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Theorem voliunlem3 25587
Description: Lemma for voliun 25589. (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 25586 . . 3 (𝜑 ran 𝐹 ∈ dom vol)
5 mblvol 25565 . . 3 ( ran 𝐹 ∈ dom vol → (vol‘ ran 𝐹) = (vol*‘ ran 𝐹))
64, 5syl 17 . 2 (𝜑 → (vol‘ ran 𝐹) = (vol*‘ ran 𝐹))
71frnd 6744 . . . . . 6 (𝜑 → ran 𝐹 ⊆ dom vol)
8 mblss 25566 . . . . . . . 8 (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ)
9 reex 11246 . . . . . . . . 9 ℝ ∈ V
109elpw2 5334 . . . . . . . 8 (𝑥 ∈ 𝒫 ℝ ↔ 𝑥 ⊆ ℝ)
118, 10sylibr 234 . . . . . . 7 (𝑥 ∈ dom vol → 𝑥 ∈ 𝒫 ℝ)
1211ssriv 3987 . . . . . 6 dom vol ⊆ 𝒫 ℝ
137, 12sstrdi 3996 . . . . 5 (𝜑 → ran 𝐹 ⊆ 𝒫 ℝ)
14 sspwuni 5100 . . . . 5 (ran 𝐹 ⊆ 𝒫 ℝ ↔ ran 𝐹 ⊆ ℝ)
1513, 14sylib 218 . . . 4 (𝜑 ran 𝐹 ⊆ ℝ)
16 ovolcl 25513 . . . 4 ( ran 𝐹 ⊆ ℝ → (vol*‘ ran 𝐹) ∈ ℝ*)
1715, 16syl 17 . . 3 (𝜑 → (vol*‘ ran 𝐹) ∈ ℝ*)
18 nnuz 12921 . . . . . . . 8 ℕ = (ℤ‘1)
19 1zzd 12648 . . . . . . . 8 (𝜑 → 1 ∈ ℤ)
20 2fveq3 6911 . . . . . . . . . . 11 (𝑛 = 𝑘 → (vol‘(𝐹𝑛)) = (vol‘(𝐹𝑘)))
21 voliunlem3.2 . . . . . . . . . . 11 𝐺 = (𝑛 ∈ ℕ ↦ (vol‘(𝐹𝑛)))
22 fvex 6919 . . . . . . . . . . 11 (vol‘(𝐹𝑘)) ∈ V
2320, 21, 22fvmpt 7016 . . . . . . . . . 10 (𝑘 ∈ ℕ → (𝐺𝑘) = (vol‘(𝐹𝑘)))
2423adantl 481 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (𝐺𝑘) = (vol‘(𝐹𝑘)))
25 voliunlem3.4 . . . . . . . . . 10 (𝜑 → ∀𝑖 ∈ ℕ (vol‘(𝐹𝑖)) ∈ ℝ)
26 2fveq3 6911 . . . . . . . . . . . 12 (𝑖 = 𝑘 → (vol‘(𝐹𝑖)) = (vol‘(𝐹𝑘)))
2726eleq1d 2826 . . . . . . . . . . 11 (𝑖 = 𝑘 → ((vol‘(𝐹𝑖)) ∈ ℝ ↔ (vol‘(𝐹𝑘)) ∈ ℝ))
2827rspccva 3621 . . . . . . . . . 10 ((∀𝑖 ∈ ℕ (vol‘(𝐹𝑖)) ∈ ℝ ∧ 𝑘 ∈ ℕ) → (vol‘(𝐹𝑘)) ∈ ℝ)
2925, 28sylan 580 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (vol‘(𝐹𝑘)) ∈ ℝ)
3024, 29eqeltrd 2841 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝐺𝑘) ∈ ℝ)
3118, 19, 30serfre 14072 . . . . . . 7 (𝜑 → seq1( + , 𝐺):ℕ⟶ℝ)
32 voliunlem3.1 . . . . . . . 8 𝑆 = seq1( + , 𝐺)
3332feq1i 6727 . . . . . . 7 (𝑆:ℕ⟶ℝ ↔ seq1( + , 𝐺):ℕ⟶ℝ)
3431, 33sylibr 234 . . . . . 6 (𝜑𝑆:ℕ⟶ℝ)
3534frnd 6744 . . . . 5 (𝜑 → ran 𝑆 ⊆ ℝ)
36 ressxr 11305 . . . . 5 ℝ ⊆ ℝ*
3735, 36sstrdi 3996 . . . 4 (𝜑 → ran 𝑆 ⊆ ℝ*)
38 supxrcl 13357 . . . 4 (ran 𝑆 ⊆ ℝ* → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
3937, 38syl 17 . . 3 (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
40 eqid 2737 . . . . 5 seq1( + , (𝑛 ∈ ℕ ↦ (vol*‘(𝐹𝑛)))) = seq1( + , (𝑛 ∈ ℕ ↦ (vol*‘(𝐹𝑛))))
41 eqid 2737 . . . . 5 (𝑛 ∈ ℕ ↦ (vol*‘(𝐹𝑛))) = (𝑛 ∈ ℕ ↦ (vol*‘(𝐹𝑛)))
421ffvelcdmda 7104 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ dom vol)
43 mblss 25566 . . . . . 6 ((𝐹𝑛) ∈ dom vol → (𝐹𝑛) ⊆ ℝ)
