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

Theorem volsup 25604
Description: The volume of the limit of an increasing sequence of measurable sets is the limit of the volumes. (Contributed by Mario Carneiro, 14-Aug-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
Assertion
Ref Expression
volsup ((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) → (vol‘ ran 𝐹) = sup((vol “ ran 𝐹), ℝ*, < ))
Distinct variable group:   𝑛,𝐹

Proof of Theorem volsup
Dummy variables 𝑗 𝑘 𝑚 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ffvelcdm 7100 . . . . . . . . . . 11 ((𝐹:ℕ⟶dom vol ∧ 𝑘 ∈ ℕ) → (𝐹𝑘) ∈ dom vol)
21ad2ant2r 747 . . . . . . . . . 10 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ (vol‘(𝐹𝑘)) ∈ ℝ)) → (𝐹𝑘) ∈ dom vol)
3 fzofi 14011 . . . . . . . . . . 11 (1..^𝑘) ∈ Fin
4 simpll 767 . . . . . . . . . . . . 13 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ (vol‘(𝐹𝑘)) ∈ ℝ)) → 𝐹:ℕ⟶dom vol)
5 elfzouz 13699 . . . . . . . . . . . . . 14 (𝑚 ∈ (1..^𝑘) → 𝑚 ∈ (ℤ‘1))
6 nnuz 12918 . . . . . . . . . . . . . 14 ℕ = (ℤ‘1)
75, 6eleqtrrdi 2849 . . . . . . . . . . . . 13 (𝑚 ∈ (1..^𝑘) → 𝑚 ∈ ℕ)
8 ffvelcdm 7100 . . . . . . . . . . . . 13 ((𝐹:ℕ⟶dom vol ∧ 𝑚 ∈ ℕ) → (𝐹𝑚) ∈ dom vol)
94, 7, 8syl2an 596 . . . . . . . . . . . 12 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ (vol‘(𝐹𝑘)) ∈ ℝ)) ∧ 𝑚 ∈ (1..^𝑘)) → (𝐹𝑚) ∈ dom vol)
109ralrimiva 3143 . . . . . . . . . . 11 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ (vol‘(𝐹𝑘)) ∈ ℝ)) → ∀𝑚 ∈ (1..^𝑘)(𝐹𝑚) ∈ dom vol)
11 finiunmbl 25592 . . . . . . . . . . 11 (((1..^𝑘) ∈ Fin ∧ ∀𝑚 ∈ (1..^𝑘)(𝐹𝑚) ∈ dom vol) → 𝑚 ∈ (1..^𝑘)(𝐹𝑚) ∈ dom vol)
123, 10, 11sylancr 587 . . . . . . . . . 10 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ (vol‘(𝐹𝑘)) ∈ ℝ)) → 𝑚 ∈ (1..^𝑘)(𝐹𝑚) ∈ dom vol)
13 difmbl 25591 . . . . . . . . . 10 (((𝐹𝑘) ∈ dom vol ∧ 𝑚 ∈ (1..^𝑘)(𝐹𝑚) ∈ dom vol) → ((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)) ∈ dom vol)
142, 12, 13syl2anc 584 . . . . . . . . 9 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ (vol‘(𝐹𝑘)) ∈ ℝ)) → ((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)) ∈ dom vol)
15 mblvol 25578 . . . . . . . . . . 11 (((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)) ∈ dom vol → (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))) = (vol*‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))
1614, 15syl 17 . . . . . . . . . 10 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))) = (vol*‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))
17 difssd 4146 . . . . . . . . . . 11 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ (vol‘(𝐹𝑘)) ∈ ℝ)) → ((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)) ⊆ (𝐹𝑘))
18 mblss 25579 . . . . . . . . . . . 12 ((𝐹𝑘) ∈ dom vol → (𝐹𝑘) ⊆ ℝ)
192, 18syl 17 . . . . . . . . . . 11 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ (vol‘(𝐹𝑘)) ∈ ℝ)) → (𝐹𝑘) ⊆ ℝ)
20 mblvol 25578 . . . . . . . . . . . . 13 ((𝐹𝑘) ∈ dom vol → (vol‘(𝐹𝑘)) = (vol*‘(𝐹𝑘)))
212, 20syl 17 . . . . . . . . . . . 12 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol‘(𝐹𝑘)) = (vol*‘(𝐹𝑘)))
22 simprr 773 . . . . . . . . . . . 12 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol‘(𝐹𝑘)) ∈ ℝ)
2321, 22eqeltrrd 2839 . . . . . . . . . . 11 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol*‘(𝐹𝑘)) ∈ ℝ)
24 ovolsscl 25534 . . . . . . . . . . 11 ((((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)) ⊆ (𝐹𝑘) ∧ (𝐹𝑘) ⊆ ℝ ∧ (vol*‘(𝐹𝑘)) ∈ ℝ) → (vol*‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))) ∈ ℝ)
2517, 19, 23, 24syl3anc 1370 . . . . . . . . . 10 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol*‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))) ∈ ℝ)
2616, 25eqeltrd 2838 . . . . . . . . 9 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))) ∈ ℝ)
2714, 26jca 511 . . . . . . . 8 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ (vol‘(𝐹𝑘)) ∈ ℝ)) → (((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)) ∈ dom vol ∧ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))) ∈ ℝ))
2827expr 456 . . . . . . 7 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ 𝑘 ∈ ℕ) → ((vol‘(𝐹𝑘)) ∈ ℝ → (((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)) ∈ dom vol ∧ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))) ∈ ℝ)))
