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Theorem volsup 23544
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 6521 . . . . . . . . . . 11 ((𝐹:ℕ⟶dom vol ∧ 𝑘 ∈ ℕ) → (𝐹𝑘) ∈ dom vol)
21ad2ant2r 800 . . . . . . . . . 10 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ (vol‘(𝐹𝑘)) ∈ ℝ)) → (𝐹𝑘) ∈ dom vol)
3 fzofi 12987 . . . . . . . . . . 11 (1..^𝑘) ∈ Fin
4 simpll 807 . . . . . . . . . . . . 13 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ (vol‘(𝐹𝑘)) ∈ ℝ)) → 𝐹:ℕ⟶dom vol)
5 elfzouz 12688 . . . . . . . . . . . . . 14 (𝑚 ∈ (1..^𝑘) → 𝑚 ∈ (ℤ‘1))
6 nnuz 11936 . . . . . . . . . . . . . 14 ℕ = (ℤ‘1)
75, 6syl6eleqr 2850 . . . . . . . . . . . . 13 (𝑚 ∈ (1..^𝑘) → 𝑚 ∈ ℕ)
8 ffvelrn 6521 . . . . . . . . . . . . 13 ((𝐹:ℕ⟶dom vol ∧ 𝑚 ∈ ℕ) → (𝐹𝑚) ∈ dom vol)
94, 7, 8syl2an 495 . . . . . . . . . . . 12 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ (vol‘(𝐹𝑘)) ∈ ℝ)) ∧ 𝑚 ∈ (1..^𝑘)) → (𝐹𝑚) ∈ dom vol)
109ralrimiva 3104 . . . . . . . . . . 11 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ (vol‘(𝐹𝑘)) ∈ ℝ)) → ∀𝑚 ∈ (1..^𝑘)(𝐹𝑚) ∈ dom vol)
11 finiunmbl 23532 . . . . . . . . . . 11 (((1..^𝑘) ∈ Fin ∧ ∀𝑚 ∈ (1..^𝑘)(𝐹𝑚) ∈ dom vol) → 𝑚 ∈ (1..^𝑘)(𝐹𝑚) ∈ dom vol)
123, 10, 11sylancr 698 . . . . . . . . . 10 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ (vol‘(𝐹𝑘)) ∈ ℝ)) → 𝑚 ∈ (1..^𝑘)(𝐹𝑚) ∈ dom vol)
13 difmbl 23531 . . . . . . . . . 10 (((𝐹𝑘) ∈ dom vol ∧ 𝑚 ∈ (1..^𝑘)(𝐹𝑚) ∈ dom vol) → ((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)) ∈ dom vol)
142, 12, 13syl2anc 696 . . . . . . . . 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 23518 . . . . . . . . . . 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 3881 . . . . . . . . . . 11 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ (vol‘(𝐹𝑘)) ∈ ℝ)) → ((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)) ⊆ (𝐹𝑘))
18 mblss 23519 . . . . . . . . . . . 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 23518 . . . . . . . . . . . . 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 813 . . . . . . . . . . . 12 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol‘(𝐹𝑘)) ∈ ℝ)
2321, 22eqeltrrd 2840 . . . . . . . . . . 11 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol*‘(𝐹𝑘)) ∈ ℝ)
24 ovolsscl 23474 . . . . . . . . . . 11 ((((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)) ⊆ (𝐹𝑘) ∧ (𝐹𝑘) ⊆ ℝ ∧ (vol*‘(𝐹𝑘)) ∈ ℝ) → (vol*‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))) ∈ ℝ)
2517, 19, 23, 24syl3anc 1477 . . . . . . . . . 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 2839 . . . . . . . . 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 555 . . . . . . . 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 644 . . . . . . 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 3100 . . . . . 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 444 . . . . 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 6353 . . . . . 6 (𝑘 = 𝑚 → (𝐹𝑘) = (𝐹𝑚))
3231iundisj2 23537 . . . . 5 Disj 𝑘 ∈ ℕ ((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))
33 eqid 2760 . . . . . 6 seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))) = seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))
34 eqid 2760 . . . . . 6 (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))) = (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))
3533, 34voliun 23542 . . . . 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 697 . . . 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 23536 . . . . . 6 𝑘 ∈ ℕ (𝐹𝑘) = 𝑘 ∈ ℕ ((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))
38 ffn 6206 . . . . . . . 8 (𝐹:ℕ⟶dom vol → 𝐹 Fn ℕ)
