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Theorem volsup 23771
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 ffvelrn 6623 . . . . . . . . . . 11 ((𝐹:ℕ⟶dom vol ∧ 𝑘 ∈ ℕ) → (𝐹𝑘) ∈ dom vol)
21ad2ant2r 737 . . . . . . . . . 10 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ (vol‘(𝐹𝑘)) ∈ ℝ)) → (𝐹𝑘) ∈ dom vol)
3 fzofi 13097 . . . . . . . . . . 11 (1..^𝑘) ∈ Fin
4 simpll 757 . . . . . . . . . . . . 13 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ (vol‘(𝐹𝑘)) ∈ ℝ)) → 𝐹:ℕ⟶dom vol)
5 elfzouz 12798 . . . . . . . . . . . . . 14 (𝑚 ∈ (1..^𝑘) → 𝑚 ∈ (ℤ‘1))
6 nnuz 12034 . . . . . . . . . . . . . 14 ℕ = (ℤ‘1)
75, 6syl6eleqr 2870 . . . . . . . . . . . . 13 (𝑚 ∈ (1..^𝑘) → 𝑚 ∈ ℕ)
8 ffvelrn 6623 . . . . . . . . . . . . 13 ((𝐹:ℕ⟶dom vol ∧ 𝑚 ∈ ℕ) → (𝐹𝑚) ∈ dom vol)
94, 7, 8syl2an 589 . . . . . . . . . . . 12 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ (vol‘(𝐹𝑘)) ∈ ℝ)) ∧ 𝑚 ∈ (1..^𝑘)) → (𝐹𝑚) ∈ dom vol)
109ralrimiva 3148 . . . . . . . . . . 11 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ (vol‘(𝐹𝑘)) ∈ ℝ)) → ∀𝑚 ∈ (1..^𝑘)(𝐹𝑚) ∈ dom vol)
11 finiunmbl 23759 . . . . . . . . . . 11 (((1..^𝑘) ∈ Fin ∧ ∀𝑚 ∈ (1..^𝑘)(𝐹𝑚) ∈ dom vol) → 𝑚 ∈ (1..^𝑘)(𝐹𝑚) ∈ dom vol)
123, 10, 11sylancr 581 . . . . . . . . . 10 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ (vol‘(𝐹𝑘)) ∈ ℝ)) → 𝑚 ∈ (1..^𝑘)(𝐹𝑚) ∈ dom vol)
13 difmbl 23758 . . . . . . . . . 10 (((𝐹𝑘) ∈ dom vol ∧ 𝑚 ∈ (1..^𝑘)(𝐹𝑚) ∈ dom vol) → ((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)) ∈ dom vol)
142, 12, 13syl2anc 579 . . . . . . . . 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 23745 . . . . . . . . . . 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 3961 . . . . . . . . . . 11 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ (vol‘(𝐹𝑘)) ∈ ℝ)) → ((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)) ⊆ (𝐹𝑘))
18 mblss 23746 . . . . . . . . . . . 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 23745 . . . . . . . . . . . . 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 763 . . . . . . . . . . . 12 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol‘(𝐹𝑘)) ∈ ℝ)
2321, 22eqeltrrd 2860 . . . . . . . . . . 11 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol*‘(𝐹𝑘)) ∈ ℝ)
24 ovolsscl 23701 . . . . . . . . . . 11 ((((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)) ⊆ (𝐹𝑘) ∧ (𝐹𝑘) ⊆ ℝ ∧ (vol*‘(𝐹𝑘)) ∈ ℝ) → (vol*‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))) ∈ ℝ)
2517, 19, 23, 24syl3anc 1439 . . . . . . . . . 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 2859 . . . . . . . . 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 507 . . . . . . . 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 450 . . . . . . 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 3144 . . . . . 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 397 . . . . 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 6448 . . . . . 6 (𝑘 = 𝑚 → (𝐹𝑘) = (𝐹𝑚))
3231iundisj2 23764 . . . . 5 Disj 𝑘 ∈ ℕ ((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))
33 eqid 2778 . . . . . 6 seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))) = seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))
34 eqid 2778 . . . . . 6 (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))) = (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))
3533, 34voliun 23769 . . . . 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 580 . . . 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 23763 . . . . . 6 𝑘 ∈ ℕ (𝐹𝑘) = 𝑘 ∈ ℕ ((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))
38 ffn 6293 . . . . . . . 8 (𝐹:ℕ⟶dom vol → 𝐹 Fn ℕ)
3938ad2antrr 716 . . . . . . 7 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → 𝐹 Fn ℕ)
40 fniunfv 6779 . . . . . . 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, 41syl5eqr 2828 . . . . 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 6452 . . . 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 11764 . . . . . . . . . . 11 1 ∈ ℤ
