Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hoidmv1lelem1 Structured version   Visualization version   GIF version

Theorem hoidmv1lelem1 42415
Description: The supremum of 𝑈 belongs to 𝑈. This is the last part of step (a) and the whole step (b) in the proof of Lemma 114B of [Fremlin1] p. 23. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
Hypotheses
Ref Expression
hoidmv1lelem1.a (𝜑𝐴 ∈ ℝ)
hoidmv1lelem1.b (𝜑𝐵 ∈ ℝ)
hoidmv1lelem1.l (𝜑𝐴 < 𝐵)
hoidmv1lelem1.c (𝜑𝐶:ℕ⟶ℝ)
hoidmv1lelem1.d (𝜑𝐷:ℕ⟶ℝ)
hoidmv1lelem1.r (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) ∈ ℝ)
hoidmv1lelem1.u 𝑈 = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))}
hoidmv1lelem1.s 𝑆 = sup(𝑈, ℝ, < )
Assertion
Ref Expression
hoidmv1lelem1 (𝜑 → (𝑆𝑈𝐴𝑈 ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥))
Distinct variable groups:   𝐴,𝑗,𝑧   𝑦,𝐴   𝑥,𝐵,𝑦   𝑧,𝐵   𝑧,𝐶   𝑧,𝐷   𝑆,𝑗,𝑧   𝑈,𝑗,𝑧   𝑥,𝑈,𝑦   𝜑,𝑗,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑗)   𝐶(𝑥,𝑦,𝑗)   𝐷(𝑥,𝑦,𝑗)   𝑆(𝑥,𝑦)

Proof of Theorem hoidmv1lelem1
StepHypRef Expression
1 hoidmv1lelem1.s . . . . . 6 𝑆 = sup(𝑈, ℝ, < )
2 hoidmv1lelem1.a . . . . . . 7 (𝜑𝐴 ∈ ℝ)
3 hoidmv1lelem1.b . . . . . . 7 (𝜑𝐵 ∈ ℝ)
4 hoidmv1lelem1.u . . . . . . . . 9 𝑈 = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))}
5 ssrab2 3977 . . . . . . . . 9 {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))} ⊆ (𝐴[,]𝐵)
64, 5eqsstri 3922 . . . . . . . 8 𝑈 ⊆ (𝐴[,]𝐵)
76a1i 11 . . . . . . 7 (𝜑𝑈 ⊆ (𝐴[,]𝐵))
82rexrd 10537 . . . . . . . . . . . 12 (𝜑𝐴 ∈ ℝ*)
93rexrd 10537 . . . . . . . . . . . 12 (𝜑𝐵 ∈ ℝ*)
10 hoidmv1lelem1.l . . . . . . . . . . . . 13 (𝜑𝐴 < 𝐵)
112, 3, 10ltled 10635 . . . . . . . . . . . 12 (𝜑𝐴𝐵)
12 lbicc2 12702 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
138, 9, 11, 12syl3anc 1364 . . . . . . . . . . 11 (𝜑𝐴 ∈ (𝐴[,]𝐵))
142recnd 10515 . . . . . . . . . . . . 13 (𝜑𝐴 ∈ ℂ)
1514subidd 10833 . . . . . . . . . . . 12 (𝜑 → (𝐴𝐴) = 0)
16 nfv 1892 . . . . . . . . . . . . 13 𝑗𝜑
17 nnex 11492 . . . . . . . . . . . . . 14 ℕ ∈ V
1817a1i 11 . . . . . . . . . . . . 13 (𝜑 → ℕ ∈ V)
19 volf 23813 . . . . . . . . . . . . . . 15 vol:dom vol⟶(0[,]+∞)
2019a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → vol:dom vol⟶(0[,]+∞))
21 hoidmv1lelem1.c . . . . . . . . . . . . . . . 16 (𝜑𝐶:ℕ⟶ℝ)
2221ffvelrnda 6716 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ∈ ℝ)
23 hoidmv1lelem1.d . . . . . . . . . . . . . . . . . 18 (𝜑𝐷:ℕ⟶ℝ)
2423ffvelrnda 6716 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗) ∈ ℝ)
252adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → 𝐴 ∈ ℝ)
2624, 25ifcld 4426 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴) ∈ ℝ)
2726rexrd 10537 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴) ∈ ℝ*)
28 icombl 23848 . . . . . . . . . . . . . . 15 (((𝐶𝑗) ∈ ℝ ∧ if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴) ∈ ℝ*) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴)) ∈ dom vol)
2922, 27, 28syl2anc 584 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴)) ∈ dom vol)
3020, 29ffvelrnd 6717 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴))) ∈ (0[,]+∞))
3116, 18, 30sge0ge0mpt 42262 . . . . . . . . . . . 12 (𝜑 → 0 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴))))))
3215, 31eqbrtrd 4984 . . . . . . . . . . 11 (𝜑 → (𝐴𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴))))))
3313, 32jca 512 . . . . . . . . . 10 (𝜑 → (𝐴 ∈ (𝐴[,]𝐵) ∧ (𝐴𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴)))))))
34 oveq1 7023 . . . . . . . . . . . 12 (𝑧 = 𝐴 → (𝑧𝐴) = (𝐴𝐴))
35 breq2 4966 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝐴 → ((𝐷𝑗) ≤ 𝑧 ↔ (𝐷𝑗) ≤ 𝐴))
36 id 22 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝐴𝑧 = 𝐴)