4442, 43syl 17 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ⊆ ℝ)
45 mblvol 25565 . . . . . . 7 ((𝐹𝑛) ∈ dom vol → (vol‘(𝐹𝑛)) = (vol*‘(𝐹𝑛)))
4642, 45syl 17 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (vol‘(𝐹𝑛)) = (vol*‘(𝐹𝑛)))
47 2fveq3 6911 . . . . . . . . 9 (𝑖 = 𝑛 → (vol‘(𝐹𝑖)) = (vol‘(𝐹𝑛)))
4847eleq1d 2826 . . . . . . . 8 (𝑖 = 𝑛 → ((vol‘(𝐹𝑖)) ∈ ℝ ↔ (vol‘(𝐹𝑛)) ∈ ℝ))
4948rspccva 3621 . . . . . . 7 ((∀𝑖 ∈ ℕ (vol‘(𝐹𝑖)) ∈ ℝ ∧ 𝑛 ∈ ℕ) → (vol‘(𝐹𝑛)) ∈ ℝ)
5025, 49sylan 580 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (vol‘(𝐹𝑛)) ∈ ℝ)
5146, 50eqeltrrd 2842 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (vol*‘(𝐹𝑛)) ∈ ℝ)
5240, 41, 44, 51ovoliun 25540 . . . 4 (𝜑 → (vol*‘ 𝑛 ∈ ℕ (𝐹𝑛)) ≤ sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol*‘(𝐹𝑛)))), ℝ*, < ))
531ffnd 6737 . . . . . 6 (𝜑𝐹 Fn ℕ)
54 fniunfv 7267 . . . . . 6 (𝐹 Fn ℕ → 𝑛 ∈ ℕ (𝐹𝑛) = ran 𝐹)
5553, 54syl 17 . . . . 5 (𝜑 𝑛 ∈ ℕ (𝐹𝑛) = ran 𝐹)
5655fveq2d 6910 . . . 4 (𝜑 → (vol*‘ 𝑛 ∈ ℕ (𝐹𝑛)) = (vol*‘ ran 𝐹))
5746mpteq2dva 5242 . . . . . . . . 9 (𝜑 → (𝑛 ∈ ℕ ↦ (vol‘(𝐹𝑛))) = (𝑛 ∈ ℕ ↦ (vol*‘(𝐹𝑛))))
5821, 57eqtrid 2789 . . . . . . . 8 (𝜑𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘(𝐹𝑛))))
5958seqeq3d 14050 . . . . . . 7 (𝜑 → seq1( + , 𝐺) = seq1( + , (𝑛 ∈ ℕ ↦ (vol*‘(𝐹𝑛)))))
6032, 59eqtr2id 2790 . . . . . 6 (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ (vol*‘(𝐹𝑛)))) = 𝑆)
6160rneqd 5949 . . . . 5 (𝜑 → ran seq1( + , (𝑛 ∈ ℕ ↦ (vol*‘(𝐹𝑛)))) = ran 𝑆)
6261supeq1d 9486 . . . 4 (𝜑 → sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol*‘(𝐹𝑛)))), ℝ*, < ) = sup(ran 𝑆, ℝ*, < ))
6352, 56, 623brtr3d 5174 . . 3 (𝜑 → (vol*‘ ran 𝐹) ≤ sup(ran 𝑆, ℝ*, < ))
64 ovolge0 25516 . . . . . . . . . 10 ( ran 𝐹 ⊆ ℝ → 0 ≤ (vol*‘ ran 𝐹))
6515, 64syl 17 . . . . . . . . 9 (𝜑 → 0 ≤ (vol*‘ ran 𝐹))
66 mnflt0 13167 . . . . . . . . . 10 -∞ < 0
67 mnfxr 11318 . . . . . . . . . . 11 -∞ ∈ ℝ*
68 0xr 11308 . . . . . . . . . . 11 0 ∈ ℝ*
69 xrltletr 13199 . . . . . . . . . . 11 ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ* ∧ (vol*‘ ran 𝐹) ∈ ℝ*) → ((-∞ < 0 ∧ 0 ≤ (vol*‘ ran 𝐹)) → -∞ < (vol*‘ ran 𝐹)))
7067, 68, 69mp3an12 1453 . . . . . . . . . 10 ((vol*‘ ran 𝐹) ∈ ℝ* → ((-∞ < 0 ∧ 0 ≤ (vol*‘ ran 𝐹)) → -∞ < (vol*‘ ran 𝐹)))
7166, 70mpani 696 . . . . . . . . 9 ((vol*‘ ran 𝐹) ∈ ℝ* → (0 ≤ (vol*‘ ran 𝐹) → -∞ < (vol*‘ ran 𝐹)))
7217, 65, 71sylc 65 . . . . . . . 8 (𝜑 → -∞ < (vol*‘ ran 𝐹))
73 xrrebnd 13210 . . . . . . . . . 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 5334 . . . . . . . . . . . 12 ( ran 𝐹 ∈ 𝒫 ℝ ↔ ran 𝐹 ⊆ ℝ)
7615, 75sylibr 234 . . . . . . . . . . 11 (𝜑 ran 𝐹 ∈ 𝒫 ℝ)
77 simpl 482 . . . . . . . . . . . . . . 15 ((𝑥 = ran 𝐹𝜑) → 𝑥 = ran 𝐹)
