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Theorem voliunlem3 24916
Description: Lemma for voliun 24918. (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))))
voliunlem3.1 𝑆 = seq1( + , 𝐺)
voliunlem3.2 𝐺 = (𝑛 ∈ ℕ ↦ (vol‘(𝐹𝑛)))
voliunlem3.4 (𝜑 → ∀𝑖 ∈ ℕ (vol‘(𝐹𝑖)) ∈ ℝ)
Assertion
Ref Expression
voliunlem3 (𝜑 → (vol‘ ran 𝐹) = sup(ran 𝑆, ℝ*, < ))
Distinct variable groups:   𝑖,𝑛,𝑥,𝐹   𝑥,𝑆   𝜑,𝑛,𝑥
Allowed substitution hints:   𝜑(𝑖)   𝑆(𝑖,𝑛)   𝐺(𝑥,𝑖,𝑛)   𝐻(𝑥,𝑖,𝑛)

Proof of Theorem voliunlem3
Dummy variables 𝑘 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 voliunlem.3 . . . 4 (𝜑𝐹:ℕ⟶dom vol)
2 voliunlem.5 . . . 4 (𝜑Disj 𝑖 ∈ ℕ (𝐹𝑖))
3 voliunlem.6 . . . 4 𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝑥 ∩ (𝐹𝑛))))
41, 2, 3voliunlem2 24915 . . 3 (𝜑 ran 𝐹 ∈ dom vol)
5 mblvol 24894 . . 3 ( ran 𝐹 ∈ dom vol → (vol‘ ran 𝐹) = (vol*‘ ran 𝐹))
64, 5syl 17 . 2 (𝜑 → (vol‘ ran 𝐹) = (vol*‘ ran 𝐹))
71frnd 6676 . . . . . 6 (𝜑 → ran 𝐹 ⊆ dom vol)
8 mblss 24895 . . . . . . . 8 (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ)
9 reex 11142 . . . . . . . . 9 ℝ ∈ V
109elpw2 5302 . . . . . . . 8 (𝑥 ∈ 𝒫 ℝ ↔ 𝑥 ⊆ ℝ)
118, 10sylibr 233 . . . . . . 7 (𝑥 ∈ dom vol → 𝑥 ∈ 𝒫 ℝ)
1211ssriv 3948 . . . . . 6 dom vol ⊆ 𝒫 ℝ
137, 12sstrdi 3956 . . . . 5 (𝜑 → ran 𝐹 ⊆ 𝒫 ℝ)
14 sspwuni 5060 . . . . 5 (ran 𝐹 ⊆ 𝒫 ℝ ↔ ran 𝐹 ⊆ ℝ)
1513, 14sylib 217 . . . 4 (𝜑 ran 𝐹 ⊆ ℝ)
16 ovolcl 24842 . . . 4 ( ran 𝐹 ⊆ ℝ → (vol*‘ ran 𝐹) ∈ ℝ*)
1715, 16syl 17 . . 3 (𝜑 → (vol*‘ ran 𝐹) ∈ ℝ*)
18 nnuz 12806 . . . . . . . 8 ℕ = (ℤ‘1)
19 1zzd 12534 . . . . . . . 8 (𝜑 → 1 ∈ ℤ)
20 2fveq3 6847 . . . . . . . . . . 11 (𝑛 = 𝑘 → (vol‘(𝐹𝑛)) = (vol‘(𝐹𝑘)))
21 voliunlem3.2 . . . . . . . . . . 11 𝐺 = (𝑛 ∈ ℕ ↦ (vol‘(𝐹𝑛)))
22 fvex 6855 . . . . . . . . . . 11 (vol‘(𝐹𝑘)) ∈ V
2320, 21, 22fvmpt 6948 . . . . . . . . . 10 (𝑘 ∈ ℕ → (𝐺𝑘) = (vol‘(𝐹𝑘)))
2423adantl 482 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (𝐺𝑘) = (vol‘(𝐹𝑘)))
25 voliunlem3.4 . . . . . . . . . 10 (𝜑 → ∀𝑖 ∈ ℕ (vol‘(𝐹𝑖)) ∈ ℝ)
26 2fveq3 6847 . . . . . . . . . . . 12 (𝑖 = 𝑘 → (vol‘(𝐹𝑖)) = (vol‘(𝐹𝑘)))
2726eleq1d 2822 . . . . . . . . . . 11 (𝑖 = 𝑘 → ((vol‘(𝐹𝑖)) ∈ ℝ ↔ (vol‘(𝐹𝑘)) ∈ ℝ))
2827rspccva 3580 . . . . . . . . . 10 ((∀𝑖 ∈ ℕ (vol‘(𝐹𝑖)) ∈ ℝ ∧ 𝑘 ∈ ℕ) → (vol‘(𝐹𝑘)) ∈ ℝ)