3735, 36ifbieq2d 4406 . . . . . . . . . . . . . . . 16 (𝑧 = 𝐴 → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴))
3837oveq2d 7032 . . . . . . . . . . . . . . 15 (𝑧 = 𝐴 → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) = ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴)))
3938fveq2d 6542 . . . . . . . . . . . . . 14 (𝑧 = 𝐴 → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))) = (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴))))
4039mpteq2dv 5056 . . . . . . . . . . . . 13 (𝑧 = 𝐴 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴)))))
4140fveq2d 6542 . . . . . . . . . . . 12 (𝑧 = 𝐴 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) = (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴))))))
4234, 41breq12d 4975 . . . . . . . . . . 11 (𝑧 = 𝐴 → ((𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ↔ (𝐴𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴)))))))
4342elrab 3618 . . . . . . . . . 10 (𝐴 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))} ↔ (𝐴 ∈ (𝐴[,]𝐵) ∧ (𝐴𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴)))))))
4433, 43sylibr 235 . . . . . . . . 9 (𝜑𝐴 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))})
4544, 4syl6eleqr 2894 . . . . . . . 8 (𝜑𝐴𝑈)
4645ne0d 4221 . . . . . . 7 (𝜑𝑈 ≠ ∅)
472, 3, 7, 46supicc 12736 . . . . . 6 (𝜑 → sup(𝑈, ℝ, < ) ∈ (𝐴[,]𝐵))
481, 47syl5eqel 2887 . . . . 5 (𝜑𝑆 ∈ (𝐴[,]𝐵))
491a1i 11 . . . . . . 7 (𝜑𝑆 = sup(𝑈, ℝ, < ))
50 nfv 1892 . . . . . . . . 9 𝑧𝜑
512, 3iccssred 41322 . . . . . . . . . . . . 13 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
527, 51sstrd 3899 . . . . . . . . . . . 12 (𝜑𝑈 ⊆ ℝ)
5352sselda 3889 . . . . . . . . . . 11 ((𝜑𝑧𝑈) → 𝑧 ∈ ℝ)
54 nfv 1892 . . . . . . . . . . . . . . . 16 𝑗(𝜑𝑧𝑈)
5517a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑈) → ℕ ∈ V)
5619a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → vol:dom vol⟶(0[,]+∞))
5722adantlr 711 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (𝐶𝑗) ∈ ℝ)
5824adantlr 711 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (𝐷𝑗) ∈ ℝ)
5953adantr 481 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → 𝑧 ∈ ℝ)
6058, 59ifcld 4426 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ∈ ℝ)
6160rexrd 10537 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ∈ ℝ*)
62 icombl 23848 . . . . . . . . . . . . . . . . . 18 (((𝐶𝑗) ∈ ℝ ∧ if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ∈ ℝ*) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ∈ dom vol)
6357, 61, 62syl2anc 584 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ∈ dom vol)
6456, 63ffvelrnd 6717 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))) ∈ (0[,]+∞))
6554, 55, 64sge0xrclmpt 42252 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ∈ ℝ*)
66 pnfxr 10541 . . . . . . . . . . . . . . . 16 +∞ ∈ ℝ*
6766a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑈) → +∞ ∈ ℝ*)
68 hoidmv1lelem1.r . . . . . . . . . . . . . . . . . 18 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) ∈ ℝ)
6968rexrd 10537 . . . . . . . . . . . . . . . . 17 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) ∈ ℝ*)
7069adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) ∈ ℝ*)
7124rexrd 10537 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗) ∈ ℝ*)
72 icombl 23848 . . . . . . . . . . . . . . . . . . . 20 (((𝐶𝑗) ∈ ℝ ∧ (𝐷𝑗) ∈ ℝ*) → ((𝐶𝑗)[,)(𝐷𝑗)) ∈ dom vol)
7322, 71, 72syl2anc 584 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)[,)(𝐷𝑗)) ∈ dom vol)
7420, 73ffvelrnd 6717 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)(𝐷𝑗))) ∈ (0[,]+∞))
7574adantlr 711 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)(𝐷𝑗))) ∈ (0[,]+∞))
7673adantlr 711 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶𝑗)[,)(𝐷𝑗)) ∈ dom vol)
