1 4180 . . . . . . . . . . . . . . . 16 (𝑥 ∩ (𝐹𝑘)) ⊆ 𝑥
38 ovolsscl 24756 . . . . . . . . . . . . . . . 16 (((𝑥 ∩ (𝐹𝑘)) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹𝑘))) ∈ ℝ)
3937, 38mp3an1 1448 . . . . . . . . . . . . . . 15 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹𝑘))) ∈ ℝ)
40393adant1 1130 . . . . . . . . . . . . . 14 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹𝑘))) ∈ ℝ)
4140adantr 482 . . . . . . . . . . . . 13 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (vol*‘(𝑥 ∩ (𝐹𝑘))) ∈ ℝ)
4236, 41eqeltrd 2838 . . . . . . . . . . . 12 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (𝐻𝑘) ∈ ℝ)
4328, 29, 42serfre 13858 . . . . . . . . . . 11 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → seq1( + , 𝐻):ℕ⟶ℝ)
4443frnd 6664 . . . . . . . . . 10 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ran seq1( + , 𝐻) ⊆ ℝ)
45 ressxr 11125 . . . . . . . . . 10 ℝ ⊆ ℝ*
4644, 45sstrdi 3948 . . . . . . . . 9 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ran seq1( + , 𝐻) ⊆ ℝ*)
47 supxrcl 13155 . . . . . . . . 9 (ran seq1( + , 𝐻) ⊆ ℝ* → sup(ran seq1( + , 𝐻), ℝ*, < ) ∈ ℝ*)
4846, 47syl 17 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → sup(ran seq1( + , 𝐻), ℝ*, < ) ∈ ℝ*)
49 simp3 1138 . . . . . . . . . 10 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) ∈ ℝ)
5049, 23resubcld 11509 . . . . . . . . 9 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹))) ∈ ℝ)
5150rexrd 11131 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹))) ∈ ℝ*)
52 iunin2 5023 . . . . . . . . . . 11 𝑛 ∈ ℕ (𝑥 ∩ (𝐹𝑛)) = (𝑥 𝑛 ∈ ℕ (𝐹𝑛))
53 ffn 6656 . . . . . . . . . . . . . 14 (𝐹:ℕ⟶dom vol → 𝐹 Fn ℕ)
54 fniunfv 7181 . . . . . . . . . . . . . 14 (𝐹 Fn ℕ → 𝑛 ∈ ℕ (𝐹𝑛) = ran 𝐹)
551, 53, 543syl 18 . . . . . . . . . . . . 13 (𝜑 𝑛 ∈ ℕ (𝐹𝑛) = ran 𝐹)
56553ad2ant1 1133 . . . . . . . . . . . 12 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝑛 ∈ ℕ (𝐹𝑛) = ran 𝐹)
5756ineq2d 4164 . . . . . . . . . . 11 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 𝑛 ∈ ℕ (𝐹𝑛)) = (𝑥 ran 𝐹))
5852, 57eqtrid 2789 . . . . . . . . . 10 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝑛 ∈ ℕ (𝑥 ∩ (𝐹𝑛)) = (𝑥 ran 𝐹))
5958fveq2d 6834 . . . . . . . . 9 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹𝑛))) = (vol*‘(𝑥 ran 𝐹)))
60 eqid 2737 . . . . . . . . . 10 seq1( + , 𝐻) = seq1( + , 𝐻)
61 inss1 4180 . . . . . . . . . . . 12 (𝑥 ∩ (𝐹𝑛)) ⊆ 𝑥
6261, 14sstrid 3947 . . . . . . . . . . 11 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ∩ (𝐹𝑛)) ⊆ ℝ)
6362adantr 482 . . . . . . . . . 10 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (𝑥 ∩ (𝐹𝑛)) ⊆ ℝ)
64 ovolsscl 24756 . . . . . . . . . . . . 13 (((𝑥 ∩ (𝐹𝑛)) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹𝑛))) ∈ ℝ)
6561, 64mp3an1 1448 . . . . . . . . . . . 12 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹𝑛))) ∈ ℝ)
66653adant1 1130 . . . . . . . . . . 11 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹𝑛))) ∈ ℝ)
6766adantr 482 . . . . . . . . . 10 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol*‘(𝑥 ∩ (𝐹𝑛))) ∈ ℝ)
6860, 33, 63, 67ovoliun 24775 . . . . . . . . 9 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹𝑛))) ≤ sup(ran seq1( + , 𝐻), ℝ*, < ))
6959, 68eqbrtrrd 5121 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ≤ sup(ran seq1( + , 𝐻), ℝ*, < ))
7013ad2ant1 1133 . . . . . . . . . . . . 13 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝐹:ℕ⟶dom vol)
71 voliunlem.5 . . . . . . . . . . . . . 14 (𝜑Disj 𝑖 ∈ ℕ (𝐹𝑖))
72713ad2ant1 1133 . . . . . . . . . . . . 13 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → Disj 𝑖 ∈ ℕ (𝐹𝑖))
7370, 72, 33, 14, 49voliunlem1 24820 . . . . . . . . . . . 12 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ran 𝐹))) ≤ (vol*‘𝑥))
7443ffvelcdmda 7022 . . . . . . . . . . . . 13 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘𝑘) ∈ ℝ)
7523adantr 482 . . . . . . . . . . . . 13 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
76 simpl3 1193 . . . . . . . . . . . . 13 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (vol*‘𝑥) ∈ ℝ)
77 leaddsub 11557 . . . . . . . . . . . . 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)))))
7874, 75, 76, 77syl3anc 1371 . . . . . . . . . . . 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)))))
7973, 78mpbid 231 . . . . . . . . . . 11 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹))))
8079ralrimiva 3140 . . . . . . . . . 10 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ∀𝑘 ∈ ℕ (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹))))
81 ffn 6656 . . . . . . . . . . 11 (seq1( + , 𝐻):ℕ⟶ℝ → seq1( + , 𝐻) Fn ℕ)
