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Theorem voliunlem3 24145
 Description: Lemma for voliun 24147. (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 24144 . . 3 (𝜑 ran 𝐹 ∈ dom vol)
5 mblvol 24123 . . 3 ( ran 𝐹 ∈ dom vol → (vol‘ ran 𝐹) = (vol*‘ ran 𝐹))
64, 5syl 17 . 2 (𝜑 → (vol‘ ran 𝐹) = (vol*‘ ran 𝐹))
71frnd 6514 . . . . . 6 (𝜑 → ran 𝐹 ⊆ dom vol)
8 mblss 24124 . . . . . . . 8 (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ)
9 reex 10620 . . . . . . . . 9 ℝ ∈ V
109elpw2 5239 . . . . . . . 8 (𝑥 ∈ 𝒫 ℝ ↔ 𝑥 ⊆ ℝ)
118, 10sylibr 236 . . . . . . 7 (𝑥 ∈ dom vol → 𝑥 ∈ 𝒫 ℝ)
1211ssriv 3969 . . . . . 6 dom vol ⊆ 𝒫 ℝ
137, 12sstrdi 3977 . . . . 5 (𝜑 → ran 𝐹 ⊆ 𝒫 ℝ)
14 sspwuni 5013 . . . . 5 (ran 𝐹 ⊆ 𝒫 ℝ ↔ ran 𝐹 ⊆ ℝ)
1513, 14sylib 220 . . . 4 (𝜑 ran 𝐹 ⊆ ℝ)
16 ovolcl 24071 . . . 4 ( ran 𝐹 ⊆ ℝ → (vol*‘ ran 𝐹) ∈ ℝ*)
1715, 16syl 17 . . 3 (𝜑 → (vol*‘ ran 𝐹) ∈ ℝ*)
18 nnuz 12273 . . . . . . . 8 ℕ = (ℤ‘1)
19 1zzd 12005 . . . . . . . 8 (𝜑 → 1 ∈ ℤ)
20 2fveq3 6668 . . . . . . . . . . 11 (𝑛 = 𝑘 → (vol‘(𝐹𝑛)) = (vol‘(𝐹𝑘)))
21 voliunlem3.2 . . . . . . . . . . 11 𝐺 = (𝑛 ∈ ℕ ↦ (vol‘(𝐹𝑛)))
22 fvex 6676 . . . . . . . . . . 11 (vol‘(𝐹𝑘)) ∈ V
2320, 21, 22fvmpt 6761 . . . . . . . . . 10 (𝑘 ∈ ℕ → (𝐺𝑘) = (vol‘(𝐹𝑘)))
2423adantl 484 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (𝐺𝑘) = (vol‘(𝐹𝑘)))
25 voliunlem3.4 . . . . . . . . . 10 (𝜑 → ∀𝑖 ∈ ℕ (vol‘(𝐹𝑖)) ∈ ℝ)
26 2fveq3 6668 . . . . . . . . . . . 12 (𝑖 = 𝑘 → (vol‘(𝐹𝑖)) = (vol‘(𝐹𝑘)))
2726eleq1d 2895 . . . . . . . . . . 11 (𝑖 = 𝑘 → ((vol‘(𝐹𝑖)) ∈ ℝ ↔ (vol‘(𝐹𝑘)) ∈ ℝ))
2827rspccva 3620 . . . . . . . . . 10 ((∀𝑖 ∈ ℕ (vol‘(𝐹𝑖)) ∈ ℝ ∧ 𝑘 ∈ ℕ) → (vol‘(𝐹𝑘)) ∈ ℝ)
2925, 28sylan 582 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (vol‘(𝐹𝑘)) ∈ ℝ)
3024, 29eqeltrd 2911 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝐺𝑘) ∈ ℝ)
3118, 19, 30serfre 13391 . . . . . . 7 (𝜑 → seq1( + , 𝐺):ℕ⟶ℝ)
32 voliunlem3.1 . . . . . . . 8 𝑆 = seq1( + , 𝐺)
3332feq1i 6498 . . . . . . 7 (𝑆:ℕ⟶ℝ ↔ seq1( + , 𝐺):ℕ⟶ℝ)
3431, 33sylibr 236 . . . . . 6 (𝜑𝑆:ℕ⟶ℝ)
3534frnd 6514 . . . . 5 (𝜑 → ran 𝑆 ⊆ ℝ)
36 ressxr 10677 . . . . 5 ℝ ⊆ ℝ*
3735, 36sstrdi 3977 . . . 4 (𝜑 → ran 𝑆 ⊆ ℝ*)
38 supxrcl 12700 . . . 4 (ran 𝑆 ⊆ ℝ* → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
3937, 38syl 17 . . 3 (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
40 eqid 2819 . . . . 5 seq1( + , (𝑛 ∈ ℕ ↦ (vol*‘(𝐹𝑛)))) = seq1( + , (𝑛 ∈ ℕ ↦ (vol*‘(𝐹𝑛))))
41 eqid 2819 . . . . 5 (𝑛 ∈ ℕ ↦ (vol*‘(𝐹𝑛))) = (𝑛 ∈ ℕ ↦ (vol*‘(𝐹𝑛)))
421ffvelrnda 6844 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ dom vol)
43 mblss 24124 . . . . . 6 ((𝐹𝑛) ∈ dom vol → (𝐹𝑛) ⊆ ℝ)
4442, 43syl 17 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ⊆ ℝ)