3938ad2antrr 764 . . . . . . 7 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → 𝐹 Fn ℕ)
40 fniunfv 6669 . . . . . . 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 2808 . . . . 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 6357 . . . 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 11619 . . . . . . . . . . 11 1 ∈ ℤ
45 seqfn 13027 . . . . . . . . . . 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 6147 . . . . . . . . . 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 221 . . . . . . . . 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 23517 . . . . . . . . . 10 vol:dom vol⟶(0[,]+∞)
51 simpll 807 . . . . . . . . . 10 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → 𝐹:ℕ⟶dom vol)
52 fco 6219 . . . . . . . . . 10 ((vol:dom vol⟶(0[,]+∞) ∧ 𝐹:ℕ⟶dom vol) → (vol ∘ 𝐹):ℕ⟶(0[,]+∞))
5350, 51, 52sylancr 698 . . . . . . . . 9 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → (vol ∘ 𝐹):ℕ⟶(0[,]+∞))
54 ffn 6206 . . . . . . . . 9 ((vol ∘ 𝐹):ℕ⟶(0[,]+∞) → (vol ∘ 𝐹) Fn ℕ)
5553, 54syl 17 . . . . . . . 8 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → (vol ∘ 𝐹) Fn ℕ)
56 fveq2 6353 . . . . . . . . . . . . 13 (𝑥 = 1 → (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘𝑥) = (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘1))
57 fveq2 6353 . . . . . . . . . . . . . 14 (𝑥 = 1 → (𝐹𝑥) = (𝐹‘1))
5857fveq2d 6357 . . . . . . . . . . . . 13 (𝑥 = 1 → (vol‘(𝐹𝑥)) = (vol‘(𝐹‘1)))
5956, 58eqeq12d 2775 . . . . . . . . . . . 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))))
6059imbi2d 329 . . . . . . . . . . 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)))))
61 fveq2 6353 . . . . . . . . . . . . 13 (𝑥 = 𝑗 → (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘𝑥) = (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘𝑗))
62 fveq2 6353 . . . . . . . . . . . . . 14 (𝑥 = 𝑗 → (𝐹𝑥) = (𝐹𝑗))
6362fveq2d 6357 . . . . . . . . . . . . 13 (𝑥 = 𝑗 → (vol‘(𝐹𝑥)) = (vol‘(𝐹𝑗)))
6461, 63eqeq12d 2775 . . . . . . . . . . . 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))))
6564imbi2d 329 . . . . . . . . . . 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)))))
66 fveq2 6353 . . . . . . . . . . . . 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)))
67 fveq2 6353 . . . . . . . . . . . . . 14 (𝑥 = (𝑗 + 1) → (𝐹𝑥) = (𝐹‘(𝑗 + 1)))
6867fveq2d 6357 . . . . . . . . . . . . 13 (𝑥 = (𝑗 + 1) → (vol‘(𝐹𝑥)) = (vol‘(𝐹‘(𝑗 + 1))))
6966, 68eqeq12d 2775 . . . . . . . . . . . 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)))))
7069imbi2d 329 . . . . . . . . . . 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))))))
71 seq1 13028 . . . . . . . . . . . . . 14 (1 ∈ ℤ → (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘1) = ((𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))‘1))
7244, 71ax-mp 5 . . . . . . . . . . . . 13 (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘1) = ((𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))‘1)
73 1nn 11243 . . . . . . . . . . . . . 14 1 ∈ ℕ
74 oveq2 6822 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 1 → (1..^𝑘) = (1..^1))
75 fzo0 12706 . . . . . . . . . . . . . . . . . . . . . 22 (1..^1) = ∅
7674, 75syl6eq 2810 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 1 → (1..^𝑘) = ∅)
7776iuneq1d 4697 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 1 → 𝑚 ∈ (1..^𝑘)(𝐹𝑚) = 𝑚 ∈ ∅ (𝐹𝑚))
78 0iun 4729 . . . . . . . . . . . . . . . . . . . 20 𝑚 ∈ ∅ (𝐹𝑚) = ∅
7977, 78syl6eq 2810 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 1 → 𝑚 ∈ (1..^𝑘)(𝐹𝑚) = ∅)