2925, 28sylan 580 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (vol‘(𝐹𝑘)) ∈ ℝ)
3024, 29eqeltrd 2838 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝐺𝑘) ∈ ℝ)
3118, 19, 30serfre 13937 . . . . . . 7 (𝜑 → seq1( + , 𝐺):ℕ⟶ℝ)
32 voliunlem3.1 . . . . . . . 8 𝑆 = seq1( + , 𝐺)
3332feq1i 6659 . . . . . . 7 (𝑆:ℕ⟶ℝ ↔ seq1( + , 𝐺):ℕ⟶ℝ)
3431, 33sylibr 233 . . . . . 6 (𝜑𝑆:ℕ⟶ℝ)
3534frnd 6676 . . . . 5 (𝜑 → ran 𝑆 ⊆ ℝ)
36 ressxr 11199 . . . . 5 ℝ ⊆ ℝ*
3735, 36sstrdi 3956 . . . 4 (𝜑 → ran 𝑆 ⊆ ℝ*)
38 supxrcl 13234 . . . 4 (ran 𝑆 ⊆ ℝ* → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
3937, 38syl 17 . . 3 (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
40 eqid 2736 . . . . 5 seq1( + , (𝑛 ∈ ℕ ↦ (vol*‘(𝐹𝑛)))) = seq1( + , (𝑛 ∈ ℕ ↦ (vol*‘(𝐹𝑛))))
41 eqid 2736 . . . . 5 (𝑛 ∈ ℕ ↦ (vol*‘(𝐹𝑛))) = (𝑛 ∈ ℕ ↦ (vol*‘(𝐹𝑛)))
421ffvelcdmda 7035 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ dom vol)
43 mblss 24895 . . . . . 6 ((𝐹𝑛) ∈ dom vol → (𝐹𝑛) ⊆ ℝ)
4442, 43syl 17 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ⊆ ℝ)
45 mblvol 24894 . . . . . . 7 ((𝐹𝑛) ∈ dom vol → (vol‘(𝐹𝑛)) = (vol*‘(𝐹𝑛)))
4642, 45syl 17 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (vol‘(𝐹𝑛)) = (vol*‘(𝐹𝑛)))
47 2fveq3 6847 . . . . . . . . 9 (𝑖 = 𝑛 → (vol‘(𝐹𝑖)) = (vol‘(𝐹𝑛)))
4847eleq1d 2822 . . . . . . . 8 (𝑖 = 𝑛 → ((vol‘(𝐹𝑖)) ∈ ℝ ↔ (vol‘(𝐹𝑛)) ∈ ℝ))
4948rspccva 3580 . . . . . . 7 ((∀𝑖 ∈ ℕ (vol‘(𝐹𝑖)) ∈ ℝ ∧ 𝑛 ∈ ℕ) → (vol‘(𝐹𝑛)) ∈ ℝ)
5025, 49sylan 580 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (vol‘(𝐹𝑛)) ∈ ℝ)
5146, 50eqeltrrd 2839 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (vol*‘(𝐹𝑛)) ∈ ℝ)
5240, 41, 44, 51ovoliun 24869 . . . 4 (𝜑 → (vol*‘ 𝑛 ∈ ℕ (𝐹𝑛)) ≤ sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol*‘(𝐹𝑛)))), ℝ*, < ))
531ffnd 6669 . . . . . 6 (𝜑𝐹 Fn ℕ)
54 fniunfv 7194 . . . . . 6 (𝐹 Fn ℕ → 𝑛 ∈ ℕ (𝐹𝑛) = ran 𝐹)
5553, 54syl 17 . . . . 5 (𝜑 𝑛 ∈ ℕ (𝐹𝑛) = ran 𝐹)
5655fveq2d 6846 . . . 4 (𝜑 → (vol*‘ 𝑛 ∈ ℕ (𝐹𝑛)) = (vol*‘ ran 𝐹))
5746mpteq2dva 5205 . . . . . . . . 9 (𝜑 → (𝑛 ∈ ℕ ↦ (vol‘(𝐹𝑛))) = (𝑛 ∈ ℕ ↦ (vol*‘(𝐹𝑛))))
5821, 57eqtrid 2788 . . . . . . . 8 (𝜑𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘(𝐹𝑛))))