45 mblvol 24123 . . . . . . 7 ((𝐹𝑛) ∈ dom vol → (vol‘(𝐹𝑛)) = (vol*‘(𝐹𝑛)))
4642, 45syl 17 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (vol‘(𝐹𝑛)) = (vol*‘(𝐹𝑛)))
47 2fveq3 6668 . . . . . . . . 9 (𝑖 = 𝑛 → (vol‘(𝐹𝑖)) = (vol‘(𝐹𝑛)))
4847eleq1d 2895 . . . . . . . 8 (𝑖 = 𝑛 → ((vol‘(𝐹𝑖)) ∈ ℝ ↔ (vol‘(𝐹𝑛)) ∈ ℝ))
4948rspccva 3620 . . . . . . 7 ((∀𝑖 ∈ ℕ (vol‘(𝐹𝑖)) ∈ ℝ ∧ 𝑛 ∈ ℕ) → (vol‘(𝐹𝑛)) ∈ ℝ)
5025, 49sylan 582 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (vol‘(𝐹𝑛)) ∈ ℝ)
5146, 50eqeltrrd 2912 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (vol*‘(𝐹𝑛)) ∈ ℝ)
5240, 41, 44, 51ovoliun 24098 . . . 4 (𝜑 → (vol*‘ 𝑛 ∈ ℕ (𝐹𝑛)) ≤ sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol*‘(𝐹𝑛)))), ℝ*, < ))
531ffnd 6508 . . . . . 6 (𝜑𝐹 Fn ℕ)
54 fniunfv 6998 . . . . . 6 (𝐹 Fn ℕ → 𝑛 ∈ ℕ (𝐹𝑛) = ran 𝐹)
5553, 54syl 17 . . . . 5 (𝜑 𝑛 ∈ ℕ (𝐹𝑛) = ran 𝐹)
5655fveq2d 6667 . . . 4 (𝜑 → (vol*‘ 𝑛 ∈ ℕ (𝐹𝑛)) = (vol*‘ ran 𝐹))
5746mpteq2dva 5152 . . . . . . . . 9 (𝜑 → (𝑛 ∈ ℕ ↦ (vol‘(𝐹𝑛))) = (𝑛 ∈ ℕ ↦ (vol*‘(𝐹𝑛))))
5821, 57syl5eq 2866 . . . . . . . 8 (𝜑𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘(𝐹𝑛))))
5958seqeq3d 13369 . . . . . . 7 (𝜑 → seq1( + , 𝐺) = seq1( + , (𝑛 ∈ ℕ ↦ (vol*‘(𝐹𝑛)))))
6032, 59syl5req 2867 . . . . . 6 (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ (vol*‘(𝐹𝑛)))) = 𝑆)
6160rneqd 5801 . . . . 5 (𝜑 → ran seq1( + , (𝑛 ∈ ℕ ↦ (vol*‘(𝐹𝑛)))) = ran 𝑆)
6261supeq1d 8902 . . . 4 (𝜑 → sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol*‘(𝐹𝑛)))), ℝ*, < ) = sup(ran 𝑆, ℝ*, < ))
6352, 56, 623brtr3d 5088 . . 3 (𝜑 → (vol*‘ ran 𝐹) ≤ sup(ran 𝑆, ℝ*, < ))
64 ovolge0 24074 . . . . . . . . . 10 ( ran 𝐹 ⊆ ℝ → 0 ≤ (vol*‘ ran 𝐹))
6515, 64syl 17 . . . . . . . . 9 (𝜑 → 0 ≤ (vol*‘ ran 𝐹))
66 mnflt0 12512 . . . . . . . . . 10 -∞ < 0
67 mnfxr 10690 . . . . . . . . . . 11 -∞ ∈ ℝ*
68 0xr 10680 . . . . . . . . . . 11 0 ∈ ℝ*
69 xrltletr 12542 . . . . . . . . . . 11 ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ* ∧ (vol*‘ ran 𝐹) ∈ ℝ*) → ((-∞ < 0 ∧ 0 ≤ (vol*‘ ran 𝐹)) → -∞ < (vol*‘ ran 𝐹)))
7067, 68, 69mp3an12 1445 . . . . . . . . . 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 12553 . . . . . . . . . 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 5239 . . . . . . . . . . . 12 ( ran 𝐹 ∈ 𝒫 ℝ ↔ ran 𝐹 ⊆ ℝ)
7615, 75sylibr 236 . . . . . . . . . . 11 (𝜑 ran 𝐹 ∈ 𝒫 ℝ)
77 simpl 485 . . . . . . . . . . . . . . 15 ((𝑥 = ran 𝐹𝜑) → 𝑥 = ran 𝐹)
7877sseq1d 3996 . . . . . . . . . . . . . 14 ((𝑥 = ran 𝐹𝜑) → (𝑥 ⊆ ℝ ↔ ran 𝐹 ⊆ ℝ))
7977fveq2d 6667 . . . . . . . . . . . . . . . 16 ((𝑥 = ran 𝐹𝜑) → (vol*‘𝑥) = (vol*‘ ran 𝐹))
8079eleq1d 2895 . . . . . . . . . . . . . . 15 ((𝑥 = ran 𝐹𝜑) → ((vol*‘𝑥) ∈ ℝ ↔ (vol*‘ ran 𝐹) ∈ ℝ))