2928ralimdva 3164 . . . . . 6 ((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) → (∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ → ∀𝑘 ∈ ℕ (((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)) ∈ dom vol ∧ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))) ∈ ℝ)))
3029imp 406 . . . . 5 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → ∀𝑘 ∈ ℕ (((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)) ∈ dom vol ∧ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))) ∈ ℝ))
31 fveq2 6906 . . . . . 6 (𝑘 = 𝑚 → (𝐹𝑘) = (𝐹𝑚))
3231iundisj2 25597 . . . . 5 Disj 𝑘 ∈ ℕ ((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))
33 eqid 2734 . . . . . 6 seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))) = seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))
34 eqid 2734 . . . . . 6 (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))) = (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))
3533, 34voliun 25602 . . . . 5 ((∀𝑘 ∈ ℕ (((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)) ∈ dom vol ∧ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))) ∈ ℝ) ∧ Disj 𝑘 ∈ ℕ ((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))) → (vol‘ 𝑘 ∈ ℕ ((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))) = sup(ran seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))), ℝ*, < ))
3630, 32, 35sylancl 586 . . . 4 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → (vol‘ 𝑘 ∈ ℕ ((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))) = sup(ran seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))), ℝ*, < ))
3731iundisj 25596 . . . . . 6 𝑘 ∈ ℕ (𝐹𝑘) = 𝑘 ∈ ℕ ((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))
38 ffn 6736 . . . . . . . 8 (𝐹:ℕ⟶dom vol → 𝐹 Fn ℕ)
3938ad2antrr 726 . . . . . . 7 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → 𝐹 Fn ℕ)
40 fniunfv 7266 . . . . . . 7 (𝐹 Fn ℕ → 𝑘 ∈ ℕ (𝐹𝑘) = ran 𝐹)
4139, 40syl 17 . . . . . 6 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → 𝑘 ∈ ℕ (𝐹𝑘) = ran 𝐹)
4237, 41eqtr3id 2788 . . . . 5 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → 𝑘 ∈ ℕ ((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)) = ran 𝐹)
4342fveq2d 6910 . . . 4 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → (vol‘ 𝑘 ∈ ℕ ((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))) = (vol‘ ran 𝐹))
44 1z 12644 . . . . . . . . . . 11 1 ∈ ℤ
45 seqfn 14050 . . . . . . . . . . 11 (1 ∈ ℤ → seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))) Fn (ℤ‘1))
4644, 45ax-mp 5 . . . . . . . . . 10 seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))) Fn (ℤ‘1)
476fneq2i 6666 . . . . . . . . . 10 (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))) Fn ℕ ↔ seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))) Fn (ℤ‘1))
4846, 47mpbir 231 . . . . . . . . 9 seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))) Fn ℕ
4948a1i 11 . . . . . . . 8 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))) Fn ℕ)
50 volf 25577 . . . . . . . . . 10 vol:dom vol⟶(0[,]+∞)
51 simpll 767 . . . . . . . . . 10 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → 𝐹:ℕ⟶dom vol)
52 fco 6760 . . . . . . . . . 10 ((vol:dom vol⟶(0[,]+∞) ∧ 𝐹:ℕ⟶dom vol) → (vol ∘ 𝐹):ℕ⟶(0[,]+∞))
5350, 51, 52sylancr 587 . . . . . . . . 9 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → (vol ∘ 𝐹):ℕ⟶(0[,]+∞))
5453ffnd 6737 . . . . . . . 8 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → (vol ∘ 𝐹) Fn ℕ)
55 fveq2 6906 . . . . . . . . . . . . 13 (𝑥 = 1 → (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘𝑥) = (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘1))
56 2fveq3 6911 . . . . . . . . . . . . 13 (𝑥 = 1 → (vol‘(𝐹𝑥)) = (vol‘(𝐹‘1)))
5755, 56eqeq12d 2750 . . . . . . . . . . . 12 (𝑥 = 1 → ((seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘𝑥) = (vol‘(𝐹𝑥)) ↔ (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘1) = (vol‘(𝐹‘1))))
5857imbi2d 340 . . . . . . . . . . 11 (𝑥 = 1 → ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘𝑥) = (vol‘(𝐹𝑥))) ↔ (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘1) = (vol‘(𝐹‘1)))))
59 fveq2 6906 . . . . . . . . . . . . 13 (𝑥 = 𝑗 → (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘𝑥) = (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘𝑗))
60 2fveq3 6911 . . . . . . . . . . . . 13 (𝑥 = 𝑗 → (vol‘(𝐹𝑥)) = (vol‘(𝐹𝑗)))
6159, 60eqeq12d 2750 . . . . . . . . . . . 12 (𝑥 = 𝑗 → ((seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘𝑥) = (vol‘(𝐹𝑥)) ↔ (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘𝑗) = (vol‘(𝐹𝑗))))