7722rexrd 10537 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ∈ ℝ*)
7877adantlr 711 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (𝐶𝑗) ∈ ℝ*)
7971adantlr 711 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (𝐷𝑗) ∈ ℝ*)
8022leidd 11054 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ≤ (𝐶𝑗))
8180adantlr 711 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (𝐶𝑗) ≤ (𝐶𝑗))
82 min1 12432 . . . . . . . . . . . . . . . . . . . 20 (((𝐷𝑗) ∈ ℝ ∧ 𝑧 ∈ ℝ) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ (𝐷𝑗))
8358, 59, 82syl2anc 584 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ (𝐷𝑗))
84 icossico 12656 . . . . . . . . . . . . . . . . . . 19 ((((𝐶𝑗) ∈ ℝ* ∧ (𝐷𝑗) ∈ ℝ*) ∧ ((𝐶𝑗) ≤ (𝐶𝑗) ∧ if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ (𝐷𝑗))) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ⊆ ((𝐶𝑗)[,)(𝐷𝑗)))
8578, 79, 81, 83, 84syl22anc 835 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ⊆ ((𝐶𝑗)[,)(𝐷𝑗)))
86 volss 23817 . . . . . . . . . . . . . . . . . 18 ((((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ∈ dom vol ∧ ((𝐶𝑗)[,)(𝐷𝑗)) ∈ dom vol ∧ ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ⊆ ((𝐶𝑗)[,)(𝐷𝑗))) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))) ≤ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))
8763, 76, 85, 86syl3anc 1364 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))) ≤ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))
8854, 55, 64, 75, 87sge0lempt 42234 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))))
8968ltpnfd 12366 . . . . . . . . . . . . . . . . 17 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) < +∞)
9089adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) < +∞)
9165, 70, 67, 88, 90xrlelttrd 12403 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) < +∞)
9265, 67, 91xrltned 41166 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ≠ +∞)
9392neneqd 2989 . . . . . . . . . . . . 13 ((𝜑𝑧𝑈) → ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) = +∞)
94 eqid 2795 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))
9564, 94fmptd 6741 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑈) → (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))):ℕ⟶(0[,]+∞))
9655, 95sge0repnf 42210 . . . . . . . . . . . . 13 ((𝜑𝑧𝑈) → ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ∈ ℝ ↔ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) = +∞))
9793, 96mpbird 258 . . . . . . . . . . . 12 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ∈ ℝ)
982adantr 481 . . . . . . . . . . . 12 ((𝜑𝑧𝑈) → 𝐴 ∈ ℝ)
9997, 98readdcld 10516 . . . . . . . . . . 11 ((𝜑𝑧𝑈) → ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) + 𝐴) ∈ ℝ)
10051, 48sseldd 3890 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑆 ∈ ℝ)
101100adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ ℕ) → 𝑆 ∈ ℝ)
10224, 101ifcld 4426 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ∈ ℝ)
103102rexrd 10537 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ∈ ℝ*)
104 icombl 23848 . . . . . . . . . . . . . . . . . . 19 (((𝐶𝑗) ∈ ℝ ∧ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ∈ ℝ*) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)) ∈ dom vol)
10522, 103, 104syl2anc 584 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)) ∈ dom vol)
10620, 105ffvelrnd 6717 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))) ∈ (0[,]+∞))
10716, 18, 106sge0xrclmpt 42252 . . . . . . . . . . . . . . . 16 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) ∈ ℝ*)
10866a1i 11 . . . . . . . . . . . . . . . 16 (𝜑 → +∞ ∈ ℝ*)
109 min1 12432 . . . . . . . . . . . . . . . . . . . . 21 (((𝐷𝑗) ∈ ℝ ∧ 𝑆 ∈ ℝ) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ≤ (𝐷𝑗))