8079difeq2d 3871 . . . . . . . . . . . . . . . . . 18 (𝑘 = 1 → ((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)) = ((𝐹𝑘) ∖ ∅))
81 dif0 4093 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑘) ∖ ∅) = (𝐹𝑘)
8280, 81syl6eq 2810 . . . . . . . . . . . . . . . . 17 (𝑘 = 1 → ((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)) = (𝐹𝑘))
83 fveq2 6353 . . . . . . . . . . . . . . . . 17 (𝑘 = 1 → (𝐹𝑘) = (𝐹‘1))
8482, 83eqtrd 2794 . . . . . . . . . . . . . . . 16 (𝑘 = 1 → ((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)) = (𝐹‘1))
8584fveq2d 6357 . . . . . . . . . . . . . . 15 (𝑘 = 1 → (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))) = (vol‘(𝐹‘1)))
86 fvex 6363 . . . . . . . . . . . . . . 15 (vol‘(𝐹‘1)) ∈ V
8785, 34, 86fvmpt 6445 . . . . . . . . . . . . . 14 (1 ∈ ℕ → ((𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))‘1) = (vol‘(𝐹‘1)))
8873, 87ax-mp 5 . . . . . . . . . . . . 13 ((𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))‘1) = (vol‘(𝐹‘1))
8972, 88eqtri 2782 . . . . . . . . . . . 12 (seq1( + , (𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)))))‘1) = (vol‘(𝐹‘1))
9089a1i 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)))
91 oveq1 6821 . . . . . . . . . . . . . 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))))
92 seqp1 13030 . . . . . . . . . . . . . . . . 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))))
9392, 6eleq2s 2857 . . . . . . . . . . . . . . . 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))))
9493adantl 473 . . . . . . . . . . . . . . 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))))
95 undif2 4188 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑗) ∪ ((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗))) = ((𝐹𝑗) ∪ (𝐹‘(𝑗 + 1)))
96 fveq2 6353 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑗 → (𝐹𝑛) = (𝐹𝑗))
97 oveq1 6821 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑗 → (𝑛 + 1) = (𝑗 + 1))
9897fveq2d 6357 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑗 → (𝐹‘(𝑛 + 1)) = (𝐹‘(𝑗 + 1)))
9996, 98sseq12d 3775 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑗 → ((𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)) ↔ (𝐹𝑗) ⊆ (𝐹‘(𝑗 + 1))))
100 simpllr 817 . . . . . . . . . . . . . . . . . . . 20 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)))
101 simpr 479 . . . . . . . . . . . . . . . . . . . 20 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
10299, 100, 101rspcdva 3455 . . . . . . . . . . . . . . . . . . 19 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (𝐹𝑗) ⊆ (𝐹‘(𝑗 + 1)))
103 ssequn1 3926 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝑗) ⊆ (𝐹‘(𝑗 + 1)) ↔ ((𝐹𝑗) ∪ (𝐹‘(𝑗 + 1))) = (𝐹‘(𝑗 + 1)))
104102, 103sylib 208 . . . . . . . . . . . . . . . . . 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)))
10595, 104syl5req 2807 . . . . . . . . . . . . . . . . 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))))
106105fveq2d 6357 . . . . . . . . . . . . . . . 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)))))
107 simplll 815 . . . . . . . . . . . . . . . . . 18 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → 𝐹:ℕ⟶dom vol)
108107, 101ffvelrnd 6524 . . . . . . . . . . . . . . . . 17 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (𝐹𝑗) ∈ dom vol)
109 peano2nn 11244 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
110109adantl 473 . . . . . . . . . . . . . . . . . . 19 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (𝑗 + 1) ∈ ℕ)
111107, 110ffvelrnd 6524 . . . . . . . . . . . . . . . . . 18 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (𝐹‘(𝑗 + 1)) ∈ dom vol)