45 seqfn 13136 . . . . . . . . . . 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 6233 . . . . . . . . . 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 223 . . . . . . . . 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 23744 . . . . . . . . . 10 vol:dom vol⟶(0[,]+∞)
51 simpll 757 . . . . . . . . . 10 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → 𝐹:ℕ⟶dom vol)
52 fco 6310 . . . . . . . . . 10 ((vol:dom vol⟶(0[,]+∞) ∧ 𝐹:ℕ⟶dom vol) → (vol ∘ 𝐹):ℕ⟶(0[,]+∞))
5350, 51, 52sylancr 581 . . . . . . . . 9 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → (vol ∘ 𝐹):ℕ⟶(0[,]+∞))
5453ffnd 6294 . . . . . . . 8 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → (vol ∘ 𝐹) Fn ℕ)
55 fveq2 6448 . . . . . . . . . . . . 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 6453 . . . . . . . . . . . . 13 (𝑥 = 1 → (vol‘(𝐹𝑥)) = (vol‘(𝐹‘1)))
5755, 56eqeq12d 2793 . . . . . . . . . . . 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 332 . . . . . . . . . . 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 6448 . . . . . . . . . . . . 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 6453 . . . . . . . . . . . . 13 (𝑥 = 𝑗 → (vol‘(𝐹𝑥)) = (vol‘(𝐹𝑗)))
6159, 60eqeq12d 2793 . . . . . . . . . . . 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 332 . . . . . . . . . . 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 6448 . . . . . . . . . . . . 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 6453 . . . . . . . . . . . . 13 (𝑥 = (𝑗 + 1) → (vol‘(𝐹𝑥)) = (vol‘(𝐹‘(𝑗 + 1))))
6563, 64eqeq12d 2793 . . . . . . . . . . . 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 332 . . . . . . . . . . 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 13137 . . . . . . . . . . . . . 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 11392 . . . . . . . . . . . . . 14 1 ∈ ℕ
70 oveq2 6932 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 1 → (1..^𝑘) = (1..^1))
71 fzo0 12816 . . . . . . . . . . . . . . . . . . . . . 22 (1..^1) = ∅
7270, 71syl6eq 2830 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 1 → (1..^𝑘) = ∅)
7372iuneq1d 4780 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 1 → 𝑚 ∈ (1..^𝑘)(𝐹𝑚) = 𝑚 ∈ ∅ (𝐹𝑚))
74 0iun 4812 . . . . . . . . . . . . . . . . . . . 20 𝑚 ∈ ∅ (𝐹𝑚) = ∅
7573, 74syl6eq 2830 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 1 → 𝑚 ∈ (1..^𝑘)(𝐹𝑚) = ∅)
7675difeq2d 3951 . . . . . . . . . . . . . . . . . 18 (𝑘 = 1 → ((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)) = ((𝐹𝑘) ∖ ∅))
77 dif0 4181 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑘) ∖ ∅) = (𝐹𝑘)
7876, 77syl6eq 2830 . . . . . . . . . . . . . . . . 17 (𝑘 = 1 → ((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)) = (𝐹𝑘))
79 fveq2 6448 . . . . . . . . . . . . . . . . 17 (𝑘 = 1 → (𝐹𝑘) = (𝐹‘1))
8078, 79eqtrd 2814 . . . . . . . . . . . . . . . 16 (𝑘 = 1 → ((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)) = (𝐹‘1))
8180fveq2d 6452 . . . . . . . . . . . . . . 15 (𝑘 = 1 → (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))) = (vol‘(𝐹‘1)))
82 fvex 6461 . . . . . . . . . . . . . . 15 (vol‘(𝐹‘1)) ∈ V
8381, 34, 82fvmpt 6544 . . . . . . . . . . . . . 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 2802 . . . . . . . . . . . 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 6931 . . . . . . . . . . . . . 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 13139 . . . . . . . . . . . . . . . . 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 2877 . . . . . . . . . . . . . . . 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 475 . . . . . . . . . . . . . . 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 4268 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑗) ∪ ((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗))) = ((𝐹𝑗) ∪ (𝐹‘(𝑗 + 1)))