11024, 101, 109syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ≤ (𝐷𝑗))
111 icossico 12656 . . . . . . . . . . . . . . . . . . . 20 ((((𝐶𝑗) ∈ ℝ* ∧ (𝐷𝑗) ∈ ℝ*) ∧ ((𝐶𝑗) ≤ (𝐶𝑗) ∧ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ≤ (𝐷𝑗))) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)) ⊆ ((𝐶𝑗)[,)(𝐷𝑗)))
11277, 71, 80, 110, 111syl22anc 835 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)) ⊆ ((𝐶𝑗)[,)(𝐷𝑗)))
113 volss 23817 . . . . . . . . . . . . . . . . . . 19 ((((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)) ∈ dom vol ∧ ((𝐶𝑗)[,)(𝐷𝑗)) ∈ dom vol ∧ ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)) ⊆ ((𝐶𝑗)[,)(𝐷𝑗))) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))) ≤ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))
114105, 73, 112, 113syl3anc 1364 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))) ≤ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))
11516, 18, 106, 74, 114sge0lempt 42234 . . . . . . . . . . . . . . . . 17 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))))
116107, 69, 108, 115, 89xrlelttrd 12403 . . . . . . . . . . . . . . . 16 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) < +∞)
117107, 108, 116xrltned 41166 . . . . . . . . . . . . . . 15 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) ≠ +∞)
118117neneqd 2989 . . . . . . . . . . . . . 14 (𝜑 → ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) = +∞)
119 eqid 2795 . . . . . . . . . . . . . . . 16 (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))
120106, 119fmptd 6741 . . . . . . . . . . . . . . 15 (𝜑 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))):ℕ⟶(0[,]+∞))
12118, 120sge0repnf 42210 . . . . . . . . . . . . . 14 (𝜑 → ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) ∈ ℝ ↔ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) = +∞))
122118, 121mpbird 258 . . . . . . . . . . . . 13 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) ∈ ℝ)
123122, 2readdcld 10516 . . . . . . . . . . . 12 (𝜑 → ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴) ∈ ℝ)
124123adantr 481 . . . . . . . . . . 11 ((𝜑𝑧𝑈) → ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴) ∈ ℝ)
1254eleq2i 2874 . . . . . . . . . . . . . . . 16 (𝑧𝑈𝑧 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))})
126125biimpi 217 . . . . . . . . . . . . . . 15 (𝑧𝑈𝑧 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))})
127126adantl 482 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑈) → 𝑧 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))})
128 rabid 3337 . . . . . . . . . . . . . 14 (𝑧 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))} ↔ (𝑧 ∈ (𝐴[,]𝐵) ∧ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))))
129127, 128sylib 219 . . . . . . . . . . . . 13 ((𝜑𝑧𝑈) → (𝑧 ∈ (𝐴[,]𝐵) ∧ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))))
130129simprd 496 . . . . . . . . . . . 12 ((𝜑𝑧𝑈) → (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))))
13153, 98, 97lesubaddd 11085 . . . . . . . . . . . 12 ((𝜑𝑧𝑈) → ((𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ↔ 𝑧 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) + 𝐴)))
132130, 131mpbid 233 . . . . . . . . . . 11 ((𝜑𝑧𝑈) → 𝑧 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) + 𝐴))
133122adantr 481 . . . . . . . . . . . 12 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) ∈ ℝ)
134106adantlr 711 . . . . . . . . . . . . 13 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))) ∈ (0[,]+∞))
135105adantlr 711 . . . . . . . . . . . . . 14 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)) ∈ dom vol)
136103adantlr 711 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ∈ ℝ*)