81 simpll 765 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑥 = ran 𝐹𝜑) ∧ 𝑛 ∈ ℕ) → 𝑥 = ran 𝐹)
8281ineq1d 4186 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑥 = ran 𝐹𝜑) ∧ 𝑛 ∈ ℕ) → (𝑥 ∩ (𝐹𝑛)) = ( ran 𝐹 ∩ (𝐹𝑛)))
83 fnfvelrn 6841 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝐹 Fn ℕ ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ∈ ran 𝐹)
8453, 83sylan 582 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ ran 𝐹)
85 elssuni 4859 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4190 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐹𝑛) ⊆ ran 𝐹 ↔ ( ran 𝐹 ∩ (𝐹𝑛)) = (𝐹𝑛))
8987, 88sylib 220 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑥 = ran 𝐹𝜑) ∧ 𝑛 ∈ ℕ) → ( ran 𝐹 ∩ (𝐹𝑛)) = (𝐹𝑛))
9082, 89eqtrd 2854 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑥 = ran 𝐹𝜑) ∧ 𝑛 ∈ ℕ) → (𝑥 ∩ (𝐹𝑛)) = (𝐹𝑛))
9190fveq2d 6667 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2857 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑥 = ran 𝐹𝜑) ∧ 𝑛 ∈ ℕ) → (vol*‘(𝑥 ∩ (𝐹𝑛))) = (vol‘(𝐹𝑛)))
9493mpteq2dva 5152 . . . . . . . . . . . . . . . . . . . . . . . . 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 2879 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → 𝐻 = 𝐺)
9796seqeq3d 13369 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → seq1( + , 𝐻) = seq1( + , 𝐺))
9897, 32syl6eqr 2872 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → seq1( + , 𝐻) = 𝑆)
9998fveq1d 6665 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → (seq1( + , 𝐻)‘𝑘) = (𝑆𝑘))
100 difeq1 4090 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = ran 𝐹 → (𝑥 ran 𝐹) = ( ran 𝐹 ran 𝐹))
101 difid 4328 . . . . . . . . . . . . . . . . . . . . . . . 24 ( ran 𝐹 ran 𝐹) = ∅
102100, 101syl6eq 2870 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = ran 𝐹 → (𝑥 ran 𝐹) = ∅)
103102fveq2d 6667 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = ran 𝐹 → (vol*‘(𝑥 ran 𝐹)) = (vol*‘∅))
104 ovol0 24086 . . . . . . . . . . . . . . . . . . . . . 22 (vol*‘∅) = 0
105103, 104syl6eq 2870 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = ran 𝐹 → (vol*‘(𝑥 ran 𝐹)) = 0)
106105adantr 483 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → (vol*‘(𝑥 ran 𝐹)) = 0)
10799, 106oveq12d 7166 . . . . . . . . . . . . . . . . . . 19 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ran 𝐹))) = ((𝑆𝑘) + 0))
10834ffvelrnda 6844 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ∈ ℝ)
109108adantl 484 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → (𝑆𝑘) ∈ ℝ)
110109recnd 10661 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → (𝑆𝑘) ∈ ℂ)