5958seqeq3d 13914 . . . . . . 7 (𝜑 → seq1( + , 𝐺) = seq1( + , (𝑛 ∈ ℕ ↦ (vol*‘(𝐹𝑛)))))
6032, 59eqtr2id 2789 . . . . . 6 (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ (vol*‘(𝐹𝑛)))) = 𝑆)
6160rneqd 5893 . . . . 5 (𝜑 → ran seq1( + , (𝑛 ∈ ℕ ↦ (vol*‘(𝐹𝑛)))) = ran 𝑆)
6261supeq1d 9382 . . . 4 (𝜑 → sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol*‘(𝐹𝑛)))), ℝ*, < ) = sup(ran 𝑆, ℝ*, < ))
6352, 56, 623brtr3d 5136 . . 3 (𝜑 → (vol*‘ ran 𝐹) ≤ sup(ran 𝑆, ℝ*, < ))
64 ovolge0 24845 . . . . . . . . . 10 ( ran 𝐹 ⊆ ℝ → 0 ≤ (vol*‘ ran 𝐹))
6515, 64syl 17 . . . . . . . . 9 (𝜑 → 0 ≤ (vol*‘ ran 𝐹))
66 mnflt0 13046 . . . . . . . . . 10 -∞ < 0
67 mnfxr 11212 . . . . . . . . . . 11 -∞ ∈ ℝ*
68 0xr 11202 . . . . . . . . . . 11 0 ∈ ℝ*
69 xrltletr 13076 . . . . . . . . . . 11 ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ* ∧ (vol*‘ ran 𝐹) ∈ ℝ*) → ((-∞ < 0 ∧ 0 ≤ (vol*‘ ran 𝐹)) → -∞ < (vol*‘ ran 𝐹)))
7067, 68, 69mp3an12 1451 . . . . . . . . . 10 ((vol*‘ ran 𝐹) ∈ ℝ* → ((-∞ < 0 ∧ 0 ≤ (vol*‘ ran 𝐹)) → -∞ < (vol*‘ ran 𝐹)))
7166, 70mpani 694 . . . . . . . . 9 ((vol*‘ ran 𝐹) ∈ ℝ* → (0 ≤ (vol*‘ ran 𝐹) → -∞ < (vol*‘ ran 𝐹)))
7217, 65, 71sylc 65 . . . . . . . 8 (𝜑 → -∞ < (vol*‘ ran 𝐹))
73 xrrebnd 13087 . . . . . . . . . 10 ((vol*‘ ran 𝐹) ∈ ℝ* → ((vol*‘ ran 𝐹) ∈ ℝ ↔ (-∞ < (vol*‘ ran 𝐹) ∧ (vol*‘ ran 𝐹) < +∞)))
7417, 73syl 17 . . . . . . . . 9 (𝜑 → ((vol*‘ ran 𝐹) ∈ ℝ ↔ (-∞ < (vol*‘ ran 𝐹) ∧ (vol*‘ ran 𝐹) < +∞)))
759elpw2 5302 . . . . . . . . . . . 12 ( ran 𝐹 ∈ 𝒫 ℝ ↔ ran 𝐹 ⊆ ℝ)
7615, 75sylibr 233 . . . . . . . . . . 11 (𝜑 ran 𝐹 ∈ 𝒫 ℝ)
77 simpl 483 . . . . . . . . . . . . . . 15 ((𝑥 = ran 𝐹𝜑) → 𝑥 = ran 𝐹)
7877sseq1d 3975 . . . . . . . . . . . . . 14 ((𝑥 = ran 𝐹𝜑) → (𝑥 ⊆ ℝ ↔ ran 𝐹 ⊆ ℝ))
7977fveq2d 6846 . . . . . . . . . . . . . . . 16 ((𝑥 = ran 𝐹𝜑) → (vol*‘𝑥) = (vol*‘ ran 𝐹))
8079eleq1d 2822 . . . . . . . . . . . . . . 15 ((𝑥 = ran 𝐹𝜑) → ((vol*‘𝑥) ∈ ℝ ↔ (vol*‘ ran 𝐹) ∈ ℝ))
81 simpll 765 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑥 = ran 𝐹𝜑) ∧ 𝑛 ∈ ℕ) → 𝑥 = ran 𝐹)
8281ineq1d 4171 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑥 = ran 𝐹𝜑) ∧ 𝑛 ∈ ℕ) → (𝑥 ∩ (𝐹𝑛)) = ( ran 𝐹 ∩ (𝐹𝑛)))
83 fnfvelrn 7031 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝐹 Fn ℕ ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ∈ ran 𝐹)