6261imbi2d 340 . . . . . . . . . . 11 (𝑥 = 𝑗 → ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘𝑥) = (vol‘(𝐹𝑥))) ↔ (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘𝑗) = (vol‘(𝐹𝑗)))))
63 fveq2 6906 . . . . . . . . . . . . 13 (𝑥 = (𝑗 + 1) → (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘𝑥) = (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘(𝑗 + 1)))
64 2fveq3 6911 . . . . . . . . . . . . 13 (𝑥 = (𝑗 + 1) → (vol‘(𝐹𝑥)) = (vol‘(𝐹‘(𝑗 + 1))))
6563, 64eqeq12d 2750 . . . . . . . . . . . 12 (𝑥 = (𝑗 + 1) → ((seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘𝑥) = (vol‘(𝐹𝑥)) ↔ (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘(𝑗 + 1)) = (vol‘(𝐹‘(𝑗 + 1)))))
6665imbi2d 340 . . . . . . . . . . 11 (𝑥 = (𝑗 + 1) → ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘𝑥) = (vol‘(𝐹𝑥))) ↔ (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘(𝑗 + 1)) = (vol‘(𝐹‘(𝑗 + 1))))))
67 seq1 14051 . . . . . . . . . . . . . 14 (1 ∈ ℤ → (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘1) = ((𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))‘1))
6844, 67ax-mp 5 . . . . . . . . . . . . 13 (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘1) = ((𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))‘1)
69 1nn 12274 . . . . . . . . . . . . . 14 1 ∈ ℕ
70 oveq2 7438 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 1 → (1..^𝑘) = (1..^1))
71 fzo0 13719 . . . . . . . . . . . . . . . . . . . . . 22 (1..^1) = ∅
7270, 71eqtrdi 2790 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 1 → (1..^𝑘) = ∅)
7372iuneq1d 5023 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 1 → 𝑚 ∈ (1..^𝑘)(𝐹𝑚) = 𝑚 ∈ ∅ (𝐹𝑚))
74 0iun 5067 . . . . . . . . . . . . . . . . . . . 20 𝑚 ∈ ∅ (𝐹𝑚) = ∅
7573, 74eqtrdi 2790 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 1 → 𝑚 ∈ (1..^𝑘)(𝐹𝑚) = ∅)
7675difeq2d 4135 . . . . . . . . . . . . . . . . . 18 (𝑘 = 1 → ((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)) = ((𝐹𝑘) ∖ ∅))
77 dif0 4383 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑘) ∖ ∅) = (𝐹𝑘)
7876, 77eqtrdi 2790 . . . . . . . . . . . . . . . . 17 (𝑘 = 1 → ((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)) = (𝐹𝑘))
79 fveq2 6906 . . . . . . . . . . . . . . . . 17 (𝑘 = 1 → (𝐹𝑘) = (𝐹‘1))
8078, 79eqtrd 2774 . . . . . . . . . . . . . . . 16 (𝑘 = 1 → ((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)) = (𝐹‘1))
8180fveq2d 6910 . . . . . . . . . . . . . . 15 (𝑘 = 1 → (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))) = (vol‘(𝐹‘1)))
82 fvex 6919 . . . . . . . . . . . . . . 15 (vol‘(𝐹‘1)) ∈ V
8381, 34, 82fvmpt 7015 . . . . . . . . . . . . . 14 (1 ∈ ℕ → ((𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))‘1) = (vol‘(𝐹‘1)))
8469, 83ax-mp 5 . . . . . . . . . . . . 13 ((𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))‘1) = (vol‘(𝐹‘1))
8568, 84eqtri 2762 . . . . . . . . . . . 12 (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘1) = (vol‘(𝐹‘1))
8685a1i 11 . . . . . . . . . . 11 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘1) = (vol‘(𝐹‘1)))
87 oveq1 7437 . . . . . . . . . . . . . 14 ((seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘𝑗) = (vol‘(𝐹𝑗)) → ((seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘𝑗) + ((𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))‘(𝑗 + 1))) = ((vol‘(𝐹𝑗)) + ((𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))‘(𝑗 + 1))))
88 seqp1 14053 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (ℤ‘1) → (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘(𝑗 + 1)) = ((seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘𝑗) + ((𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))‘(𝑗 + 1))))
8988, 6eleq2s 2856 . . . . . . . . . . . . . . . 16 (𝑗 ∈ ℕ → (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘(𝑗 + 1)) = ((seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘𝑗) + ((𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))‘(𝑗 + 1))))
9089adantl 481 . . . . . . . . . . . . . . 15 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘(𝑗 + 1)) = ((seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘𝑗) + ((𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))‘(𝑗 + 1))))
91 undif2 4482 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑗) ∪ ((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗))) = ((𝐹𝑗) ∪ (𝐹‘(𝑗 + 1)))