111110addid1d 10832 . . . . . . . . . . . . . . . . . . 19 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → ((𝑆𝑘) + 0) = (𝑆𝑘))
112107, 111eqtrd 2854 . . . . . . . . . . . . . . . . . 18 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ran 𝐹))) = (𝑆𝑘))
113 fveq2 6663 . . . . . . . . . . . . . . . . . . 19 (𝑥 = ran 𝐹 → (vol*‘𝑥) = (vol*‘ ran 𝐹))
114113adantr 483 . . . . . . . . . . . . . . . . . 18 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → (vol*‘𝑥) = (vol*‘ ran 𝐹))
115112, 114breq12d 5070 . . . . . . . . . . . . . . . . 17 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → (((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ran 𝐹))) ≤ (vol*‘𝑥) ↔ (𝑆𝑘) ≤ (vol*‘ ran 𝐹)))
116115expr 459 . . . . . . . . . . . . . . . 16 ((𝑥 = ran 𝐹𝜑) → (𝑘 ∈ ℕ → (((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ran 𝐹))) ≤ (vol*‘𝑥) ↔ (𝑆𝑘) ≤ (vol*‘ ran 𝐹))))
117116pm5.74d 275 . . . . . . . . . . . . . . 15 ((𝑥 = ran 𝐹𝜑) → ((𝑘 ∈ ℕ → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ran 𝐹))) ≤ (vol*‘𝑥)) ↔ (𝑘 ∈ ℕ → (𝑆𝑘) ≤ (vol*‘ ran 𝐹))))
11880, 117imbi12d 347 . . . . . . . . . . . . . 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 347 . . . . . . . . . . . . 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 1128 . . . . . . . . . . . . . 14 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝐹:ℕ⟶dom vol)
12223ad2ant1 1128 . . . . . . . . . . . . . 14 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → Disj 𝑖 ∈ ℕ (𝐹𝑖))
123 simp2 1132 . . . . . . . . . . . . . 14 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝑥 ⊆ ℝ)
124 simp3 1133 . . . . . . . . . . . . . 14 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) ∈ ℝ)
125121, 122, 3, 123, 124voliunlem1 24143 . . . . . . . . . . . . 13 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ran 𝐹))) ≤ (vol*‘𝑥))
1261253exp1 1347 . . . . . . . . . . . 12 (𝜑 → (𝑥 ⊆ ℝ → ((vol*‘𝑥) ∈ ℝ → (𝑘 ∈ ℕ → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ran 𝐹))) ≤ (vol*‘𝑥)))))
127120, 126vtoclg 3566 . . . . . . . . . . 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 262 . . . . . . . 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 12549 . . . . . . . . 9 ((vol*‘ ran 𝐹) ∈ ℝ* → ((vol*‘ ran 𝐹) = +∞ ↔ ¬ (vol*‘ ran 𝐹) < +∞))
13317, 132syl 17 . . . . . . . 8 (𝜑 → ((vol*‘ ran 𝐹) = +∞ ↔ ¬ (vol*‘ ran 𝐹) < +∞))
134 rexr 10679 . . . . . . . . . . 11 ((𝑆𝑘) ∈ ℝ → (𝑆𝑘) ∈ ℝ*)
135 pnfge 12517 . . . . . . . . . . 11 ((𝑆𝑘) ∈ ℝ* → (𝑆𝑘) ≤ +∞)
136108, 134, 1353syl 18 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ≤ +∞)
137136ex 415 . . . . . . . . 9 (𝜑 → (𝑘 ∈ ℕ → (𝑆𝑘) ≤ +∞))
138 breq2 5061 . . . . . . . . . 10 ((vol*‘ ran 𝐹) = +∞ → ((𝑆𝑘) ≤ (vol*‘ ran 𝐹) ↔ (𝑆𝑘) ≤ +∞))