82 breq1 5100 . . . . . . . . . . . 12 (𝑧 = (seq1( + , 𝐻)‘𝑘) → (𝑧 ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹))) ↔ (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹)))))
8382ralrn 7025 . . . . . . . . . . 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)))))
8443, 81, 833syl 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)))))
8580, 84mpbird 257 . . . . . . . . 9 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ∀𝑧 ∈ ran seq1( + , 𝐻)𝑧 ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹))))
86 supxrleub 13166 . . . . . . . . . 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)))))
8746, 51, 86syl2anc 585 . . . . . . . . 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)))))
8885, 87mpbird 257 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → sup(ran seq1( + , 𝐻), ℝ*, < ) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹))))
8927, 48, 51, 69, 88xrletrd 13002 . . . . . . 7 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹))))
90 leaddsub 11557 . . . . . . . 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)))))
9118, 23, 49, 90syl3anc 1371 . . . . . . 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)))))
9289, 91mpbird 257 . . . . . 6 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹))) ≤ (vol*‘𝑥))
9318, 23readdcld 11110 . . . . . . 7 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹))) ∈ ℝ)
9449, 93letri3d 11223 . . . . . 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))))
9526, 92, 94mpbir2and 711 . . . . 5 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹))))
96953expia 1121 . . . 4 ((𝜑𝑥 ⊆ ℝ) → ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹)))))
9710, 96sylan2 594 . . 3 ((𝜑𝑥 ∈ 𝒫 ℝ) → ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹)))))
9897ralrimiva 3140 . 2 (𝜑 → ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹)))))
99 ismbl 24796 . 2 ( ran 𝐹 ∈ dom vol ↔ ( ran 𝐹 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹))))))
1009, 98, 99sylanbrc 584 1 (𝜑 ran 𝐹 ∈ dom vol)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1087   = wceq 1541  wcel 2106  wral 3062  cdif 3899  cun 3900  cin 3901  wss 3902  𝒫 cpw 4552   cuni 4857   ciun 4946  Disj wdisj 5062   class class class wbr 5097  cmpt 5180  dom cdm 5625  ran crn 5626   Fn wfn 6479  wf 6480  cfv 6484  (class class class)co 7342  supcsup 9302  cr 10976  1c1 10978   + caddc 10980  *cxr 11114   < clt 11115  cle 11116  cmin 11311  cn 12079  seqcseq 13827  vol*covol 24732  volcvol 24733
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 2708  ax-rep 5234  ax-sep 5248  ax-nul 5255  ax-pow 5313  ax-pr 5377  ax-un 7655  ax-inf2 9503  ax-cc 10297  ax-cnex 11033  ax-resscn 11034  ax-1cn 11035  ax-icn 11036  ax-addcl 11037  ax-addrcl 11038  ax-mulcl 11039  ax-mulrcl 11040  ax-mulcom 11041  ax-addass 11042  ax-mulass 11043  ax-distr 11044  ax-i2m1 11045  ax-1ne0 11046  ax-1rid 11047  ax-rnegex 11048  ax-rrecex 11049  ax-cnre 11050  ax-pre-lttri 11051  ax-pre-lttrn 11052  ax-pre-ltadd 11053  ax-pre-mulgt0 11054  ax-pre-sup 11055
This theorem depends on definitions:  df-bi 206  df-an 398  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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3732  df-csb 3848  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3921  df-nul 4275  df-if 4479  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4858  df-int 4900  df-iun 4948  df-disj 5063  df-br 5098  df-opab 5160  df-mpt 5181  df-tr 5215  df-id 5523  df-eprel 5529  df-po 5537  df-so 5538  df-fr 5580  df-se 5581  df-we 5582  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6243  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6436  df-fun 6486  df-fn 6487  df-f 6488  df-f1 6489  df-fo 6490  df-f1o 6491  df-fv 6492  df-isom 6493  df-riota 7298  df-ov 7345  df-oprab 7346  df-mpo 7347  df-om 7786  df-1st 7904  df-2nd 7905  df-frecs 8172  df-wrecs 8203  df-recs 8277  df-rdg 8316  df-1o 8372  df-er 8574  df-map 8693  df-pm 8694  df-en 8810  df-dom 8811  df-sdom 8812  df-fin 8813  df-sup 9304  df-inf 9305  df-oi 9372  df-card 9801  df-pnf 11117  df-mnf 11118  df-xr 11119  df-ltxr 11120  df-le 11121  df-sub 11313  df-neg 11314  df-div 11739  df-nn 12080  df-2 12142  df-3 12143  df-n0 12340  df-z 12426  df-uz 12689  df-q 12795  df-rp 12837  df-ioo 13189  df-ico 13191  df-icc 13192  df-fz 13346  df-fzo 13489  df-fl 13618  df-seq 13828  df-exp 13889  df-hash 14151  df-cj 14910  df-re 14911  df-im 14912  df-sqrt 15046  df-abs 15047  df-clim 15297  df-rlim 15298  df-sum 15498  df-ovol 24734  df-vol 24735
This theorem is referenced by:  voliunlem3  24822  iunmbl  24823
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