MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  voliunlem2 Structured version   Visualization version   GIF version

Theorem voliunlem2 23365
Description: Lemma for voliun 23368. (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 6091 . . . . 5 (𝐹:ℕ⟶dom vol → ran 𝐹 ⊆ dom vol)
31, 2syl 17 . . . 4 (𝜑 → ran 𝐹 ⊆ dom vol)
4 mblss 23345 . . . . . 6 (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ)
5 selpw 4198 . . . . . 6 (𝑥 ∈ 𝒫 ℝ ↔ 𝑥 ⊆ ℝ)
64, 5sylibr 224 . . . . 5 (𝑥 ∈ dom vol → 𝑥 ∈ 𝒫 ℝ)
76ssriv 3640 . . . 4 dom vol ⊆ 𝒫 ℝ
83, 7syl6ss 3648 . . 3 (𝜑 → ran 𝐹 ⊆ 𝒫 ℝ)
9 sspwuni 4643 . . 3 (ran 𝐹 ⊆ 𝒫 ℝ ↔ ran 𝐹 ⊆ ℝ)
108, 9sylib 208 . 2 (𝜑 ran 𝐹 ⊆ ℝ)
11 elpwi 4201 . . . 4 (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
12 inundif 4079 . . . . . . . 8 ((𝑥 ran 𝐹) ∪ (𝑥 ran 𝐹)) = 𝑥
1312fveq2i 6232 . . . . . . 7 (vol*‘((𝑥 ran 𝐹) ∪ (𝑥 ran 𝐹))) = (vol*‘𝑥)
14 inss1 3866 . . . . . . . . 9 (𝑥 ran 𝐹) ⊆ 𝑥
15 simp2 1082 . . . . . . . . 9 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝑥 ⊆ ℝ)
1614, 15syl5ss 3647 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ran 𝐹) ⊆ ℝ)
17 ovolsscl 23300 . . . . . . . . . 10 (((𝑥 ran 𝐹) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
1814, 17mp3an1 1451 . . . . . . . . 9 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
19183adant1 1099 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
20 difss 3770 . . . . . . . . 9 (𝑥 ran 𝐹) ⊆ 𝑥
2120, 15syl5ss 3647 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ran 𝐹) ⊆ ℝ)
22 ovolsscl 23300 . . . . . . . . . 10 (((𝑥 ran 𝐹) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
2320, 22mp3an1 1451 . . . . . . . . 9 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
24233adant1 1099 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
25 ovolun 23313 . . . . . . . 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 1367 . . . . . . 7 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘((𝑥 ran 𝐹) ∪ (𝑥 ran 𝐹))) ≤ ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹))))
2713, 26syl5eqbrr 4721 . . . . . 6 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) ≤ ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹))))
2819rexrd 10127 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ*)
29 nnuz 11761 . . . . . . . . . . . 12 ℕ = (ℤ‘1)
30 1zzd 11446 . . . . . . . . . . . 12 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 1 ∈ ℤ)
31 fveq2 6229 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
3231ineq2d 3847 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (𝑥 ∩ (𝐹𝑛)) = (𝑥 ∩ (𝐹𝑘)))
3332fveq2d 6233 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → (vol*‘(𝑥 ∩ (𝐹𝑛))) = (vol*‘(𝑥 ∩ (𝐹𝑘))))
34 voliunlem.6 . . . . . . . . . . . . . . 15 𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝑥 ∩ (𝐹𝑛))))
35 fvex 6239 . . . . . . . . . . . . . . 15 (vol*‘(𝑥 ∩ (𝐹𝑘))) ∈ V
3633, 34, 35fvmpt 6321 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → (𝐻𝑘) = (vol*‘(𝑥 ∩ (𝐹𝑘))))
3736adantl 481 . . . . . . . . . . . . 13 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (𝐻𝑘) = (vol*‘(𝑥 ∩ (𝐹𝑘))))
38 inss1 3866 . . . . . . . . . . . . . . . 16 (𝑥 ∩ (𝐹𝑘)) ⊆ 𝑥
39 ovolsscl 23300 . . . . . . . . . . . . . . . 16 (((𝑥 ∩ (𝐹𝑘)) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹𝑘))) ∈ ℝ)
4038, 39mp3an1 1451 . . . . . . . . . . . . . . 15 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹𝑘))) ∈ ℝ)
41403adant1 1099 . . . . . . . . . . . . . 14 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹𝑘))) ∈ ℝ)
4241adantr 480 . . . . . . . . . . . . 13 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (vol*‘(𝑥 ∩ (𝐹𝑘))) ∈ ℝ)
4337, 42eqeltrd 2730 . . . . . . . . . . . 12 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (𝐻𝑘) ∈ ℝ)
4429, 30, 43serfre 12870 . . . . . . . . . . 11 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → seq1( + , 𝐻):ℕ⟶ℝ)
45 frn 6091 . . . . . . . . . . 11 (seq1( + , 𝐻):ℕ⟶ℝ → ran seq1( + , 𝐻) ⊆ ℝ)
4644, 45syl 17 . . . . . . . . . 10 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ran seq1( + , 𝐻) ⊆ ℝ)
47 ressxr 10121 . . . . . . . . . 10 ℝ ⊆ ℝ*
4846, 47syl6ss 3648 . . . . . . . . 9 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ran seq1( + , 𝐻) ⊆ ℝ*)
49 supxrcl 12183 . . . . . . . . 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 1083 . . . . . . . . . 10 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) ∈ ℝ)
5251, 24resubcld 10496 . . . . . . . . 9 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹))) ∈ ℝ)
5352rexrd 10127 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹))) ∈ ℝ*)
54 iunin2 4616 . . . . . . . . . . 11 𝑛 ∈ ℕ (𝑥 ∩ (𝐹𝑛)) = (𝑥 𝑛 ∈ ℕ (𝐹𝑛))
55 ffn 6083 . . . . . . . . . . . . . 14 (𝐹:ℕ⟶dom vol → 𝐹 Fn ℕ)
56 fniunfv 6545 . . . . . . . . . . . . . 14 (𝐹 Fn ℕ → 𝑛 ∈ ℕ (𝐹𝑛) = ran 𝐹)
571, 55, 563syl 18 . . . . . . . . . . . . 13 (𝜑 𝑛 ∈ ℕ (𝐹𝑛) = ran 𝐹)
