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Theorem voliunlem2 25568
Description: Lemma for voliun 25571. (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 6728 . . . 4 (𝜑 → ran 𝐹 ⊆ dom vol)
3 mblss 25548 . . . . . 6 (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ)
4 velpw 4602 . . . . . 6 (𝑥 ∈ 𝒫 ℝ ↔ 𝑥 ⊆ ℝ)
53, 4sylibr 233 . . . . 5 (𝑥 ∈ dom vol → 𝑥 ∈ 𝒫 ℝ)
65ssriv 3982 . . . 4 dom vol ⊆ 𝒫 ℝ
72, 6sstrdi 3991 . . 3 (𝜑 → ran 𝐹 ⊆ 𝒫 ℝ)
8 sspwuni 5100 . . 3 (ran 𝐹 ⊆ 𝒫 ℝ ↔ ran 𝐹 ⊆ ℝ)
97, 8sylib 217 . 2 (𝜑 ran 𝐹 ⊆ ℝ)
10 elpwi 4604 . . . 4 (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
11 inundif 4473 . . . . . . . 8 ((𝑥 ran 𝐹) ∪ (𝑥 ran 𝐹)) = 𝑥
1211fveq2i 6896 . . . . . . 7 (vol*‘((𝑥 ran 𝐹) ∪ (𝑥 ran 𝐹))) = (vol*‘𝑥)
13 inss1 4227 . . . . . . . . 9 (𝑥 ran 𝐹) ⊆ 𝑥
14 simp2 1134 . . . . . . . . 9 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝑥 ⊆ ℝ)
1513, 14sstrid 3990 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ran 𝐹) ⊆ ℝ)
16 ovolsscl 25503 . . . . . . . . . 10 (((𝑥 ran 𝐹) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
1713, 16mp3an1 1445 . . . . . . . . 9 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
18173adant1 1127 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
19 difss 4128 . . . . . . . . 9 (𝑥 ran 𝐹) ⊆ 𝑥
2019, 14sstrid 3990 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ran 𝐹) ⊆ ℝ)
21 ovolsscl 25503 . . . . . . . . . 10 (((𝑥 ran 𝐹) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
2219, 21mp3an1 1445 . . . . . . . . 9 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
23223adant1 1127 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
24 ovolun 25516 . . . . . . . 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 5181 . . . . . 6 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) ≤ ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹))))
2718rexrd 11305 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ*)
28 nnuz 12911 . . . . . . . . . . . 12 ℕ = (ℤ‘1)
29 1zzd 12639 . . . . . . . . . . . 12 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 1 ∈ ℤ)
30 fveq2 6893 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
3130ineq2d 4210 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (𝑥 ∩ (𝐹𝑛)) = (𝑥 ∩ (𝐹𝑘)))
3231fveq2d 6897 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → (vol*‘(𝑥 ∩ (𝐹𝑛))) = (vol*‘(𝑥 ∩ (𝐹𝑘))))
33 voliunlem.6 . . . . . . . . . . . . . . 15 𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝑥 ∩ (𝐹𝑛))))
34 fvex 6906 . . . . . . . . . . . . . . 15 (vol*‘(𝑥 ∩ (𝐹𝑘))) ∈ V
3532, 33, 34fvmpt 7001 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → (𝐻𝑘) = (vol*‘(𝑥 ∩ (𝐹𝑘))))
3635adantl 480 . . . . . . . . . . . . 13 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (𝐻𝑘) = (vol*‘(𝑥 ∩ (𝐹𝑘))))
37 inss1 4227 . . . . . . . . . . . . . . . 16 (𝑥 ∩ (𝐹𝑘)) ⊆ 𝑥
38 ovolsscl 25503 . . . . . . . . . . . . . . . 16 (((𝑥 ∩ (𝐹𝑘)) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹𝑘))) ∈ ℝ)
3937, 38mp3an1 1445 . . . . . . . . . . . . . . 15 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹𝑘))) ∈ ℝ)
40393adant1 1127 . . . . . . . . . . . . . 14 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹𝑘))) ∈ ℝ)
4140adantr 479 . . . . . . . . . . . . 13 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (vol*‘(𝑥 ∩ (𝐹𝑘))) ∈ ℝ)
4236, 41eqeltrd 2826 . . . . . . . . . . . 12 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (𝐻𝑘) ∈ ℝ)
4328, 29, 42serfre 14045 . . . . . . . . . . 11 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → seq1( + , 𝐻):ℕ⟶ℝ)
4443frnd 6728 . . . . . . . . . 10 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ran seq1( + , 𝐻) ⊆ ℝ)