92 fveq2 6906 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑗 → (𝐹𝑛) = (𝐹𝑗))
93 fvoveq1 7453 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑗 → (𝐹‘(𝑛 + 1)) = (𝐹‘(𝑗 + 1)))
9492, 93sseq12d 4028 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑗 → ((𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)) ↔ (𝐹𝑗) ⊆ (𝐹‘(𝑗 + 1))))
95 simpllr 776 . . . . . . . . . . . . . . . . . . . 20 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)))
96 simpr 484 . . . . . . . . . . . . . . . . . . . 20 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
9794, 95, 96rspcdva 3622 . . . . . . . . . . . . . . . . . . 19 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (𝐹𝑗) ⊆ (𝐹‘(𝑗 + 1)))
98 ssequn1 4195 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝑗) ⊆ (𝐹‘(𝑗 + 1)) ↔ ((𝐹𝑗) ∪ (𝐹‘(𝑗 + 1))) = (𝐹‘(𝑗 + 1)))
9997, 98sylib 218 . . . . . . . . . . . . . . . . . 18 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝐹𝑗) ∪ (𝐹‘(𝑗 + 1))) = (𝐹‘(𝑗 + 1)))
10091, 99eqtr2id 2787 . . . . . . . . . . . . . . . . 17 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (𝐹‘(𝑗 + 1)) = ((𝐹𝑗) ∪ ((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗))))
101100fveq2d 6910 . . . . . . . . . . . . . . . 16 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (vol‘(𝐹‘(𝑗 + 1))) = (vol‘((𝐹𝑗) ∪ ((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗)))))
102 simplll 775 . . . . . . . . . . . . . . . . . 18 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → 𝐹:ℕ⟶dom vol)
103102, 96ffvelcdmd 7104 . . . . . . . . . . . . . . . . 17 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (𝐹𝑗) ∈ dom vol)
104 peano2nn 12275 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
105104adantl 481 . . . . . . . . . . . . . . . . . . 19 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (𝑗 + 1) ∈ ℕ)
106102, 105ffvelcdmd 7104 . . . . . . . . . . . . . . . . . 18 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (𝐹‘(𝑗 + 1)) ∈ dom vol)
107 difmbl 25591 . . . . . . . . . . . . . . . . . 18 (((𝐹‘(𝑗 + 1)) ∈ dom vol ∧ (𝐹𝑗) ∈ dom vol) → ((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗)) ∈ dom vol)
108106, 103, 107syl2anc 584 . . . . . . . . . . . . . . . . 17 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗)) ∈ dom vol)
109 disjdif 4477 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑗) ∩ ((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗))) = ∅
110109a1i 11 . . . . . . . . . . . . . . . . 17 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝐹𝑗) ∩ ((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗))) = ∅)
111 2fveq3 6911 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑗 → (vol‘(𝐹𝑘)) = (vol‘(𝐹𝑗)))
112111eleq1d 2823 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑗 → ((vol‘(𝐹𝑘)) ∈ ℝ ↔ (vol‘(𝐹𝑗)) ∈ ℝ))
113 simplr 769 . . . . . . . . . . . . . . . . . 18 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ)
114112, 113, 96rspcdva 3622 . . . . . . . . . . . . . . . . 17 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (vol‘(𝐹𝑗)) ∈ ℝ)
115 mblvol 25578 . . . . . . . . . . . . . . . . . . 19 (((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗)) ∈ dom vol → (vol‘((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗))) = (vol*‘((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗))))
116108, 115syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗))) = (vol*‘((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗))))
117 difssd 4146 . . . . . . . . . . . . . . . . . . 19 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗)) ⊆ (𝐹‘(𝑗 + 1)))
118 mblss 25579 . . . . . . . . . . . . . . . . . . . 20 ((𝐹‘(𝑗 + 1)) ∈ dom vol → (𝐹‘(𝑗 + 1)) ⊆ ℝ)
119106, 118syl 17 . . . . . . . . . . . . . . . . . . 19 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (𝐹‘(𝑗 + 1)) ⊆ ℝ)
120 mblvol 25578 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹‘(𝑗 + 1)) ∈ dom vol → (vol‘(𝐹‘(𝑗 + 1))) = (vol*‘(𝐹‘(𝑗 + 1))))
121106, 120syl 17 . . . . . . . . . . . . . . . . . . . 20 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (vol‘(𝐹‘(𝑗 + 1))) = (vol*‘(𝐹‘(𝑗 + 1))))
122 2fveq3 6911 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = (𝑗 + 1) → (vol‘(𝐹𝑘)) = (vol‘(𝐹‘(𝑗 + 1))))
123122eleq1d 2823 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = (𝑗 + 1) → ((vol‘(𝐹𝑘)) ∈ ℝ ↔ (vol‘(𝐹‘(𝑗 + 1))) ∈ ℝ))