8453, 83sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ ran 𝐹)
85 elssuni 4898 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝐹𝑛) ∈ ran 𝐹 → (𝐹𝑛) ⊆ ran 𝐹)
8684, 85syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ⊆ ran 𝐹)
8786adantll 712 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑥 = ran 𝐹𝜑) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ⊆ ran 𝐹)
88 sseqin2 4175 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐹𝑛) ⊆ ran 𝐹 ↔ ( ran 𝐹 ∩ (𝐹𝑛)) = (𝐹𝑛))
8987, 88sylib 217 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑥 = ran 𝐹𝜑) ∧ 𝑛 ∈ ℕ) → ( ran 𝐹 ∩ (𝐹𝑛)) = (𝐹𝑛))
9082, 89eqtrd 2776 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑥 = ran 𝐹𝜑) ∧ 𝑛 ∈ ℕ) → (𝑥 ∩ (𝐹𝑛)) = (𝐹𝑛))
9190fveq2d 6846 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑥 = ran 𝐹𝜑) ∧ 𝑛 ∈ ℕ) → (vol*‘(𝑥 ∩ (𝐹𝑛))) = (vol*‘(𝐹𝑛)))
9246adantll 712 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑥 = ran 𝐹𝜑) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐹𝑛)) = (vol*‘(𝐹𝑛)))
9391, 92eqtr4d 2779 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑥 = ran 𝐹𝜑) ∧ 𝑛 ∈ ℕ) → (vol*‘(𝑥 ∩ (𝐹𝑛))) = (vol‘(𝐹𝑛)))
9493mpteq2dva 5205 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 = ran 𝐹𝜑) → (𝑛 ∈ ℕ ↦ (vol*‘(𝑥 ∩ (𝐹𝑛)))) = (𝑛 ∈ ℕ ↦ (vol‘(𝐹𝑛))))
9594adantrr 715 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → (𝑛 ∈ ℕ ↦ (vol*‘(𝑥 ∩ (𝐹𝑛)))) = (𝑛 ∈ ℕ ↦ (vol‘(𝐹𝑛))))
9695, 3, 213eqtr4g 2801 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → 𝐻 = 𝐺)
9796seqeq3d 13914 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → seq1( + , 𝐻) = seq1( + , 𝐺))
9897, 32eqtr4di 2794 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → seq1( + , 𝐻) = 𝑆)
9998fveq1d 6844 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → (seq1( + , 𝐻)‘𝑘) = (𝑆𝑘))
100 difeq1 4075 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = ran 𝐹 → (𝑥 ran 𝐹) = ( ran 𝐹 ran 𝐹))
101 difid 4330 . . . . . . . . . . . . . . . . . . . . . . . 24 ( ran 𝐹 ran 𝐹) = ∅
102100, 101eqtrdi 2792 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = ran 𝐹 → (𝑥 ran 𝐹) = ∅)
103102fveq2d 6846 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = ran 𝐹 → (vol*‘(𝑥 ran 𝐹)) = (vol*‘∅))