92 fveq2 6448 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑗 → (𝐹𝑛) = (𝐹𝑗))
93 fvoveq1 6947 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑗 → (𝐹‘(𝑛 + 1)) = (𝐹‘(𝑗 + 1)))
9492, 93sseq12d 3853 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑗 → ((𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)) ↔ (𝐹𝑗) ⊆ (𝐹‘(𝑗 + 1))))
95 simpllr 766 . . . . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . . . . 20 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
9794, 95, 96rspcdva 3517 . . . . . . . . . . . . . . . . . . 19 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (𝐹𝑗) ⊆ (𝐹‘(𝑗 + 1)))
98 ssequn1 4006 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝑗) ⊆ (𝐹‘(𝑗 + 1)) ↔ ((𝐹𝑗) ∪ (𝐹‘(𝑗 + 1))) = (𝐹‘(𝑗 + 1)))
9997, 98sylib 210 . . . . . . . . . . . . . . . . . 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, 99syl5req 2827 . . . . . . . . . . . . . . . . 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 6452 . . . . . . . . . . . . . . . 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 765 . . . . . . . . . . . . . . . . . 18 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → 𝐹:ℕ⟶dom vol)
103102, 96ffvelrnd 6626 . . . . . . . . . . . . . . . . 17 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (𝐹𝑗) ∈ dom vol)
104 peano2nn 11393 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
105104adantl 475 . . . . . . . . . . . . . . . . . . 19 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (𝑗 + 1) ∈ ℕ)
106102, 105ffvelrnd 6626 . . . . . . . . . . . . . . . . . 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 23758 . . . . . . . . . . . . . . . . . 18 (((𝐹‘(𝑗 + 1)) ∈ dom vol ∧ (𝐹𝑗) ∈ dom vol) → ((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗)) ∈ dom vol)
108106, 103, 107syl2anc 579 . . . . . . . . . . . . . . . . 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 4264 . . . . . . . . . . . . . . . . . 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 6453 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑗 → (vol‘(𝐹𝑘)) = (vol‘(𝐹𝑗)))
112111eleq1d 2844 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑗 → ((vol‘(𝐹𝑘)) ∈ ℝ ↔ (vol‘(𝐹𝑗)) ∈ ℝ))
113 simplr 759 . . . . . . . . . . . . . . . . . 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 3517 . . . . . . . . . . . . . . . . 17 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (vol‘(𝐹𝑗)) ∈ ℝ)
115 mblvol 23745 . . . . . . . . . . . . . . . . . . 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 3961 . . . . . . . . . . . . . . . . . . 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 23746 . . . . . . . . . . . . . . . . . . . 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 23745 . . . . . . . . . . . . . . . . . . . . 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 6453 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = (𝑗 + 1) → (vol‘(𝐹𝑘)) = (vol‘(𝐹‘(𝑗 + 1))))
123122eleq1d 2844 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = (𝑗 + 1) → ((vol‘(𝐹𝑘)) ∈ ℝ ↔ (vol‘(𝐹‘(𝑗 + 1))) ∈ ℝ))
124123, 113, 105rspcdva 3517 . . . . . . . . . . . . . . . . . . . 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 2860 . . . . . . . . . . . . . . . . . . 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 23701 . . . . . . . . . . . . . . . . . . 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 1439 . . . . . . . . . . . . . . . . . 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 2859 . . . . . . . . . . . . . . . . 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 23760 . . . . . . . . . . . . . . . . 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 1446 . . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . . . . . . . . . . . . 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 12692 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑚 ∈ (1...𝑗) → 𝑚 ∈ ℕ)
133132adantl 475 . . . . . . . . . . . . . . . . . . . . . . . . 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 12661 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑚 ∈ (1...𝑗) → 𝑗 ∈ (ℤ𝑚))