124123, 113, 105rspcdva 3622 . . . . . . . . . . . . . . . . . . . 20 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (vol‘(𝐹‘(𝑗 + 1))) ∈ ℝ)
125121, 124eqeltrrd 2839 . . . . . . . . . . . . . . . . . . 19 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (vol*‘(𝐹‘(𝑗 + 1))) ∈ ℝ)
126 ovolsscl 25534 . . . . . . . . . . . . . . . . . . 19 ((((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗)) ⊆ (𝐹‘(𝑗 + 1)) ∧ (𝐹‘(𝑗 + 1)) ⊆ ℝ ∧ (vol*‘(𝐹‘(𝑗 + 1))) ∈ ℝ) → (vol*‘((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗))) ∈ ℝ)
127117, 119, 125, 126syl3anc 1370 . . . . . . . . . . . . . . . . . 18 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (vol*‘((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗))) ∈ ℝ)
128116, 127eqeltrd 2838 . . . . . . . . . . . . . . . . 17 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗))) ∈ ℝ)
129 volun 25593 . . . . . . . . . . . . . . . . 17 ((((𝐹𝑗) ∈ dom vol ∧ ((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗)) ∈ dom vol ∧ ((𝐹𝑗) ∩ ((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗))) = ∅) ∧ ((vol‘(𝐹𝑗)) ∈ ℝ ∧ (vol‘((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗))) ∈ ℝ)) → (vol‘((𝐹𝑗) ∪ ((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗)))) = ((vol‘(𝐹𝑗)) + (vol‘((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗)))))
130103, 108, 110, 114, 128, 129syl32anc 1377 . . . . . . . . . . . . . . . 16 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐹𝑗) ∪ ((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗)))) = ((vol‘(𝐹𝑗)) + (vol‘((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗)))))
13195adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑗)) → ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)))
132 elfznn 13589 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑚 ∈ (1...𝑗) → 𝑚 ∈ ℕ)
133132adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑗)) → 𝑚 ∈ ℕ)
134 elfzuz3 13557 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑚 ∈ (1...𝑗) → 𝑗 ∈ (ℤ𝑚))
135134adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑗)) → 𝑗 ∈ (ℤ𝑚))
136 volsuplem 25603 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)) ∧ (𝑚 ∈ ℕ ∧ 𝑗 ∈ (ℤ𝑚))) → (𝐹𝑚) ⊆ (𝐹𝑗))
137131, 133, 135, 136syl12anc 837 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑗)) → (𝐹𝑚) ⊆ (𝐹𝑗))
138137ralrimiva 3143 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ∀𝑚 ∈ (1...𝑗)(𝐹𝑚) ⊆ (𝐹𝑗))
139 iunss 5049 . . . . . . . . . . . . . . . . . . . . . . 23 ( 𝑚 ∈ (1...𝑗)(𝐹𝑚) ⊆ (𝐹𝑗) ↔ ∀𝑚 ∈ (1...𝑗)(𝐹𝑚) ⊆ (𝐹𝑗))
140138, 139sylibr 234 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → 𝑚 ∈ (1...𝑗)(𝐹𝑚) ⊆ (𝐹𝑗))
14196, 6eleqtrdi 2848 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ (ℤ‘1))
142 eluzfz2 13568 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (ℤ‘1) → 𝑗 ∈ (1...𝑗))
143141, 142syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ (1...𝑗))
144 fveq2 6906 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 = 𝑗 → (𝐹𝑚) = (𝐹𝑗))
145144ssiun2s 5052 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ (1...𝑗) → (𝐹𝑗) ⊆ 𝑚 ∈ (1...𝑗)(𝐹𝑚))
146143, 145syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (𝐹𝑗) ⊆ 𝑚 ∈ (1...𝑗)(𝐹𝑚))
147140, 146eqssd 4012 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → 𝑚 ∈ (1...𝑗)(𝐹𝑚) = (𝐹𝑗))
14896nnzd 12637 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℤ)
149 fzval3 13769 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ ℤ → (1...𝑗) = (1..^(𝑗 + 1)))
150148, 149syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (1...𝑗) = (1..^(𝑗 + 1)))
151150iuneq1d 5023 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → 𝑚 ∈ (1...𝑗)(𝐹𝑚) = 𝑚 ∈ (1..^(𝑗 + 1))(𝐹𝑚))
152147, 151eqtr3d 2776 . . . . . . . . . . . . . . . . . . . 20 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (𝐹𝑗) = 𝑚 ∈ (1..^(𝑗 + 1))(𝐹𝑚))
153152difeq2d 4135 . . . . . . . . . . . . . . . . . . 19 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗)) = ((𝐹‘(𝑗 + 1)) ∖ 𝑚 ∈ (1..^(𝑗 + 1))(𝐹𝑚)))
154153fveq2d 6910 . . . . . . . . . . . . . . . . . 18 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗))) = (vol‘((𝐹‘(𝑗 + 1)) ∖ 𝑚 ∈ (1..^(𝑗 + 1))(𝐹𝑚))))
155 fveq2 6906 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = (𝑗 + 1) → (𝐹𝑘) = (𝐹‘(𝑗 + 1)))
156 oveq2 7438 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = (𝑗 + 1) → (1..^𝑘) = (1..^(𝑗 + 1)))