112 difmbl 23531 . . . . . . . . . . . . . . . . . 18 (((𝐹‘(𝑗 + 1)) ∈ dom vol ∧ (𝐹𝑗) ∈ dom vol) → ((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗)) ∈ dom vol)
113111, 108, 112syl2anc 696 . . . . . . . . . . . . . . . . 17 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗)) ∈ dom vol)
114 disjdif 4184 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑗) ∩ ((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗))) = ∅
115114a1i 11 . . . . . . . . . . . . . . . . 17 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((𝐹𝑗) ∩ ((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗))) = ∅)
116 fveq2 6353 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑗 → (𝐹𝑘) = (𝐹𝑗))
117116fveq2d 6357 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑗 → (vol‘(𝐹𝑘)) = (vol‘(𝐹𝑗)))
118117eleq1d 2824 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑗 → ((vol‘(𝐹𝑘)) ∈ ℝ ↔ (vol‘(𝐹𝑗)) ∈ ℝ))
119 simplr 809 . . . . . . . . . . . . . . . . . 18 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ)
120118, 119, 101rspcdva 3455 . . . . . . . . . . . . . . . . 17 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (vol‘(𝐹𝑗)) ∈ ℝ)
121 mblvol 23518 . . . . . . . . . . . . . . . . . . 19 (((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗)) ∈ dom vol → (vol‘((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗))) = (vol*‘((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗))))
122113, 121syl 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))))
123 difssd 3881 . . . . . . . . . . . . . . . . . . 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)))
124 mblss 23519 . . . . . . . . . . . . . . . . . . . 20 ((𝐹‘(𝑗 + 1)) ∈ dom vol → (𝐹‘(𝑗 + 1)) ⊆ ℝ)
125111, 124syl 17 . . . . . . . . . . . . . . . . . . 19 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (𝐹‘(𝑗 + 1)) ⊆ ℝ)
126 mblvol 23518 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹‘(𝑗 + 1)) ∈ dom vol → (vol‘(𝐹‘(𝑗 + 1))) = (vol*‘(𝐹‘(𝑗 + 1))))
127111, 126syl 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))))
128 fveq2 6353 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = (𝑗 + 1) → (𝐹𝑘) = (𝐹‘(𝑗 + 1)))
129128fveq2d 6357 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = (𝑗 + 1) → (vol‘(𝐹𝑘)) = (vol‘(𝐹‘(𝑗 + 1))))
130129eleq1d 2824 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = (𝑗 + 1) → ((vol‘(𝐹𝑘)) ∈ ℝ ↔ (vol‘(𝐹‘(𝑗 + 1))) ∈ ℝ))
131130, 119, 110rspcdva 3455 . . . . . . . . . . . . . . . . . . . 20 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (vol‘(𝐹‘(𝑗 + 1))) ∈ ℝ)
132127, 131eqeltrrd 2840 . . . . . . . . . . . . . . . . . . 19 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (vol*‘(𝐹‘(𝑗 + 1))) ∈ ℝ)
133 ovolsscl 23474 . . . . . . . . . . . . . . . . . . 19 ((((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗)) ⊆ (𝐹‘(𝑗 + 1)) ∧ (𝐹‘(𝑗 + 1)) ⊆ ℝ ∧ (vol*‘(𝐹‘(𝑗 + 1))) ∈ ℝ) → (vol*‘((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗))) ∈ ℝ)
134123, 125, 132, 133syl3anc 1477 . . . . . . . . . . . . . . . . . 18 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (vol*‘((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗))) ∈ ℝ)
135122, 134eqeltrd 2839 . . . . . . . . . . . . . . . . 17 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐹‘(𝑗 + 1)) ∖ (𝐹𝑗))) ∈ ℝ)
136 volun 23533 . . . . . . . . . . . . . . . . 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)))))
137108, 113, 115, 120, 135, 136syl32anc 1485 . . . . . . . . . . . . . . . 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)))))
138100adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . 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)))