45 ressxr 11299 . . . . . . . . . 10 ℝ ⊆ ℝ*
4644, 45sstrdi 3991 . . . . . . . . 9 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ran seq1( + , 𝐻) ⊆ ℝ*)
47 supxrcl 13342 . . . . . . . . 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 1135 . . . . . . . . . 10 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) ∈ ℝ)
5049, 23resubcld 11683 . . . . . . . . 9 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹))) ∈ ℝ)
5150rexrd 11305 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹))) ∈ ℝ*)
52 iunin2 5071 . . . . . . . . . . 11 𝑛 ∈ ℕ (𝑥 ∩ (𝐹𝑛)) = (𝑥 𝑛 ∈ ℕ (𝐹𝑛))
53 ffn 6720 . . . . . . . . . . . . . 14 (𝐹:ℕ⟶dom vol → 𝐹 Fn ℕ)
54 fniunfv 7254 . . . . . . . . . . . . . 14 (𝐹 Fn ℕ → 𝑛 ∈ ℕ (𝐹𝑛) = ran 𝐹)
551, 53, 543syl 18 . . . . . . . . . . . . 13 (𝜑 𝑛 ∈ ℕ (𝐹𝑛) = ran 𝐹)
56553ad2ant1 1130 . . . . . . . . . . . 12 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝑛 ∈ ℕ (𝐹𝑛) = ran 𝐹)
5756ineq2d 4210 . . . . . . . . . . 11 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 𝑛 ∈ ℕ (𝐹𝑛)) = (𝑥 ran 𝐹))
5852, 57eqtrid 2778 . . . . . . . . . 10 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝑛 ∈ ℕ (𝑥 ∩ (𝐹𝑛)) = (𝑥 ran 𝐹))
5958fveq2d 6897 . . . . . . . . 9 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹𝑛))) = (vol*‘(𝑥 ran 𝐹)))
60 eqid 2726 . . . . . . . . . 10 seq1( + , 𝐻) = seq1( + , 𝐻)
61 inss1 4227 . . . . . . . . . . . 12 (𝑥 ∩ (𝐹𝑛)) ⊆ 𝑥
6261, 14sstrid 3990 . . . . . . . . . . 11 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ∩ (𝐹𝑛)) ⊆ ℝ)
6362adantr 479 . . . . . . . . . 10 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (𝑥 ∩ (𝐹𝑛)) ⊆ ℝ)
64 ovolsscl 25503 . . . . . . . . . . . . 13 (((𝑥 ∩ (𝐹𝑛)) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹𝑛))) ∈ ℝ)
6561, 64mp3an1 1445 . . . . . . . . . . . 12 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹𝑛))) ∈ ℝ)
66653adant1 1127 . . . . . . . . . . 11 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹𝑛))) ∈ ℝ)
6766adantr 479 . . . . . . . . . 10 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol*‘(𝑥 ∩ (𝐹𝑛))) ∈ ℝ)
6860, 33, 63, 67ovoliun 25522 . . . . . . . . 9 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹𝑛))) ≤ sup(ran seq1( + , 𝐻), ℝ*, < ))
6959, 68eqbrtrrd 5169 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ≤ sup(ran seq1( + , 𝐻), ℝ*, < ))
7013ad2ant1 1130 . . . . . . . . . . . . 13 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝐹:ℕ⟶dom vol)
71 voliunlem.5 . . . . . . . . . . . . . 14 (𝜑Disj 𝑖 ∈ ℕ (𝐹𝑖))
72713ad2ant1 1130 . . . . . . . . . . . . 13 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → Disj 𝑖 ∈ ℕ (𝐹𝑖))
7370, 72, 33, 14, 49voliunlem1 25567 . . . . . . . . . . . 12 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ran 𝐹))) ≤ (vol*‘𝑥))
7443ffvelcdmda 7090 . . . . . . . . . . . . 13 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘𝑘) ∈ ℝ)
7523adantr 479 . . . . . . . . . . . . 13 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
76 simpl3 1190 . . . . . . . . . . . . 13 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (vol*‘𝑥) ∈ ℝ)
77 leaddsub 11731 . . . . . . . . . . . . 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 1368 . . . . . . . . . . . 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 3136 . . . . . . . . . 10 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ∀𝑘 ∈ ℕ (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹))))
81 ffn 6720 . . . . . . . . . . 11 (seq1( + , 𝐻):ℕ⟶ℝ → seq1( + , 𝐻) Fn ℕ)