58573ad2ant1 1102 . . . . . . . . . . . 12 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝑛 ∈ ℕ (𝐹𝑛) = ran 𝐹)
5958ineq2d 3847 . . . . . . . . . . 11 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 𝑛 ∈ ℕ (𝐹𝑛)) = (𝑥 ran 𝐹))
6054, 59syl5eq 2697 . . . . . . . . . 10 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝑛 ∈ ℕ (𝑥 ∩ (𝐹𝑛)) = (𝑥 ran 𝐹))
6160fveq2d 6233 . . . . . . . . 9 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹𝑛))) = (vol*‘(𝑥 ran 𝐹)))
62 eqid 2651 . . . . . . . . . 10 seq1( + , 𝐻) = seq1( + , 𝐻)
63 inss1 3866 . . . . . . . . . . . 12 (𝑥 ∩ (𝐹𝑛)) ⊆ 𝑥
6463, 15syl5ss 3647 . . . . . . . . . . 11 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ∩ (𝐹𝑛)) ⊆ ℝ)
6564adantr 480 . . . . . . . . . 10 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (𝑥 ∩ (𝐹𝑛)) ⊆ ℝ)
66 ovolsscl 23300 . . . . . . . . . . . . 13 (((𝑥 ∩ (𝐹𝑛)) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹𝑛))) ∈ ℝ)
6763, 66mp3an1 1451 . . . . . . . . . . . 12 ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹𝑛))) ∈ ℝ)
68673adant1 1099 . . . . . . . . . . 11 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ (𝐹𝑛))) ∈ ℝ)
6968adantr 480 . . . . . . . . . 10 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol*‘(𝑥 ∩ (𝐹𝑛))) ∈ ℝ)
7062, 34, 65, 69ovoliun 23319 . . . . . . . . 9 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹𝑛))) ≤ sup(ran seq1( + , 𝐻), ℝ*, < ))
7161, 70eqbrtrrd 4709 . . . . . . . 8 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ≤ sup(ran seq1( + , 𝐻), ℝ*, < ))
7213ad2ant1 1102 . . . . . . . . . . . . 13 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝐹:ℕ⟶dom vol)
73 voliunlem.5 . . . . . . . . . . . . . 14 (𝜑Disj 𝑖 ∈ ℕ (𝐹𝑖))
74733ad2ant1 1102 . . . . . . . . . . . . 13 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → Disj 𝑖 ∈ ℕ (𝐹𝑖))
7572, 74, 34, 15, 51voliunlem1 23364 . . . . . . . . . . . 12 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ran 𝐹))) ≤ (vol*‘𝑥))
7644ffvelrnda 6399 . . . . . . . . . . . . 13 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘𝑘) ∈ ℝ)
7724adantr 480 . . . . . . . . . . . . 13 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (vol*‘(𝑥 ran 𝐹)) ∈ ℝ)
78 simpl3 1086 . . . . . . . . . . . . 13 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (vol*‘𝑥) ∈ ℝ)
79 leaddsub 10542 . . . . . . . . . . . . 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 1366 . . . . . . . . . . . 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 2995 . . . . . . . . . 10 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ∀𝑘 ∈ ℕ (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹))))
83 ffn 6083 . . . . . . . . . . 11 (seq1( + , 𝐻):ℕ⟶ℝ → seq1( + , 𝐻) Fn ℕ)
84 breq1 4688 . . . . . . . . . . . 12 (𝑧 = (seq1( + , 𝐻)‘𝑘) → (𝑧 ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹))) ↔ (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹)))))
8584ralrn 6402 . . . . . . . . . . 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 12194 . . . . . . . . . 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 694 . . . . . . . . 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 12031 . . . . . . 7 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ran 𝐹)) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ran 𝐹))))
92 leaddsub 10542 . . . . . . . 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 1366 . . . . . . 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 10107 . . . . . . 7 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹))) ∈ ℝ)
9651, 95letri3d 10217 . . . . . 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 977 . . . . 5 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹))))
98973expia 1286 . . . 4 ((𝜑𝑥 ⊆ ℝ) → ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹)))))
9911, 98sylan2 490 . . 3 ((𝜑𝑥 ∈ 𝒫 ℝ) → ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹)))))
10099ralrimiva 2995 . 2 (𝜑 → ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ran 𝐹)) + (vol*‘(𝑥 ran 𝐹)))))
101 ismbl 23340 . 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 699 1 (𝜑 ran 𝐹 ∈ dom vol)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  cdif 3604  cun 3605  cin 3606  wss 3607  𝒫 cpw 4191   cuni 4468   ciun 4552  Disj wdisj 4652   class class class wbr 4685  cmpt 4762  dom cdm 5143  ran crn 5144   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  supcsup 8387  cr 9973  1c1 9975   + caddc 9977  *cxr 10111   < clt 10112  cle 10113  cmin 10304  cn 11058  seqcseq 12841  vol*covol 23277  volcvol 23278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cc 9295  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-disj 4653  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-ioo 12217  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-rlim 14264  df-sum 14461  df-ovol 23279  df-vol 23280
This theorem is referenced by:  voliunlem3  23366  iunmbl  23367
  Copyright terms: Public domain W3C validator