139 elfznn 12583 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑚 ∈ (1...𝑗) → 𝑚 ∈ ℕ)
140139adantl 473 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑗)) → 𝑚 ∈ ℕ)
141 elfzuz3 12552 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑚 ∈ (1...𝑗) → 𝑗 ∈ (ℤ𝑚))
142141adantl 473 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑗)) → 𝑗 ∈ (ℤ𝑚))
143 volsuplem 23543 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)) ∧ (𝑚 ∈ ℕ ∧ 𝑗 ∈ (ℤ𝑚))) → (𝐹𝑚) ⊆ (𝐹𝑗))
144138, 140, 142, 143syl12anc 1475 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑗)) → (𝐹𝑚) ⊆ (𝐹𝑗))
145144ralrimiva 3104 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ∀𝑚 ∈ (1...𝑗)(𝐹𝑚) ⊆ (𝐹𝑗))
146 iunss 4713 . . . . . . . . . . . . . . . . . . . . . . 23 ( 𝑚 ∈ (1...𝑗)(𝐹𝑚) ⊆ (𝐹𝑗) ↔ ∀𝑚 ∈ (1...𝑗)(𝐹𝑚) ⊆ (𝐹𝑗))
147145, 146sylibr 224 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → 𝑚 ∈ (1...𝑗)(𝐹𝑚) ⊆ (𝐹𝑗))
148101, 6syl6eleq 2849 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ (ℤ‘1))
149 eluzfz2 12562 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (ℤ‘1) → 𝑗 ∈ (1...𝑗))
150148, 149syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ (1...𝑗))
151 fveq2 6353 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 = 𝑗 → (𝐹𝑚) = (𝐹𝑗))
152151ssiun2s 4716 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ (1...𝑗) → (𝐹𝑗) ⊆ 𝑚 ∈ (1...𝑗)(𝐹𝑚))
153150, 152syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (𝐹𝑗) ⊆ 𝑚 ∈ (1...𝑗)(𝐹𝑚))
154147, 153eqssd 3761 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → 𝑚 ∈ (1...𝑗)(𝐹𝑚) = (𝐹𝑗))
155101nnzd 11693 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℤ)
156 fzval3 12751 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ ℤ → (1...𝑗) = (1..^(𝑗 + 1)))
157155, 156syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (1...𝑗) = (1..^(𝑗 + 1)))
158157iuneq1d 4697 . . . . . . . . . . . . . . . . . . . . 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))
159154, 158eqtr3d 2796 . . . . . . . . . . . . . . . . . . . 20 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → (𝐹𝑗) = 𝑚 ∈ (1..^(𝑗 + 1))(𝐹𝑚))
160159difeq2d 3871 . . . . . . . . . . . . . . . . . . 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)))
161160fveq2d 6357 . . . . . . . . . . . . . . . . . 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))))
162 oveq2 6822 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = (𝑗 + 1) → (1..^𝑘) = (1..^(𝑗 + 1)))
163162iuneq1d 4697 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = (𝑗 + 1) → 𝑚 ∈ (1..^𝑘)(𝐹𝑚) = 𝑚 ∈ (1..^(𝑗 + 1))(𝐹𝑚))
164128, 163difeq12d 3872 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = (𝑗 + 1) → ((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚)) = ((𝐹‘(𝑗 + 1)) ∖ 𝑚 ∈ (1..^(𝑗 + 1))(𝐹𝑚)))
165164fveq2d 6357 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = (𝑗 + 1) → (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))) = (vol‘((𝐹‘(𝑗 + 1)) ∖ 𝑚 ∈ (1..^(𝑗 + 1))(𝐹𝑚))))
166 fvex 6363 . . . . . . . . . . . . . . . . . . . 20 (vol‘((𝐹‘(𝑗 + 1)) ∖ 𝑚 ∈ (1..^(𝑗 + 1))(𝐹𝑚))) ∈ V
167165, 34, 166fvmpt 6445 . . . . . . . . . . . . . . . . . . 19 ((𝑗 + 1) ∈ ℕ → ((𝑘 ∈ ℕ ↦ (vol‘((𝐹𝑘) ∖ 𝑚 ∈ (1..^𝑘)(𝐹𝑚))))‘(𝑗 + 1)) = (vol‘((𝐹‘(𝑗 + 1)) ∖ 𝑚 ∈ (1..^(𝑗 + 1))(𝐹𝑚))))
168110, 167syl 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))))
169161, 168eqtr4d 2797 . . . . . . . . . . . . . . . . 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)))
170169oveq2d 6830 . . . . . . . . . . . . . . . 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))))
171106, 137, 1703eqtrd 2798 . . . . . . . . . . . . . . 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))))