7877sseq1d 4015 . . . . . . . . . . . . . 14 ((𝑥 = ran 𝐹𝜑) → (𝑥 ⊆ ℝ ↔ ran 𝐹 ⊆ ℝ))
7977fveq2d 6910 . . . . . . . . . . . . . . . 16 ((𝑥 = ran 𝐹𝜑) → (vol*‘𝑥) = (vol*‘ ran 𝐹))
8079eleq1d 2826 . . . . . . . . . . . . . . 15 ((𝑥 = ran 𝐹𝜑) → ((vol*‘𝑥) ∈ ℝ ↔ (vol*‘ ran 𝐹) ∈ ℝ))
81 simpll 767 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑥 = ran 𝐹𝜑) ∧ 𝑛 ∈ ℕ) → 𝑥 = ran 𝐹)
8281ineq1d 4219 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑥 = ran 𝐹𝜑) ∧ 𝑛 ∈ ℕ) → (𝑥 ∩ (𝐹𝑛)) = ( ran 𝐹 ∩ (𝐹𝑛)))
83 fnfvelrn 7100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝐹 Fn ℕ ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ∈ ran 𝐹)
8453, 83sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ ran 𝐹)
85 elssuni 4937 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝐹𝑛) ∈ ran 𝐹 → (𝐹𝑛) ⊆ ran 𝐹)
8684, 85syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ⊆ ran 𝐹)
8786adantll 714 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑥 = ran 𝐹𝜑) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ⊆ ran 𝐹)
88 sseqin2 4223 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐹𝑛) ⊆ ran 𝐹 ↔ ( ran 𝐹 ∩ (𝐹𝑛)) = (𝐹𝑛))
8987, 88sylib 218 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑥 = ran 𝐹𝜑) ∧ 𝑛 ∈ ℕ) → ( ran 𝐹 ∩ (𝐹𝑛)) = (𝐹𝑛))
9082, 89eqtrd 2777 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑥 = ran 𝐹𝜑) ∧ 𝑛 ∈ ℕ) → (𝑥 ∩ (𝐹𝑛)) = (𝐹𝑛))
9190fveq2d 6910 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑥 = ran 𝐹𝜑) ∧ 𝑛 ∈ ℕ) → (vol*‘(𝑥 ∩ (𝐹𝑛))) = (vol*‘(𝐹𝑛)))
9246adantll 714 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑥 = ran 𝐹𝜑) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐹𝑛)) = (vol*‘(𝐹𝑛)))
9391, 92eqtr4d 2780 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑥 = ran 𝐹𝜑) ∧ 𝑛 ∈ ℕ) → (vol*‘(𝑥 ∩ (𝐹𝑛))) = (vol‘(𝐹𝑛)))
9493mpteq2dva 5242 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 = ran 𝐹𝜑) → (𝑛 ∈ ℕ ↦ (vol*‘(𝑥 ∩ (𝐹𝑛)))) = (𝑛 ∈ ℕ ↦ (vol‘(𝐹𝑛))))
9594adantrr 717 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → (𝑛 ∈ ℕ ↦ (vol*‘(𝑥 ∩ (𝐹𝑛)))) = (𝑛 ∈ ℕ ↦ (vol‘(𝐹𝑛))))
9695, 3, 213eqtr4g 2802 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → 𝐻 = 𝐺)
9796seqeq3d 14050 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → seq1( + , 𝐻) = seq1( + , 𝐺))
9897, 32eqtr4di 2795 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → seq1( + , 𝐻) = 𝑆)
9998fveq1d 6908 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → (seq1( + , 𝐻)‘𝑘) = (𝑆𝑘))
100 difeq1 4119 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = ran 𝐹 → (𝑥 ran 𝐹) = ( ran 𝐹 ran 𝐹))
101 difid 4376 . . . . . . . . . . . . . . . . . . . . . . . 24 ( ran 𝐹 ran 𝐹) = ∅