157156iuneq1d 5023 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = (𝑗 + 1) → 𝑚 ∈ (1..^𝑘)(𝐹𝑚) = 𝑚 ∈ (1..^(𝑗 + 1))(𝐹𝑚))
158155, 157difeq12d 4136 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = (𝑗 + 1) → ((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)) = ((𝐹‘(𝑗 + 1)) ∖ 𝑚 ∈ (1..^(𝑗 + 1))(𝐹𝑚)))
159158fveq2d 6910 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = (𝑗 + 1) → (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))) = (vol‘((𝐹‘(𝑗 + 1)) ∖ 𝑚 ∈ (1..^(𝑗 + 1))(𝐹𝑚))))
160 fvex 6919 . . . . . . . . . . . . . . . . . . . 20 (vol‘((𝐹‘(𝑗 + 1)) ∖ 𝑚 ∈ (1..^(𝑗 + 1))(𝐹𝑚))) ∈ V
161159, 34, 160fvmpt 7015 . . . . . . . . . . . . . . . . . . 19 ((𝑗 + 1) ∈ ℕ → ((𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))‘(𝑗 + 1)) = (vol‘((𝐹‘(𝑗 + 1)) ∖ 𝑚 ∈ (1..^(𝑗 + 1))(𝐹𝑚))))
162105, 161syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))‘(𝑗 + 1)) = (vol‘((𝐹‘(𝑗 + 1)) ∖ 𝑚 ∈ (1..^(𝑗 + 1))(𝐹𝑚))))
163154, 162eqtr4d 2777 . . . . . . . . . . . . . . . . 17 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗))) = ((𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))‘(𝑗 + 1)))
164163oveq2d 7446 . . . . . . . . . . . . . . . 16 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((vol‘(𝐹𝑗)) + (vol‘((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗)))) = ((vol‘(𝐹𝑗)) + ((𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))‘(𝑗 + 1))))
165101, 130, 1643eqtrd 2778 . . . . . . . . . . . . . . 15 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (vol‘(𝐹‘(𝑗 + 1))) = ((vol‘(𝐹𝑗)) + ((𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))‘(𝑗 + 1))))
16690, 165eqeq12d 2750 . . . . . . . . . . . . . 14 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘(𝑗 + 1)) = (vol‘(𝐹‘(𝑗 + 1))) ↔ ((seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘𝑗) + ((𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))‘(𝑗 + 1))) = ((vol‘(𝐹𝑗)) + ((𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))‘(𝑗 + 1)))))
16787, 166imbitrrid 246 . . . . . . . . . . . . 13 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘𝑗) = (vol‘(𝐹𝑗)) → (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘(𝑗 + 1)) = (vol‘(𝐹‘(𝑗 + 1)))))
168167expcom 413 . . . . . . . . . . . 12 (𝑗 ∈ ℕ → (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → ((seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘𝑗) = (vol‘(𝐹𝑗)) → (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘(𝑗 + 1)) = (vol‘(𝐹‘(𝑗 + 1))))))
169168a2d 29 . . . . . . . . . . 11 (𝑗 ∈ ℕ → ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘𝑗) = (vol‘(𝐹𝑗))) → (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘(𝑗 + 1)) = (vol‘(𝐹‘(𝑗 + 1))))))
17058, 62, 66, 62, 86, 169nnind 12281 . . . . . . . . . 10 (𝑗 ∈ ℕ → (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘𝑗) = (vol‘(𝐹𝑗))))
171170impcom 407 . . . . . . . . 9 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘𝑗) = (vol‘(𝐹𝑗)))
172 fvco3 7007 . . . . . . . . . 10 ((𝐹:ℕ⟶dom vol ∧ 𝑗 ∈ ℕ) → ((vol ∘ 𝐹)‘𝑗) = (vol‘(𝐹𝑗)))
17351, 172sylan 580 . . . . . . . . 9 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((vol ∘ 𝐹)‘𝑗) = (vol‘(𝐹𝑗)))
174171, 173eqtr4d 2777 . . . . . . . 8 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘𝑗) = ((vol ∘ 𝐹)‘𝑗))
17549, 54, 174eqfnfvd 7053 . . . . . . 7 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))) = (vol ∘ 𝐹))
176175rneqd 5951 . . . . . 6 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → ran seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))) = ran (vol ∘ 𝐹))
177 rnco2 6274 . . . . . 6 ran (vol ∘ 𝐹) = (vol “ ran 𝐹)
178176, 177eqtrdi 2790 . . . . 5 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → ran seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))) = (vol “ ran 𝐹))
179178supeq1d 9483 . . . 4 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → sup(ran seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))), ℝ*, < ) = sup((vol “ ran 𝐹), ℝ*, < ))
18036, 43, 1793eqtr3d 2782 . . 3 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → (vol‘ ran 𝐹) = sup((vol “ ran 𝐹), ℝ*, < ))
181180ex 412 . 2 ((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) → (∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ → (vol‘ ran 𝐹) = sup((vol “ ran 𝐹), ℝ*, < )))
182 rexnal 3097 . . 3 (∃𝑘 ∈ ℕ ¬ (vol‘(𝐹𝑘)) ∈ ℝ ↔ ¬ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ)