82 breq1 5148 . . . . . . . . . . . 12 (𝑧 = (seq1( + , 𝐻)‘𝑘) → (𝑧 ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹))) ↔ (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹)))))
8382ralrn 7094 . . . . . . . . . . 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 256 . . . . . . . . 9 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ∀𝑧 ∈ ran seq1( + , 𝐻)𝑧 ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹))))
86 supxrleub 13353 . . . . . . . . . 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 582 . . . . . . . . 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 256 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → sup(ran seq1( + , 𝐻), ℝ*, < ) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹))))
8927, 48, 51, 69, 88xrletrd 13189 . . . . . . 7 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹))))
90 leaddsub 11731 . . . . . . . 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 1368 . . . . . . 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 256 . . . . . 6 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹))) ≤ (vol*‘𝑥))
9318, 23readdcld 11284 . . . . . . 7 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹))) ∈ ℝ)
9449, 93letri3d 11397 . . . . . 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 1118 . . . 4 ((𝜑𝑥 ⊆ ℝ) → ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹)))))
9710, 96sylan2 591 . . 3 ((𝜑𝑥 ∈ 𝒫 ℝ) → ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹)))))
9897ralrimiva 3136 . 2 (𝜑 → ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹)))))
99 ismbl 25543 . 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 581 1 (𝜑 ran 𝐹 ∈ dom vol)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1534  wcel 2099  wral 3051  cdif 3943  cun 3944  cin 3945  wss 3946  𝒫 cpw 4597   cuni 4905   ciun 4993  Disj wdisj 5110   class class class wbr 5145  cmpt 5228  dom cdm 5674  ran crn 5675   Fn wfn 6541  wf 6542  cfv 6546  (class class class)co 7416  supcsup 9476  cr 11148  1c1 11150   + caddc 11152  *cxr 11288   < clt 11289  cle 11290  cmin 11485  cn 12258  seqcseq 14015  vol*covol 25479  volcvol 25480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738  ax-inf2 9677  ax-cc 10469  ax-cnex 11205  ax-resscn 11206  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-addrcl 11210  ax-mulcl 11211  ax-mulrcl 11212  ax-mulcom 11213  ax-addass 11214  ax-mulass 11215  ax-distr 11216  ax-i2m1 11217  ax-1ne0 11218  ax-1rid 11219  ax-rnegex 11220  ax-rrecex 11221  ax-cnre 11222  ax-pre-lttri 11223  ax-pre-lttrn 11224  ax-pre-ltadd 11225  ax-pre-mulgt0 11226  ax-pre-sup 11227
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-int 4947  df-iun 4995  df-disj 5111  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-se 5630  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-isom 6555  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7995  df-2nd 7996  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-er 8726  df-map 8849  df-pm 8850  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-sup 9478  df-inf 9479  df-oi 9546  df-card 9975  df-pnf 11291  df-mnf 11292  df-xr 11293  df-ltxr 11294  df-le 11295  df-sub 11487  df-neg 11488  df-div 11913  df-nn 12259  df-2 12321  df-3 12322  df-n0 12519  df-z 12605  df-uz 12869  df-q 12979  df-rp 13023  df-ioo 13376  df-ico 13378  df-icc 13379  df-fz 13533  df-fzo 13676  df-fl 13806  df-seq 14016  df-exp 14076  df-hash 14343  df-cj 15099  df-re 15100  df-im 15101  df-sqrt 15235  df-abs 15236  df-clim 15485  df-rlim 15486  df-sum 15686  df-ovol 25481  df-vol 25482
This theorem is referenced by:  voliunlem3  25569  iunmbl  25570
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