139138imbi2d 343 . . . . . . . . 9 ((vol*‘ ran 𝐹) = +∞ → ((𝑘 ∈ ℕ → (𝑆𝑘) ≤ (vol*‘ ran 𝐹)) ↔ (𝑘 ∈ ℕ → (𝑆𝑘) ≤ +∞)))
140137, 139syl5ibrcom 249 . . . . . . . 8 (𝜑 → ((vol*‘ ran 𝐹) = +∞ → (𝑘 ∈ ℕ → (𝑆𝑘) ≤ (vol*‘ ran 𝐹))))
141133, 140sylbird 262 . . . . . . 7 (𝜑 → (¬ (vol*‘ ran 𝐹) < +∞ → (𝑘 ∈ ℕ → (𝑆𝑘) ≤ (vol*‘ ran 𝐹))))
142131, 141pm2.61d 181 . . . . . 6 (𝜑 → (𝑘 ∈ ℕ → (𝑆𝑘) ≤ (vol*‘ ran 𝐹)))
143142ralrimiv 3179 . . . . 5 (𝜑 → ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ (vol*‘ ran 𝐹))
14434ffnd 6508 . . . . . 6 (𝜑𝑆 Fn ℕ)
145 breq1 5060 . . . . . . 7 (𝑧 = (𝑆𝑘) → (𝑧 ≤ (vol*‘ ran 𝐹) ↔ (𝑆𝑘) ≤ (vol*‘ ran 𝐹)))
146145ralrn 6847 . . . . . 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 259 . . . 4 (𝜑 → ∀𝑧 ∈ ran 𝑆 𝑧 ≤ (vol*‘ ran 𝐹))
149 supxrleub 12711 . . . . 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 586 . . . 4 (𝜑 → (sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘ ran 𝐹) ↔ ∀𝑧 ∈ ran 𝑆 𝑧 ≤ (vol*‘ ran 𝐹)))
151148, 150mpbird 259 . . 3 (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘ ran 𝐹))
15217, 39, 63, 151xrletrid 12540 . 2 (𝜑 → (vol*‘ ran 𝐹) = sup(ran 𝑆, ℝ*, < ))
1536, 152eqtrd 2854 1 (𝜑 → (vol‘ ran 𝐹) = sup(ran 𝑆, ℝ*, < ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1082   = wceq 1531   ∈ wcel 2108  ∀wral 3136   ∖ cdif 3931   ∩ cin 3933   ⊆ wss 3934  ∅c0 4289  𝒫 cpw 4537  ∪ cuni 4830  ∪ ciun 4910  Disj wdisj 5022   class class class wbr 5057   ↦ cmpt 5137  dom cdm 5548  ran crn 5549   Fn wfn 6343  ⟶wf 6344  ‘cfv 6348  (class class class)co 7148  supcsup 8896  ℝcr 10528  0cc0 10529  1c1 10530   + caddc 10532  +∞cpnf 10664  -∞cmnf 10665  ℝ*cxr 10666   < clt 10667   ≤ cle 10668  ℕcn 11630  seqcseq 13361  vol*covol 24055  volcvol 24056 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 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-inf2 9096  ax-cc 9849  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-fal 1544  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-disj 5023  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-of 7401  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-er 8281  df-map 8400  df-pm 8401  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-sup 8898  df-inf 8899  df-oi 8966  df-dju 9322  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-q 12341  df-rp 12382  df-xadd 12500  df-ioo 12734  df-ico 12736  df-icc 12737  df-fz 12885  df-fzo 13026  df-fl 13154  df-seq 13362  df-exp 13422  df-hash 13683  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-clim 14837  df-rlim 14838  df-sum 15035  df-xmet 20530  df-met 20531  df-ovol 24057  df-vol 24058 This theorem is referenced by:  voliun  24147
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