17294, 171eqeq12d 2775 . . . . . . . . . . . . . 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)))))
17391, 172syl5ibr 236 . . . . . . . . . . . . 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)))))
174173expcom 450 . . . . . . . . . . . 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))))))
175174a2d 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))))))
17660, 65, 70, 65, 90, 175nnind 11250 . . . . . . . . . 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))))
177176impcom 445 . . . . . . . . 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)))
178 fvco3 6438 . . . . . . . . . 10 ((𝐹:ℕ⟶dom vol ∧ 𝑗 ∈ ℕ) → ((vol ∘ 𝐹)‘𝑗) = (vol‘(𝐹𝑗)))
17951, 178sylan 489 . . . . . . . . 9 ((((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) ∧ 𝑗 ∈ ℕ) → ((vol ∘ 𝐹)‘𝑗) = (vol‘(𝐹𝑗)))
180177, 179eqtr4d 2797 . . . . . . . 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))
18149, 55, 180eqfnfvd 6478 . . . . . . 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))
182181rneqd 5508 . . . . . 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))
183 rnco2 5803 . . . . . 6 ran (vol ∘ 𝐹) = (vol “ ran 𝐹)
184182, 183syl6eq 2810 . . . . 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))
185184supeq1d 8519 . . . 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*, < ))
18636, 43, 1853eqtr3d 2802 . . 3 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ) → (vol‘ ran 𝐹) = sup((vol “ ran 𝐹), ℝ*, < ))
187186ex 449 . 2 ((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) → (∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ → (vol‘ ran 𝐹) = sup((vol “ ran 𝐹), ℝ*, < )))
188 rexnal 3133 . . 3 (∃𝑘 ∈ ℕ ¬ (vol‘(𝐹𝑘)) ∈ ℝ ↔ ¬ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ)
189 fniunfv 6669 . . . . . . . . . . . 12 (𝐹 Fn ℕ → 𝑛 ∈ ℕ (𝐹𝑛) = ran 𝐹)
19038, 189syl 17 . . . . . . . . . . 11 (𝐹:ℕ⟶dom vol → 𝑛 ∈ ℕ (𝐹𝑛) = ran 𝐹)
191 ffvelrn 6521 . . . . . . . . . . . . 13 ((𝐹:ℕ⟶dom vol ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ∈ dom vol)
192191ralrimiva 3104 . . . . . . . . . . . 12 (𝐹:ℕ⟶dom vol → ∀𝑛 ∈ ℕ (𝐹𝑛) ∈ dom vol)
193 iunmbl 23541 . . . . . . . . . . . 12 (∀𝑛 ∈ ℕ (𝐹𝑛) ∈ dom vol → 𝑛 ∈ ℕ (𝐹𝑛) ∈ dom vol)
194192, 193syl 17 . . . . . . . . . . 11 (𝐹:ℕ⟶dom vol → 𝑛 ∈ ℕ (𝐹𝑛) ∈ dom vol)
195190, 194eqeltrrd 2840 . . . . . . . . . 10 (𝐹:ℕ⟶dom vol → ran 𝐹 ∈ dom vol)
196195ad2antrr 764 . . . . . . . . 9 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → ran 𝐹 ∈ dom vol)
197 mblss 23519 . . . . . . . . 9 ( ran 𝐹 ∈ dom vol → ran 𝐹 ⊆ ℝ)
198196, 197syl 17 . . . . . . . 8 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → ran 𝐹 ⊆ ℝ)
199 ovolcl 23466 . . . . . . . 8 ( ran 𝐹 ⊆ ℝ → (vol*‘ ran 𝐹) ∈ ℝ*)
200198, 199syl 17 . . . . . . 7 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol*‘ ran 𝐹) ∈ ℝ*)
201 pnfge 12177 . . . . . . 7 ((vol*‘ ran 𝐹) ∈ ℝ* → (vol*‘ ran 𝐹) ≤ +∞)
202200, 201syl 17 . . . . . 6 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol*‘ ran 𝐹) ≤ +∞)
203 simprr 813 . . . . . . . . 9 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → ¬ (vol‘(𝐹𝑘)) ∈ ℝ)
2041ad2ant2r 800 . . . . . . . . . . . . 13 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (𝐹𝑘) ∈ dom vol)
205204, 18syl 17 . . . . . . . . . . . 12 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (𝐹𝑘) ⊆ ℝ)
206 ovolcl 23466 . . . . . . . . . . . 12 ((𝐹𝑘) ⊆ ℝ → (vol*‘(𝐹𝑘)) ∈ ℝ*)
207205, 206syl 17 . . . . . . . . . . 11 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol*‘(𝐹𝑘)) ∈ ℝ*)
208 xrrebnd 12212 . . . . . . . . . . 11 ((vol*‘(𝐹𝑘)) ∈ ℝ* → ((vol*‘(𝐹𝑘)) ∈ ℝ ↔ (-∞ < (vol*‘(𝐹𝑘)) ∧ (vol*‘(𝐹𝑘)) < +∞)))