104 ovol0 24857 . . . . . . . . . . . . . . . . . . . . . 22 (vol*‘∅) = 0
105103, 104eqtrdi 2792 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = ran 𝐹 → (vol*‘(𝑥 ran 𝐹)) = 0)
106105adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → (vol*‘(𝑥 ran 𝐹)) = 0)
10799, 106oveq12d 7375 . . . . . . . . . . . . . . . . . . 19 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ran 𝐹))) = ((𝑆𝑘) + 0))
10834ffvelcdmda 7035 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ∈ ℝ)
109108adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → (𝑆𝑘) ∈ ℝ)
110109recnd 11183 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → (𝑆𝑘) ∈ ℂ)
111110addid1d 11355 . . . . . . . . . . . . . . . . . . 19 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → ((𝑆𝑘) + 0) = (𝑆𝑘))
112107, 111eqtrd 2776 . . . . . . . . . . . . . . . . . 18 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ran 𝐹))) = (𝑆𝑘))
113 fveq2 6842 . . . . . . . . . . . . . . . . . . 19 (𝑥 = ran 𝐹 → (vol*‘𝑥) = (vol*‘ ran 𝐹))
114113adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → (vol*‘𝑥) = (vol*‘ ran 𝐹))
115112, 114breq12d 5118 . . . . . . . . . . . . . . . . 17 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → (((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ran 𝐹))) ≤ (vol*‘𝑥) ↔ (𝑆𝑘) ≤ (vol*‘ ran 𝐹)))
116115expr 457 . . . . . . . . . . . . . . . 16 ((𝑥 = ran 𝐹𝜑) → (𝑘 ∈ ℕ → (((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ran 𝐹))) ≤ (vol*‘𝑥) ↔ (𝑆𝑘) ≤ (vol*‘ ran 𝐹))))
117116pm5.74d 272 . . . . . . . . . . . . . . 15 ((𝑥 = ran 𝐹𝜑) → ((𝑘 ∈ ℕ → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ran 𝐹))) ≤ (vol*‘𝑥)) ↔ (𝑘 ∈ ℕ → (𝑆𝑘) ≤ (vol*‘ ran 𝐹))))
11880, 117imbi12d 344 . . . . . . . . . . . . . 14 ((𝑥 = ran 𝐹𝜑) → (((vol*‘𝑥) ∈ ℝ → (𝑘 ∈ ℕ → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ran 𝐹))) ≤ (vol*‘𝑥))) ↔ ((vol*‘ ran 𝐹) ∈ ℝ → (𝑘 ∈ ℕ → (𝑆𝑘) ≤ (vol*‘ ran 𝐹)))))
11978, 118imbi12d 344 . . . . . . . . . . . . 13 ((𝑥 = ran 𝐹𝜑) → ((𝑥 ⊆ ℝ → ((vol*‘𝑥) ∈ ℝ → (𝑘 ∈ ℕ → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ran 𝐹))) ≤ (vol*‘𝑥)))) ↔ ( ran 𝐹 ⊆ ℝ → ((vol*‘ ran 𝐹) ∈ ℝ → (𝑘 ∈ ℕ → (𝑆𝑘) ≤ (vol*‘ ran 𝐹))))))