102100, 101eqtrdi 2793 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = ran 𝐹 → (𝑥 ran 𝐹) = ∅)
103102fveq2d 6910 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = ran 𝐹 → (vol*‘(𝑥 ran 𝐹)) = (vol*‘∅))
104 ovol0 25528 . . . . . . . . . . . . . . . . . . . . . 22 (vol*‘∅) = 0
105103, 104eqtrdi 2793 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = ran 𝐹 → (vol*‘(𝑥 ran 𝐹)) = 0)
106105adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → (vol*‘(𝑥 ran 𝐹)) = 0)
10799, 106oveq12d 7449 . . . . . . . . . . . . . . . . . . 19 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ran 𝐹))) = ((𝑆𝑘) + 0))
10834ffvelcdmda 7104 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ∈ ℝ)
109108adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → (𝑆𝑘) ∈ ℝ)
110109recnd 11289 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → (𝑆𝑘) ∈ ℂ)
111110addridd 11461 . . . . . . . . . . . . . . . . . . 19 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → ((𝑆𝑘) + 0) = (𝑆𝑘))
112107, 111eqtrd 2777 . . . . . . . . . . . . . . . . . 18 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ran 𝐹))) = (𝑆𝑘))
113 fveq2 6906 . . . . . . . . . . . . . . . . . . 19 (𝑥 = ran 𝐹 → (vol*‘𝑥) = (vol*‘ ran 𝐹))
114113adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → (vol*‘𝑥) = (vol*‘ ran 𝐹))
115112, 114breq12d 5156 . . . . . . . . . . . . . . . . 17 ((𝑥 = ran 𝐹 ∧ (𝜑𝑘 ∈ ℕ)) → (((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ran 𝐹))) ≤ (vol*‘𝑥) ↔ (𝑆𝑘) ≤ (vol*‘ ran 𝐹)))
116115expr 456 . . . . . . . . . . . . . . . 16 ((𝑥 = ran 𝐹𝜑) → (𝑘 ∈ ℕ → (((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ran 𝐹))) ≤ (vol*‘𝑥) ↔ (𝑆𝑘) ≤ (vol*‘ ran 𝐹))))
117116pm5.74d 273 . . . . . . . . . . . . . . 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 804 . . . . . . . . . . . 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 1134 . . . . . . . . . . . . . 14 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝐹:ℕ⟶dom vol)
12223ad2ant1 1134 . . . . . . . . . . . . . 14 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → Disj 𝑖 ∈ ℕ (𝐹𝑖))
123 simp2 1138 . . . . . . . . . . . . . 14 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝑥 ⊆ ℝ)
124 simp3 1139 . . . . . . . . . . . . . 14 ((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) ∈ ℝ)
125121, 122, 3, 123, 124voliunlem1 25585 . . . . . . . . . . . . 13 (((𝜑𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ran 𝐹))) ≤ (vol*‘𝑥))
1261253exp1 1353 . . . . . . . . . . . 12 (𝜑 → (𝑥 ⊆ ℝ → ((vol*‘𝑥) ∈ ℝ → (𝑘 ∈ ℕ → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ran 𝐹))) ≤ (vol*‘𝑥)))))