135134adantl 475 . . . . . . . . . . . . . . . . . . . . . . . . 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 23770 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)) ∧ (𝑚 ∈ ℕ ∧ 𝑗 ∈ (ℤ𝑚))) → (𝐹𝑚) ⊆ (𝐹𝑗))
137131, 133, 135, 136syl12anc 827 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑗)) → (𝐹𝑚) ⊆ (𝐹𝑗))
138137ralrimiva 3148 . . . . . . . . . . . . . . . . . . . . . . 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 4796 . . . . . . . . . . . . . . . . . . . . . . 23 ( 𝑚 ∈ (1...𝑗)(𝐹𝑚) ⊆ (𝐹𝑗) ↔ ∀𝑚 ∈ (1...𝑗)(𝐹𝑚) ⊆ (𝐹𝑗))
140138, 139sylibr 226 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → 𝑚 ∈ (1...𝑗)(𝐹𝑚) ⊆ (𝐹𝑗))
14196, 6syl6eleq 2869 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ (ℤ‘1))
142 eluzfz2 12671 . . . . . . . . . . . . . . . . . . . . . . . 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 6448 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 = 𝑗 → (𝐹𝑚) = (𝐹𝑗))
145144ssiun2s 4799 . . . . . . . . . . . . . . . . . . . . . . 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 3838 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → 𝑚 ∈ (1...𝑗)(𝐹𝑚) = (𝐹𝑗))
14896nnzd 11838 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℤ)
149 fzval3 12861 . . . . . . . . . . . . . . . . . . . . . . 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 4780 . . . . . . . . . . . . . . . . . . . . 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 2816 . . . . . . . . . . . . . . . . . . . 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 3951 . . . . . . . . . . . . . . . . . . 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 6452 . . . . . . . . . . . . . . . . . 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 6448 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = (𝑗 + 1) → (𝐹𝑘) = (𝐹‘(𝑗 + 1)))
156 oveq2 6932 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = (𝑗 + 1) → (1..^𝑘) = (1..^(𝑗 + 1)))
157156iuneq1d 4780 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = (𝑗 + 1) → 𝑚 ∈ (1..^𝑘)(𝐹𝑚) = 𝑚 ∈ (1..^(𝑗 + 1))(𝐹𝑚))
158155, 157difeq12d 3952 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = (𝑗 + 1) → ((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)) = ((𝐹‘(𝑗 + 1)) ∖ 𝑚 ∈ (1..^(𝑗 + 1))(𝐹𝑚)))
159158fveq2d 6452 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = (𝑗 + 1) → (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))) = (vol‘((𝐹‘(𝑗 + 1)) ∖ 𝑚 ∈ (1..^(𝑗 + 1))(𝐹𝑚))))
160 fvex 6461 . . . . . . . . . . . . . . . . . . . 20 (vol‘((𝐹‘(𝑗 + 1)) ∖ 𝑚 ∈ (1..^(𝑗 + 1))(𝐹𝑚))) ∈ V
161159, 34, 160fvmpt 6544 . . . . . . . . . . . . . . . . . . 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 2817 . . . . . . . . . . . . . . . . 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 6940 . . . . . . . . . . . . . . . 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 2818 . . . . . . . . . . . . . . 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 2793 . . . . . . . . . . . . . 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, 166syl5ibr 238 . . . . . . . . . . . . 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 404 . . . . . . . . . . . 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 11399 . . . . . . . . . 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 398 . . . . . . . . 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 6537 . . . . . . . . . 10 ((𝐹:ℕ⟶dom vol ∧ 𝑗 ∈ ℕ) → ((vol ∘ 𝐹)‘𝑗) = (vol‘(𝐹𝑗)))
17351, 172sylan 575 . . . . . . . . 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 2817 . . . . . . . 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 6579 . . . . . . 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 5600 . . . . . 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 5898 . . . . . 6 ran (vol ∘ 𝐹) = (vol “ ran 𝐹)