120119pm5.74da 802 . . . . . . . . . . . 12 (𝑥 = ran 𝐹 → ((𝜑 → (𝑥 ⊆ ℝ → ((vol*‘𝑥) ∈ ℝ → (𝑘 ∈ ℕ → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ran 𝐹))) ≤ (vol*‘𝑥))))) ↔ (𝜑 → ( ran 𝐹 ⊆ ℝ → ((vol*‘ ran 𝐹) ∈ ℝ → (𝑘 ∈ ℕ → (𝑆𝑘) ≤ (vol*‘ ran 𝐹)))))))
12113ad2ant1 1133 . . . . . . . . . . . . . 14 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝐹:ℕ⟶dom vol)
12223ad2ant1 1133 . . . . . . . . . . . . . 14 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → Disj 𝑖 ∈ ℕ (𝐹𝑖))
123 simp2 1137 . . . . . . . . . . . . . 14 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝑥 ⊆ ℝ)
124 simp3 1138 . . . . . . . . . . . . . 14 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) ∈ ℝ)
125121, 122, 3, 123, 124voliunlem1 24914 . . . . . . . . . . . . 13 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ran 𝐹))) ≤ (vol*‘𝑥))
1261253exp1 1352 . . . . . . . . . . . 12 (𝜑 → (𝑥 ⊆ ℝ → ((vol*‘𝑥) ∈ ℝ → (𝑘 ∈ ℕ → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ran 𝐹))) ≤ (vol*‘𝑥)))))
127120, 126vtoclg 3525 . . . . . . . . . . 11 ( ran 𝐹 ∈ 𝒫 ℝ → (𝜑 → ( ran 𝐹 ⊆ ℝ → ((vol*‘ ran 𝐹) ∈ ℝ → (𝑘 ∈ ℕ → (𝑆𝑘) ≤ (vol*‘ ran 𝐹))))))
12876, 127mpcom 38 . . . . . . . . . 10 (𝜑 → ( ran 𝐹 ⊆ ℝ → ((vol*‘ ran 𝐹) ∈ ℝ → (𝑘 ∈ ℕ → (𝑆𝑘) ≤ (vol*‘ ran 𝐹)))))
12915, 128mpd 15 . . . . . . . . 9 (𝜑 → ((vol*‘ ran 𝐹) ∈ ℝ → (𝑘 ∈ ℕ → (𝑆𝑘) ≤ (vol*‘ ran 𝐹))))
13074, 129sylbird 259 . . . . . . . 8 (𝜑 → ((-∞ < (vol*‘ ran 𝐹) ∧ (vol*‘ ran 𝐹) < +∞) → (𝑘 ∈ ℕ → (𝑆𝑘) ≤ (vol*‘ ran 𝐹))))
13172, 130mpand 693 . . . . . . 7 (𝜑 → ((vol*‘ ran 𝐹) < +∞ → (𝑘 ∈ ℕ → (𝑆𝑘) ≤ (vol*‘ ran 𝐹))))
132 nltpnft 13083 . . . . . . . . 9 ((vol*‘ ran 𝐹) ∈ ℝ* → ((vol*‘ ran 𝐹) = +∞ ↔ ¬ (vol*‘ ran 𝐹) < +∞))
13317, 132syl 17 . . . . . . . 8 (𝜑 → ((vol*‘ ran 𝐹) = +∞ ↔ ¬ (vol*‘ ran 𝐹) < +∞))
134 rexr 11201 . . . . . . . . . . 11 ((𝑆𝑘) ∈ ℝ → (𝑆𝑘) ∈ ℝ*)
135 pnfge 13051 . . . . . . . . . . 11 ((𝑆𝑘) ∈ ℝ* → (𝑆𝑘) ≤ +∞)
136108, 134, 1353syl 18 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ≤ +∞)
137136ex 413 . . . . . . . . 9 (𝜑 → (𝑘 ∈ ℕ → (𝑆𝑘) ≤ +∞))
138 breq2 5109 . . . . . . . . . 10 ((vol*‘ ran 𝐹) = +∞ → ((𝑆𝑘) ≤ (vol*‘ ran 𝐹) ↔ (𝑆𝑘) ≤ +∞))
139138imbi2d 340 . . . . . . . . 9 ((vol*‘ ran 𝐹) = +∞ → ((𝑘 ∈ ℕ → (𝑆𝑘) ≤ (vol*‘ ran 𝐹)) ↔ (𝑘 ∈ ℕ → (𝑆𝑘) ≤ +∞)))
140137, 139syl5ibrcom 246 . . . . . . . 8 (𝜑 → ((vol*‘ ran 𝐹) = +∞ → (𝑘 ∈ ℕ → (𝑆𝑘) ≤ (vol*‘ ran 𝐹))))