183 fniunfv 7266 . . . . . . . . . . . 12 (𝐹 Fn ℕ → 𝑛 ∈ ℕ (𝐹𝑛) = ran 𝐹)
18438, 183syl 17 . . . . . . . . . . 11 (𝐹:ℕ⟶dom vol → 𝑛 ∈ ℕ (𝐹𝑛) = ran 𝐹)
185 ffvelcdm 7100 . . . . . . . . . . . . 13 ((𝐹:ℕ⟶dom vol ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ∈ dom vol)
186185ralrimiva 3143 . . . . . . . . . . . 12 (𝐹:ℕ⟶dom vol → ∀𝑛 ∈ ℕ (𝐹𝑛) ∈ dom vol)
187 iunmbl 25601 . . . . . . . . . . . 12 (∀𝑛 ∈ ℕ (𝐹𝑛) ∈ dom vol → 𝑛 ∈ ℕ (𝐹𝑛) ∈ dom vol)
188186, 187syl 17 . . . . . . . . . . 11 (𝐹:ℕ⟶dom vol → 𝑛 ∈ ℕ (𝐹𝑛) ∈ dom vol)
189184, 188eqeltrrd 2839 . . . . . . . . . 10 (𝐹:ℕ⟶dom vol → ran 𝐹 ∈ dom vol)
190189ad2antrr 726 . . . . . . . . 9 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → ran 𝐹 ∈ dom vol)
191 mblss 25579 . . . . . . . . 9 ( ran 𝐹 ∈ dom vol → ran 𝐹 ⊆ ℝ)
192190, 191syl 17 . . . . . . . 8 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → ran 𝐹 ⊆ ℝ)
193 ovolcl 25526 . . . . . . . 8 ( ran 𝐹 ⊆ ℝ → (vol*‘ ran 𝐹) ∈ ℝ*)
194192, 193syl 17 . . . . . . 7 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol*‘ ran 𝐹) ∈ ℝ*)
195 pnfge 13169 . . . . . . 7 ((vol*‘ ran 𝐹) ∈ ℝ* → (vol*‘ ran 𝐹) ≤ +∞)
196194, 195syl 17 . . . . . 6 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol*‘ ran 𝐹) ≤ +∞)
197 simprr 773 . . . . . . . . 9 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → ¬ (vol‘(𝐹𝑘)) ∈ ℝ)
1981ad2ant2r 747 . . . . . . . . . . . . 13 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (𝐹𝑘) ∈ dom vol)
199198, 18syl 17 . . . . . . . . . . . 12 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (𝐹𝑘) ⊆ ℝ)
200 ovolcl 25526 . . . . . . . . . . . 12 ((𝐹𝑘) ⊆ ℝ → (vol*‘(𝐹𝑘)) ∈ ℝ*)
201199, 200syl 17 . . . . . . . . . . 11 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol*‘(𝐹𝑘)) ∈ ℝ*)
202 xrrebnd 13206 . . . . . . . . . . 11 ((vol*‘(𝐹𝑘)) ∈ ℝ* → ((vol*‘(𝐹𝑘)) ∈ ℝ ↔ (-∞ < (vol*‘(𝐹𝑘)) ∧ (vol*‘(𝐹𝑘)) < +∞)))
203201, 202syl 17 . . . . . . . . . 10 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → ((vol*‘(𝐹𝑘)) ∈ ℝ ↔ (-∞ < (vol*‘(𝐹𝑘)) ∧ (vol*‘(𝐹𝑘)) < +∞)))
204198, 20syl 17 . . . . . . . . . . 11 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol‘(𝐹𝑘)) = (vol*‘(𝐹𝑘)))
205204eleq1d 2823 . . . . . . . . . 10 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → ((vol‘(𝐹𝑘)) ∈ ℝ ↔ (vol*‘(𝐹𝑘)) ∈ ℝ))
206 ovolge0 25529 . . . . . . . . . . . . 13 ((𝐹𝑘) ⊆ ℝ → 0 ≤ (vol*‘(𝐹𝑘)))
207 mnflt0 13164 . . . . . . . . . . . . . 14 -∞ < 0
208 mnfxr 11315 . . . . . . . . . . . . . . 15 -∞ ∈ ℝ*
209 0xr 11305 . . . . . . . . . . . . . . 15 0 ∈ ℝ*
210 xrltletr 13195 . . . . . . . . . . . . . . 15 ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ* ∧ (vol*‘(𝐹𝑘)) ∈ ℝ*) → ((-∞ < 0 ∧ 0 ≤ (vol*‘(𝐹𝑘))) → -∞ < (vol*‘(𝐹𝑘))))
211208, 209, 210mp3an12 1450 . . . . . . . . . . . . . 14 ((vol*‘(𝐹𝑘)) ∈ ℝ* → ((-∞ < 0 ∧ 0 ≤ (vol*‘(𝐹𝑘))) → -∞ < (vol*‘(𝐹𝑘))))
212207, 211mpani 696 . . . . . . . . . . . . 13 ((vol*‘(𝐹𝑘)) ∈ ℝ* → (0 ≤ (vol*‘(𝐹𝑘)) → -∞ < (vol*‘(𝐹𝑘))))
213200, 206, 212sylc 65 . . . . . . . . . . . 12 ((𝐹𝑘) ⊆ ℝ → -∞ < (vol*‘(𝐹𝑘)))
214199, 213syl 17 . . . . . . . . . . 11 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → -∞ < (vol*‘(𝐹𝑘)))
215214biantrurd 532 . . . . . . . . . 10 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → ((vol*‘(𝐹𝑘)) < +∞ ↔ (-∞ < (vol*‘(𝐹𝑘)) ∧ (vol*‘(𝐹𝑘)) < +∞)))
216203, 205, 2153bitr4d 311 . . . . . . . . 9 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → ((vol‘(𝐹𝑘)) ∈ ℝ ↔ (vol*‘(𝐹𝑘)) < +∞))
217197, 216mtbid 324 . . . . . . . 8 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → ¬ (vol*‘(𝐹𝑘)) < +∞)
218 nltpnft 13202 . . . . . . . . 9 ((vol*‘(𝐹𝑘)) ∈ ℝ* → ((vol*‘(𝐹𝑘)) = +∞ ↔ ¬ (vol*‘(𝐹𝑘)) < +∞))
219201, 218syl 17 . . . . . . . 8 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → ((vol*‘(𝐹𝑘)) = +∞ ↔ ¬ (vol*‘(𝐹𝑘)) < +∞))
220217, 219mpbird 257 . . . . . . 7 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol*‘(𝐹𝑘)) = +∞)
22138ad2antrr 726 . . . . . . . . . 10 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → 𝐹 Fn ℕ)
222 simprl 771 . . . . . . . . . 10 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → 𝑘 ∈ ℕ)
223 fnfvelrn 7099 . . . . . . . . . 10 ((𝐹 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝐹𝑘) ∈ ran 𝐹)
224221, 222, 223syl2anc 584 . . . . . . . . 9 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (𝐹𝑘) ∈ ran 𝐹)
225 elssuni 4941 . . . . . . . . 9 ((𝐹𝑘) ∈ ran 𝐹 → (𝐹𝑘) ⊆ ran 𝐹)