178176, 177syl6eq 2830 . . . . 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 8642 . . . 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 2822 . . 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 403 . 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 3176 . . 3 (∃𝑘 ∈ ℕ ¬ (vol‘(𝐹𝑘)) ∈ ℝ ↔ ¬ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ)
183 fniunfv 6779 . . . . . . . . . . . 12 (𝐹 Fn ℕ → 𝑛 ∈ ℕ (𝐹𝑛) = ran 𝐹)
18438, 183syl 17 . . . . . . . . . . 11 (𝐹:ℕ⟶dom vol → 𝑛 ∈ ℕ (𝐹𝑛) = ran 𝐹)
185 ffvelrn 6623 . . . . . . . . . . . . 13 ((𝐹:ℕ⟶dom vol ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ∈ dom vol)
186185ralrimiva 3148 . . . . . . . . . . . 12 (𝐹:ℕ⟶dom vol → ∀𝑛 ∈ ℕ (𝐹𝑛) ∈ dom vol)
187 iunmbl 23768 . . . . . . . . . . . 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 2860 . . . . . . . . . 10 (𝐹:ℕ⟶dom vol → ran 𝐹 ∈ dom vol)
190189ad2antrr 716 . . . . . . . . 9 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → ran 𝐹 ∈ dom vol)
191 mblss 23746 . . . . . . . . 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 23693 . . . . . . . 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 12280 . . . . . . 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 763 . . . . . . . . 9 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → ¬ (vol‘(𝐹𝑘)) ∈ ℝ)
1981ad2ant2r 737 . . . . . . . . . . . . 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 23693 . . . . . . . . . . . 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 12316 . . . . . . . . . . 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 2844 . . . . . . . . . 10 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → ((vol‘(𝐹𝑘)) ∈ ℝ ↔ (vol*‘(𝐹𝑘)) ∈ ℝ))
206 ovolge0 23696 . . . . . . . . . . . . 13 ((𝐹𝑘) ⊆ ℝ → 0 ≤ (vol*‘(𝐹𝑘)))
207 mnflt0 12275 . . . . . . . . . . . . . 14 -∞ < 0
208 mnfxr 10436 . . . . . . . . . . . . . . 15 -∞ ∈ ℝ*
209 0xr 10425 . . . . . . . . . . . . . . 15 0 ∈ ℝ*
210 xrltletr 12305 . . . . . . . . . . . . . . 15 ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ* ∧ (vol*‘(𝐹𝑘)) ∈ ℝ*) → ((-∞ < 0 ∧ 0 ≤ (vol*‘(𝐹𝑘))) → -∞ < (vol*‘(𝐹𝑘))))
211208, 209, 210mp3an12 1524 . . . . . . . . . . . . . 14 ((vol*‘(𝐹𝑘)) ∈ ℝ* → ((-∞ < 0 ∧ 0 ≤ (vol*‘(𝐹𝑘))) → -∞ < (vol*‘(𝐹𝑘))))
212207, 211mpani 686 . . . . . . . . . . . . 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 528 . . . . . . . . . 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 303 . . . . . . . . 9 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → ((vol‘(𝐹𝑘)) ∈ ℝ ↔ (vol*‘(𝐹𝑘)) < +∞))
217197, 216mtbid 316 . . . . . . . 8 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → ¬ (vol*‘(𝐹𝑘)) < +∞)
218 nltpnft 12312 . . . . . . . . 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 249 . . . . . . 7 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol*‘(𝐹𝑘)) = +∞)
22138ad2antrr 716 . . . . . . . . . 10 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → 𝐹 Fn ℕ)
222 simprl 761 . . . . . . . . . 10 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → 𝑘 ∈ ℕ)
223 fnfvelrn 6622 . . . . . . . . . 10 ((𝐹 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝐹𝑘) ∈ ran 𝐹)
224221, 222, 223syl2anc 579 . . . . . . . . 9 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (𝐹𝑘) ∈ ran 𝐹)
225 elssuni 4704 . . . . . . . . 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 23700 . . . . . . . 8 (((𝐹𝑘) ⊆ ran 𝐹 ran 𝐹 ⊆ ℝ) → (vol*‘(𝐹𝑘)) ≤ (vol*‘ ran 𝐹))
228226, 192, 227syl2anc 579 . . . . . . 7 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol*‘(𝐹𝑘)) ≤ (vol*‘ ran 𝐹))
229220, 228eqbrtrrd 4912 . . . . . 6 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → +∞ ≤ (vol*‘ ran 𝐹))
230 pnfxr 10432 . . . . . . 7 +∞ ∈ ℝ*
231 xrletri3 12302 . . . . . . 7 (((vol*‘ ran 𝐹) ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((vol*‘ ran 𝐹) = +∞ ↔ ((vol*‘ ran 𝐹) ≤ +∞ ∧ +∞ ≤ (vol*‘ ran 𝐹))))
232194, 230, 231sylancl 580 . . . . . 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 703 . . . . 5 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol*‘ ran 𝐹) = +∞)