13760adantr 481 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ∈ ℝ)
138 eqidd 2796 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → (𝐷𝑗) = (𝐷𝑗))
139 iftrue 4387 . . . . . . . . . . . . . . . . . . 19 ((𝐷𝑗) ≤ 𝑧 → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = (𝐷𝑗))
140139adantl 482 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = (𝐷𝑗))
14158adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → (𝐷𝑗) ∈ ℝ)
14259adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → 𝑧 ∈ ℝ)
143100ad3antrrr 726 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → 𝑆 ∈ ℝ)
144 simpr 485 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → (𝐷𝑗) ≤ 𝑧)
14552adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑧𝑈) → 𝑈 ⊆ ℝ)
14646adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑧𝑈) → 𝑈 ≠ ∅)
1472, 3jca 512 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
148 iccsupr 12680 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑈 ⊆ (𝐴[,]𝐵) ∧ 𝐴𝑈) → (𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥))
149147, 7, 45, 148syl3anc 1364 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥))
150149simp3d 1137 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥)
151150adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑧𝑈) → ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥)
152127, 125sylibr 235 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑧𝑈) → 𝑧𝑈)
153 suprub 11450 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥) ∧ 𝑧𝑈) → 𝑧 ≤ sup(𝑈, ℝ, < ))
154145, 146, 151, 152, 153syl31anc 1366 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑧𝑈) → 𝑧 ≤ sup(𝑈, ℝ, < ))
155154, 1syl6breqr 5004 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑧𝑈) → 𝑧𝑆)
156155ad2antrr 722 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → 𝑧𝑆)
157141, 142, 143, 144, 156letrd 10644 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → (𝐷𝑗) ≤ 𝑆)
158157iftrued 4389 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) = (𝐷𝑗))
159138, 140, 1583eqtr4d 2841 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
160137, 159eqled 10590 . . . . . . . . . . . . . . . 16 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
16159adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → 𝑧 ∈ ℝ)
16258adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → (𝐷𝑗) ∈ ℝ)
163 simpr 485 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → ¬ (𝐷𝑗) ≤ 𝑧)
164161, 162ltnled 10634 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → (𝑧 < (𝐷𝑗) ↔ ¬ (𝐷𝑗) ≤ 𝑧))
165163, 164mpbird 258 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → 𝑧 < (𝐷𝑗))
166161, 162, 165ltled 10635 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → 𝑧 ≤ (𝐷𝑗))
167166adantr 481 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ (𝐷𝑗) ≤ 𝑆) → 𝑧 ≤ (𝐷𝑗))
168 iffalse 4390 . . . . . . . . . . . . . . . . . . . 20 (¬ (𝐷𝑗) ≤ 𝑧 → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = 𝑧)
169168ad2antlr 723 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ (𝐷𝑗) ≤ 𝑆) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = 𝑧)
170 iftrue 4387 . . . . . . . . . . . . . . . . . . . 20 ((𝐷𝑗) ≤ 𝑆 → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) = (𝐷𝑗))
171170adantl 482 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ (𝐷𝑗) ≤ 𝑆) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) = (𝐷𝑗))
172169, 171breq12d 4975 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ (𝐷𝑗) ≤ 𝑆) → (if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ↔ 𝑧 ≤ (𝐷𝑗)))
173167, 172mpbird 258 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ (𝐷𝑗) ≤ 𝑆) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
174155ad3antrrr 726 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ ¬ (𝐷𝑗) ≤ 𝑆) → 𝑧𝑆)