127120, 126vtoclg 3554 . . . . . . . . . . 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 260 . . . . . . . 8 (𝜑 → ((-∞ < (vol*‘ ran 𝐹) ∧ (vol*‘ ran 𝐹) < +∞) → (𝑘 ∈ ℕ → (𝑆𝑘) ≤ (vol*‘ ran 𝐹))))
13172, 130mpand 695 . . . . . . 7 (𝜑 → ((vol*‘ ran 𝐹) < +∞ → (𝑘 ∈ ℕ → (𝑆𝑘) ≤ (vol*‘ ran 𝐹))))
132 nltpnft 13206 . . . . . . . . 9 ((vol*‘ ran 𝐹) ∈ ℝ* → ((vol*‘ ran 𝐹) = +∞ ↔ ¬ (vol*‘ ran 𝐹) < +∞))
13317, 132syl 17 . . . . . . . 8 (𝜑 → ((vol*‘ ran 𝐹) = +∞ ↔ ¬ (vol*‘ ran 𝐹) < +∞))
134 rexr 11307 . . . . . . . . . . 11 ((𝑆𝑘) ∈ ℝ → (𝑆𝑘) ∈ ℝ*)
135 pnfge 13172 . . . . . . . . . . 11 ((𝑆𝑘) ∈ ℝ* → (𝑆𝑘) ≤ +∞)
136108, 134, 1353syl 18 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → (𝑆𝑘) ≤ +∞)
137136ex 412 . . . . . . . . 9 (𝜑 → (𝑘 ∈ ℕ → (𝑆𝑘) ≤ +∞))
138 breq2 5147 . . . . . . . . . 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 247 . . . . . . . 8 (𝜑 → ((vol*‘ ran 𝐹) = +∞ → (𝑘 ∈ ℕ → (𝑆𝑘) ≤ (vol*‘ ran 𝐹))))
141133, 140sylbird 260 . . . . . . 7 (𝜑 → (¬ (vol*‘ ran 𝐹) < +∞ → (𝑘 ∈ ℕ → (𝑆𝑘) ≤ (vol*‘ ran 𝐹))))
142131, 141pm2.61d 179 . . . . . 6 (𝜑 → (𝑘 ∈ ℕ → (𝑆𝑘) ≤ (vol*‘ ran 𝐹)))
143142ralrimiv 3145 . . . . 5 (𝜑 → ∀𝑘 ∈ ℕ (𝑆𝑘) ≤ (vol*‘ ran 𝐹))
14434ffnd 6737 . . . . . 6 (𝜑𝑆 Fn ℕ)
145 breq1 5146 . . . . . . 7 (𝑧 = (𝑆𝑘) → (𝑧 ≤ (vol*‘ ran 𝐹) ↔ (𝑆𝑘) ≤ (vol*‘ ran 𝐹)))
146145ralrn 7108 . . . . . 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 257 . . . 4 (𝜑 → ∀𝑧 ∈ ran 𝑆 𝑧 ≤ (vol*‘ ran 𝐹))
149 supxrleub 13368 . . . . 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 257 . . 3 (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘ ran 𝐹))
15217, 39, 63, 151xrletrid 13197 . 2 (𝜑 → (vol*‘ ran 𝐹) = sup(ran 𝑆, ℝ*, < ))
1536, 152eqtrd 2777 1 (𝜑 → (vol‘ ran 𝐹) = sup(ran 𝑆, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  cdif 3948  cin 3950  wss 3951  c0 4333  𝒫 cpw 4600   cuni 4907   ciun 4991  Disj wdisj 5110   class class class wbr 5143  cmpt 5225  dom cdm 5685  ran crn 5686   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  supcsup 9480  cr 11154  0cc0 11155  1c1 11156   + caddc 11158  +∞cpnf 11292  -∞cmnf 11293  *cxr 11294   < clt 11295  cle 11296  cn 12266  seqcseq 14042  vol*covol 25497  volcvol 25498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cc 10475  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-disj 5111  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xadd 13155  df-ioo 13391  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-rlim 15525  df-sum 15723  df-xmet 21357  df-met 21358  df-ovol 25499  df-vol 25500
This theorem is referenced by:  voliun  25589
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