141133, 140sylbird 259 . . . . . . 7 (𝜑 → (¬ (vol*‘ ran 𝐹) < +∞ → (𝑘 ∈ ℕ → (𝑆𝑘) ≤ (vol*‘ ran 𝐹))))
142131, 141pm2.61d 179 . . . . . 6 (𝜑 → (𝑘 ∈ ℕ → (𝑆𝑘) ≤ (vol*‘ ran 𝐹)))
143142ralrimiv 3142 . . . . 5 (𝜑 → ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ (vol*‘ ran 𝐹))
14434ffnd 6669 . . . . . 6 (𝜑𝑆 Fn ℕ)
145 breq1 5108 . . . . . . 7 (𝑧 = (𝑆𝑘) → (𝑧 ≤ (vol*‘ ran 𝐹) ↔ (𝑆𝑘) ≤ (vol*‘ ran 𝐹)))
146145ralrn 7038 . . . . . 6 (𝑆 Fn ℕ → (∀𝑧 ∈ ran 𝑆 𝑧 ≤ (vol*‘ ran 𝐹) ↔ ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ (vol*‘ ran 𝐹)))
147144, 146syl 17 . . . . 5 (𝜑 → (∀𝑧 ∈ ran 𝑆 𝑧 ≤ (vol*‘ ran 𝐹) ↔ ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ (vol*‘ ran 𝐹)))
148143, 147mpbird 256 . . . 4 (𝜑 → ∀𝑧 ∈ ran 𝑆 𝑧 ≤ (vol*‘ ran 𝐹))
149 supxrleub 13245 . . . . 5 ((ran 𝑆 ⊆ ℝ* ∧ (vol*‘ ran 𝐹) ∈ ℝ*) → (sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘ ran 𝐹) ↔ ∀𝑧 ∈ ran 𝑆 𝑧 ≤ (vol*‘ ran 𝐹)))
15037, 17, 149syl2anc 584 . . . 4 (𝜑 → (sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘ ran 𝐹) ↔ ∀𝑧 ∈ ran 𝑆 𝑧 ≤ (vol*‘ ran 𝐹)))
151148, 150mpbird 256 . . 3 (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘ ran 𝐹))
15217, 39, 63, 151xrletrid 13074 . 2 (𝜑 → (vol*‘ ran 𝐹) = sup(ran 𝑆, ℝ*, < ))
1536, 152eqtrd 2776 1 (𝜑 → (vol‘ ran 𝐹) = sup(ran 𝑆, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  cdif 3907  cin 3909  wss 3910  c0 4282  𝒫 cpw 4560   cuni 4865   ciun 4954  Disj wdisj 5070   class class class wbr 5105  cmpt 5188  dom cdm 5633  ran crn 5634   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  supcsup 9376  cr 11050  0cc0 11051  1c1 11052   + caddc 11054  +∞cpnf 11186  -∞cmnf 11187  *cxr 11188   < clt 11189  cle 11190  cn 12153  seqcseq 13906  vol*covol 24826  volcvol 24827
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cc 10371  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-disj 5071  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xadd 13034  df-ioo 13268  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-rlim 15371  df-sum 15571  df-xmet 20789  df-met 20790  df-ovol 24828  df-vol 24829
This theorem is referenced by:  voliun  24918
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