234 mblvol 23745 . . . . . 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 5733 . . . . . . 7 (vol “ ran 𝐹) ⊆ ran vol
237 frn 6299 . . . . . . . . 9 (vol:dom vol⟶(0[,]+∞) → ran vol ⊆ (0[,]+∞))
23850, 237ax-mp 5 . . . . . . . 8 ran vol ⊆ (0[,]+∞)
239 iccssxr 12573 . . . . . . . 8 (0[,]+∞) ⊆ ℝ*
240238, 239sstri 3830 . . . . . . 7 ran vol ⊆ ℝ*
241236, 240sstri 3830 . . . . . 6 (vol “ ran 𝐹) ⊆ ℝ*
242204, 220eqtrd 2814 . . . . . . 7 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol‘(𝐹𝑘)) = +∞)
243 simpll 757 . . . . . . . 8 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → 𝐹:ℕ⟶dom vol)
244 ffun 6296 . . . . . . . . . 10 (vol:dom vol⟶(0[,]+∞) → Fun vol)
24550, 244ax-mp 5 . . . . . . . . 9 Fun vol
246 frn 6299 . . . . . . . . 9 (𝐹:ℕ⟶dom vol → ran 𝐹 ⊆ dom vol)
247 funfvima2 6767 . . . . . . . . 9 ((Fun vol ∧ ran 𝐹 ⊆ dom vol) → ((𝐹𝑘) ∈ ran 𝐹 → (vol‘(𝐹𝑘)) ∈ (vol “ ran 𝐹)))
248245, 246, 247sylancr 581 . . . . . . . 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 2860 . . . . . 6 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → +∞ ∈ (vol “ ran 𝐹))
251 supxrpnf 12465 . . . . . 6 (((vol “ ran 𝐹) ⊆ ℝ* ∧ +∞ ∈ (vol “ ran 𝐹)) → sup((vol “ ran 𝐹), ℝ*, < ) = +∞)
252241, 250, 251sylancr 581 . . . . 5 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → sup((vol “ ran 𝐹), ℝ*, < ) = +∞)
253233, 235, 2523eqtr4d 2824 . . . 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 3214 . . 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, 254syl5bir 235 . 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 172 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 198  wa 386   = wceq 1601  wcel 2107  wral 3090  wrex 3091  cdif 3789  cun 3790  cin 3791  wss 3792  c0 4141   cuni 4673   ciun 4755  Disj wdisj 4856   class class class wbr 4888  cmpt 4967  dom cdm 5357  ran crn 5358  cima 5360  ccom 5361  Fun wfun 6131   Fn wfn 6132  wf 6133  cfv 6137  (class class class)co 6924  Fincfn 8243  supcsup 8636  cr 10273  0cc0 10274  1c1 10275   + caddc 10277  +∞cpnf 10410  -∞cmnf 10411  *cxr 10412   < clt 10413  cle 10414  cn 11379  cz 11733  cuz 11997  [,]cicc 12495  ...cfz 12648  ..^cfzo 12789  seqcseq 13124  vol*covol 23677  volcvol 23678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-inf2 8837  ax-cc 9594  ax-cnex 10330  ax-resscn 10331  ax-1cn 10332  ax-icn 10333  ax-addcl 10334  ax-addrcl 10335  ax-mulcl 10336  ax-mulrcl 10337  ax-mulcom 10338  ax-addass 10339  ax-mulass 10340  ax-distr 10341  ax-i2m1 10342  ax-1ne0 10343  ax-1rid 10344  ax-rnegex 10345  ax-rrecex 10346  ax-cnre 10347  ax-pre-lttri 10348  ax-pre-lttrn 10349  ax-pre-ltadd 10350  ax-pre-mulgt0 10351  ax-pre-sup 10352
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-disj 4857  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-se 5317  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-isom 6146  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-of 7176  df-om 7346  df-1st 7447  df-2nd 7448  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-2o 7846  df-oadd 7849  df-er 8028  df-map 8144  df-pm 8145  df-en 8244  df-dom 8245  df-sdom 8246  df-fin 8247  df-sup 8638  df-inf 8639  df-oi 8706  df-card 9100  df-cda 9327  df-pnf 10415  df-mnf 10416  df-xr 10417  df-ltxr 10418  df-le 10419  df-sub 10610  df-neg 10611  df-div 11036  df-nn 11380  df-2 11443  df-3 11444  df-n0 11648  df-z 11734  df-uz 11998  df-q 12101  df-rp 12143  df-xadd 12263  df-ioo 12496  df-ico 12498  df-icc 12499  df-fz 12649  df-fzo 12790  df-fl 12917  df-seq 13125  df-exp 13184  df-hash 13442  df-cj 14252  df-re 14253  df-im 14254  df-sqrt 14388  df-abs 14389  df-clim 14636  df-rlim 14637  df-sum 14834  df-xmet 20146  df-met 20147  df-ovol 23679  df-vol 23680
This theorem is referenced by:  volsup2  23820  itg1climres  23929  itg2gt0  23975
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