226224, 225syl 17 . . . . . . . 8 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (𝐹𝑘) ⊆ ran 𝐹)
227 ovolss 25533 . . . . . . . 8 (((𝐹𝑘) ⊆ ran 𝐹 ran 𝐹 ⊆ ℝ) → (vol*‘(𝐹𝑘)) ≤ (vol*‘ ran 𝐹))
228226, 192, 227syl2anc 584 . . . . . . 7 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol*‘(𝐹𝑘)) ≤ (vol*‘ ran 𝐹))
229220, 228eqbrtrrd 5171 . . . . . 6 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → +∞ ≤ (vol*‘ ran 𝐹))
230 pnfxr 11312 . . . . . . 7 +∞ ∈ ℝ*
231 xrletri3 13192 . . . . . . 7 (((vol*‘ ran 𝐹) ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((vol*‘ ran 𝐹) = +∞ ↔ ((vol*‘ ran 𝐹) ≤ +∞ ∧ +∞ ≤ (vol*‘ ran 𝐹))))
232194, 230, 231sylancl 586 . . . . . 6 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → ((vol*‘ ran 𝐹) = +∞ ↔ ((vol*‘ ran 𝐹) ≤ +∞ ∧ +∞ ≤ (vol*‘ ran 𝐹))))
233196, 229, 232mpbir2and 713 . . . . 5 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol*‘ ran 𝐹) = +∞)
234 mblvol 25578 . . . . . 6 ( ran 𝐹 ∈ dom vol → (vol‘ ran 𝐹) = (vol*‘ ran 𝐹))
235190, 234syl 17 . . . . 5 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol‘ ran 𝐹) = (vol*‘ ran 𝐹))
236 imassrn 6090 . . . . . . 7 (vol “ ran 𝐹) ⊆ ran vol
237 frn 6743 . . . . . . . . 9 (vol:dom vol⟶(0[,]+∞) → ran vol ⊆ (0[,]+∞))
23850, 237ax-mp 5 . . . . . . . 8 ran vol ⊆ (0[,]+∞)
239 iccssxr 13466 . . . . . . . 8 (0[,]+∞) ⊆ ℝ*
240238, 239sstri 4004 . . . . . . 7 ran vol ⊆ ℝ*
241236, 240sstri 4004 . . . . . 6 (vol “ ran 𝐹) ⊆ ℝ*
242204, 220eqtrd 2774 . . . . . . 7 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol‘(𝐹𝑘)) = +∞)
243 simpll 767 . . . . . . . 8 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → 𝐹:ℕ⟶dom vol)
244 ffun 6739 . . . . . . . . . 10 (vol:dom vol⟶(0[,]+∞) → Fun vol)
24550, 244ax-mp 5 . . . . . . . . 9 Fun vol
246 frn 6743 . . . . . . . . 9 (𝐹:ℕ⟶dom vol → ran 𝐹 ⊆ dom vol)
247 funfvima2 7250 . . . . . . . . 9 ((Fun vol ∧ ran 𝐹 ⊆ dom vol) → ((𝐹𝑘) ∈ ran 𝐹 → (vol‘(𝐹𝑘)) ∈ (vol “ ran 𝐹)))
248245, 246, 247sylancr 587 . . . . . . . 8 (𝐹:ℕ⟶dom vol → ((𝐹𝑘) ∈ ran 𝐹 → (vol‘(𝐹𝑘)) ∈ (vol “ ran 𝐹)))
249243, 224, 248sylc 65 . . . . . . 7 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol‘(𝐹𝑘)) ∈ (vol “ ran 𝐹))
250242, 249eqeltrrd 2839 . . . . . 6 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → +∞ ∈ (vol “ ran 𝐹))
251 supxrpnf 13356 . . . . . 6 (((vol “ ran 𝐹) ⊆ ℝ* ∧ +∞ ∈ (vol “ ran 𝐹)) → sup((vol “ ran 𝐹), ℝ*, < ) = +∞)
252241, 250, 251sylancr 587 . . . . 5 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → sup((vol “ ran 𝐹), ℝ*, < ) = +∞)
253233, 235, 2523eqtr4d 2784 . . . 4 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol‘ ran 𝐹) = sup((vol “ ran 𝐹), ℝ*, < ))
254253rexlimdvaa 3153 . . 3 ((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) → (∃𝑘 ∈ ℕ ¬ (vol‘(𝐹𝑘)) ∈ ℝ → (vol‘ ran 𝐹) = sup((vol “ ran 𝐹), ℝ*, < )))
255182, 254biimtrrid 243 . 2 ((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) → (¬ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ → (vol‘ ran 𝐹) = sup((vol “ ran 𝐹), ℝ*, < )))
256181, 255pm2.61d 179 1 ((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) → (vol‘ ran 𝐹) = sup((vol “ ran 𝐹), ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wral 3058  wrex 3067  cdif 3959  cun 3960  cin 3961  wss 3962  c0 4338   cuni 4911   ciun 4995  Disj wdisj 5114   class class class wbr 5147  cmpt 5230  dom cdm 5688  ran crn 5689  cima 5691  ccom 5692  Fun wfun 6556   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  Fincfn 8983  supcsup 9477  cr 11151  0cc0 11152  1c1 11153   + caddc 11155  +∞cpnf 11289  -∞cmnf 11290  *cxr 11291   < clt 11292  cle 11293  cn 12263  cz 12610  cuz 12875  [,]cicc 13386  ...cfz 13543  ..^cfzo 13690  seqcseq 14038  vol*covol 25510  volcvol 25511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cc 10472  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-disj 5115  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-map 8866  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-oi 9547  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-q 12988  df-rp 13032  df-xadd 13152  df-ioo 13387  df-ico 13389  df-icc 13390  df-fz 13544  df-fzo 13691  df-fl 13828  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-rlim 15521  df-sum 15719  df-xmet 21374  df-met 21375  df-ovol 25512  df-vol 25513
This theorem is referenced by:  volsup2  25653  itg1climres  25763  itg2gt0  25809
  Copyright terms: Public domain W3C validator