209207, 208syl 17 . . . . . . . . . 10 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → ((vol*‘(𝐹𝑘)) ∈ ℝ ↔ (-∞ < (vol*‘(𝐹𝑘)) ∧ (vol*‘(𝐹𝑘)) < +∞)))
210204, 20syl 17 . . . . . . . . . . 11 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol‘(𝐹𝑘)) = (vol*‘(𝐹𝑘)))
211210eleq1d 2824 . . . . . . . . . 10 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → ((vol‘(𝐹𝑘)) ∈ ℝ ↔ (vol*‘(𝐹𝑘)) ∈ ℝ))
212 ovolge0 23469 . . . . . . . . . . . . 13 ((𝐹𝑘) ⊆ ℝ → 0 ≤ (vol*‘(𝐹𝑘)))
213 mnflt0 12172 . . . . . . . . . . . . . 14 -∞ < 0
214 mnfxr 10308 . . . . . . . . . . . . . . 15 -∞ ∈ ℝ*
215 0xr 10298 . . . . . . . . . . . . . . 15 0 ∈ ℝ*
216 xrltletr 12201 . . . . . . . . . . . . . . 15 ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ* ∧ (vol*‘(𝐹𝑘)) ∈ ℝ*) → ((-∞ < 0 ∧ 0 ≤ (vol*‘(𝐹𝑘))) → -∞ < (vol*‘(𝐹𝑘))))
217214, 215, 216mp3an12 1563 . . . . . . . . . . . . . 14 ((vol*‘(𝐹𝑘)) ∈ ℝ* → ((-∞ < 0 ∧ 0 ≤ (vol*‘(𝐹𝑘))) → -∞ < (vol*‘(𝐹𝑘))))
218213, 217mpani 714 . . . . . . . . . . . . 13 ((vol*‘(𝐹𝑘)) ∈ ℝ* → (0 ≤ (vol*‘(𝐹𝑘)) → -∞ < (vol*‘(𝐹𝑘))))
219206, 212, 218sylc 65 . . . . . . . . . . . 12 ((𝐹𝑘) ⊆ ℝ → -∞ < (vol*‘(𝐹𝑘)))
220205, 219syl 17 . . . . . . . . . . 11 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → -∞ < (vol*‘(𝐹𝑘)))
221220biantrurd 530 . . . . . . . . . 10 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → ((vol*‘(𝐹𝑘)) < +∞ ↔ (-∞ < (vol*‘(𝐹𝑘)) ∧ (vol*‘(𝐹𝑘)) < +∞)))
222209, 211, 2213bitr4d 300 . . . . . . . . 9 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → ((vol‘(𝐹𝑘)) ∈ ℝ ↔ (vol*‘(𝐹𝑘)) < +∞))
223203, 222mtbid 313 . . . . . . . 8 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → ¬ (vol*‘(𝐹𝑘)) < +∞)
224 nltpnft 12208 . . . . . . . . 9 ((vol*‘(𝐹𝑘)) ∈ ℝ* → ((vol*‘(𝐹𝑘)) = +∞ ↔ ¬ (vol*‘(𝐹𝑘)) < +∞))
225207, 224syl 17 . . . . . . . 8 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → ((vol*‘(𝐹𝑘)) = +∞ ↔ ¬ (vol*‘(𝐹𝑘)) < +∞))
226223, 225mpbird 247 . . . . . . 7 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol*‘(𝐹𝑘)) = +∞)
22738ad2antrr 764 . . . . . . . . . 10 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → 𝐹 Fn ℕ)
228 simprl 811 . . . . . . . . . 10 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → 𝑘 ∈ ℕ)
229 fnfvelrn 6520 . . . . . . . . . 10 ((𝐹 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝐹𝑘) ∈ ran 𝐹)
230227, 228, 229syl2anc 696 . . . . . . . . 9 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (𝐹𝑘) ∈ ran 𝐹)
231 elssuni 4619 . . . . . . . . 9 ((𝐹𝑘) ∈ ran 𝐹 → (𝐹𝑘) ⊆ ran 𝐹)
232230, 231syl 17 . . . . . . . 8 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (𝐹𝑘) ⊆ ran 𝐹)
233 ovolss 23473 . . . . . . . 8 (((𝐹𝑘) ⊆ ran 𝐹 ran 𝐹 ⊆ ℝ) → (vol*‘(𝐹𝑘)) ≤ (vol*‘ ran 𝐹))
234232, 198, 233syl2anc 696 . . . . . . 7 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol*‘(𝐹𝑘)) ≤ (vol*‘ ran 𝐹))
235226, 234eqbrtrrd 4828 . . . . . 6 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → +∞ ≤ (vol*‘ ran 𝐹))
236 pnfxr 10304 . . . . . . 7 +∞ ∈ ℝ*
237 xrletri3 12198 . . . . . . 7 (((vol*‘ ran 𝐹) ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((vol*‘ ran 𝐹) = +∞ ↔ ((vol*‘ ran 𝐹) ≤ +∞ ∧ +∞ ≤ (vol*‘ ran 𝐹))))
238200, 236, 237sylancl 697 . . . . . 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))))
239202, 235, 238mpbir2and 995 . . . . 5 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol*‘ ran 𝐹) = +∞)
240 mblvol 23518 . . . . . 6 ( ran 𝐹 ∈ dom vol → (vol‘ ran 𝐹) = (vol*‘ ran 𝐹))
241196, 240syl 17 . . . . 5 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol‘ ran 𝐹) = (vol*‘ ran 𝐹))
242 imassrn 5635 . . . . . . 7 (vol “ ran 𝐹) ⊆ ran vol
243 frn 6214 . . . . . . . . 9 (vol:dom vol⟶(0[,]+∞) → ran vol ⊆ (0[,]+∞))
24450, 243ax-mp 5 . . . . . . . 8 ran vol ⊆ (0[,]+∞)
245 iccssxr 12469 . . . . . . . 8 (0[,]+∞) ⊆ ℝ*
246244, 245sstri 3753 . . . . . . 7 ran vol ⊆ ℝ*
247242, 246sstri 3753 . . . . . 6 (vol “ ran 𝐹) ⊆ ℝ*
248210, 226eqtrd 2794 . . . . . . 7 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol‘(𝐹𝑘)) = +∞)
249 simpll 807 . . . . . . . 8 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → 𝐹:ℕ⟶dom vol)
250 ffun 6209 . . . . . . . . . 10 (vol:dom vol⟶(0[,]+∞) → Fun vol)
25150, 250ax-mp 5 . . . . . . . . 9 Fun vol
252 frn 6214 . . . . . . . . 9 (𝐹:ℕ⟶dom vol → ran 𝐹 ⊆ dom vol)
253 funfvima2 6657 . . . . . . . . 9 ((Fun vol ∧ ran 𝐹 ⊆ dom vol) → ((𝐹𝑘) ∈ ran 𝐹 → (vol‘(𝐹𝑘)) ∈ (vol “ ran 𝐹)))
254251, 252, 253sylancr 698 . . . . . . . 8 (𝐹:ℕ⟶dom vol → ((𝐹𝑘) ∈ ran 𝐹 → (vol‘(𝐹𝑘)) ∈ (vol “ ran 𝐹)))
255249, 230, 254sylc 65 . . . . . . 7 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol‘(𝐹𝑘)) ∈ (vol “ ran 𝐹))
256248, 255eqeltrrd 2840 . . . . . 6 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → +∞ ∈ (vol “ ran 𝐹))
257 supxrpnf 12361 . . . . . 6 (((vol “ ran 𝐹) ⊆ ℝ* ∧ +∞ ∈ (vol “ ran 𝐹)) → sup((vol “ ran 𝐹), ℝ*, < ) = +∞)
258247, 256, 257sylancr 698 . . . . 5 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → sup((vol “ ran 𝐹), ℝ*, < ) = +∞)
259239, 241, 2583eqtr4d 2804 . . . 4 (((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) ∧ (𝑘 ∈ ℕ ∧ ¬ (vol‘(𝐹𝑘)) ∈ ℝ)) → (vol‘ ran 𝐹) = sup((vol “ ran 𝐹), ℝ*, < ))
260259rexlimdvaa 3170 . . 3 ((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) → (∃𝑘 ∈ ℕ ¬ (vol‘(𝐹𝑘)) ∈ ℝ → (vol‘ ran 𝐹) = sup((vol “ ran 𝐹), ℝ*, < )))
261188, 260syl5bir 233 . 2 ((𝐹:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1))) → (¬ ∀𝑘 ∈ ℕ (vol‘(𝐹𝑘)) ∈ ℝ → (vol‘ ran 𝐹) = sup((vol “ ran 𝐹), ℝ*, < )))
262187, 261pm2.61d 170 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 196  wa 383   = wceq 1632  wcel 2139  wral 3050  wrex 3051  cdif 3712  cun 3713  cin 3714  wss 3715  c0 4058   cuni 4588   ciun 4672  Disj wdisj 4772   class class class wbr 4804  cmpt 4881  dom cdm 5266  ran crn 5267  cima 5269  ccom 5270  Fun wfun 6043   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6814  Fincfn 8123  supcsup 8513  cr 10147  0cc0 10148  1c1 10149   + caddc 10151  +∞cpnf 10283  -∞cmnf 10284  *cxr 10285   < clt 10286  cle 10287  cn 11232  cz 11589  cuz 11899  [,]cicc 12391  ...cfz 12539  ..^cfzo 12679  seqcseq 13015  vol*covol 23451  volcvol 23452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-inf2 8713  ax-cc 9469  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225  ax-pre-sup 10226
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-disj 4773  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-of 7063  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-2o 7731  df-oadd 7734  df-er 7913  df-map 8027  df-pm 8028  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-sup 8515  df-inf 8516  df-oi 8582  df-card 8975  df-cda 9202  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-div 10897  df-nn 11233  df-2 11291  df-3 11292  df-n0 11505  df-z 11590  df-uz 11900  df-q 12002  df-rp 12046  df-xadd 12160  df-ioo 12392  df-ico 12394  df-icc 12395  df-fz 12540  df-fzo 12680  df-fl 12807  df-seq 13016  df-exp 13075  df-hash 13332  df-cj 14058  df-re 14059  df-im 14060  df-sqrt 14194  df-abs 14195  df-clim 14438  df-rlim 14439  df-sum 14636  df-xmet 19961  df-met 19962  df-ovol 23453  df-vol 23454
This theorem is referenced by:  volsup2  23593  itg1climres  23700  itg2gt0  23746
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