175168ad2antlr 723 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ ¬ (𝐷𝑗) ≤ 𝑆) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = 𝑧)
176 iffalse 4390 . . . . . . . . . . . . . . . . . . . 20 (¬ (𝐷𝑗) ≤ 𝑆 → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) = 𝑆)
177176adantl 482 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ ¬ (𝐷𝑗) ≤ 𝑆) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) = 𝑆)
178175, 177breq12d 4975 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ ¬ (𝐷𝑗) ≤ 𝑆) → (if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ↔ 𝑧𝑆))
179174, 178mpbird 258 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ ¬ (𝐷𝑗) ≤ 𝑆) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
180173, 179pm2.61dan 809 . . . . . . . . . . . . . . . 16 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
181160, 180pm2.61dan 809 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
182 icossico 12656 . . . . . . . . . . . . . . 15 ((((𝐶𝑗) ∈ ℝ* ∧ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ∈ ℝ*) ∧ ((𝐶𝑗) ≤ (𝐶𝑗) ∧ if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ⊆ ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))
18378, 136, 81, 181, 182syl22anc 835 . . . . . . . . . . . . . 14 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ⊆ ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))
184 volss 23817 . . . . . . . . . . . . . 14 ((((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ∈ dom vol ∧ ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)) ∈ dom vol ∧ ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ⊆ ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))) ≤ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))
18563, 135, 183, 184syl3anc 1364 . . . . . . . . . . . . 13 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))) ≤ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))
18654, 55, 64, 134, 185sge0lempt 42234 . . . . . . . . . . . 12 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))))
18797, 133, 98, 186leadd1dd 11102 . . . . . . . . . . 11 ((𝜑𝑧𝑈) → ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) + 𝐴) ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴))
18853, 99, 124, 132, 187letrd 10644 . . . . . . . . . 10 ((𝜑𝑧𝑈) → 𝑧 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴))
189188ex 413 . . . . . . . . 9 (𝜑 → (𝑧𝑈𝑧 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴)))
19050, 189ralrimi 3183 . . . . . . . 8 (𝜑 → ∀𝑧𝑈 𝑧 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴))
191 suprleub 11455 . . . . . . . . 9 (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥) ∧ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴) ∈ ℝ) → (sup(𝑈, ℝ, < ) ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴) ↔ ∀𝑧𝑈 𝑧 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴)))
19252, 46, 150, 123, 191syl31anc 1366 . . . . . . . 8 (𝜑 → (sup(𝑈, ℝ, < ) ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴) ↔ ∀𝑧𝑈 𝑧 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴)))
193190, 192mpbird 258 . . . . . . 7 (𝜑 → sup(𝑈, ℝ, < ) ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴))
19449, 193eqbrtrd 4984 . . . . . 6 (𝜑𝑆 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴))
195100, 2, 122lesubaddd 11085 . . . . . 6 (𝜑 → ((𝑆𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) ↔ 𝑆 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴)))
196194, 195mpbird 258 . . . . 5 (𝜑 → (𝑆𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))))
19748, 196jca 512 . . . 4 (𝜑 → (𝑆 ∈ (𝐴[,]𝐵) ∧ (𝑆𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))))))
198 oveq1 7023 . . . . . 6 (𝑧 = 𝑆 → (𝑧𝐴) = (𝑆𝐴))
199 breq2 4966 . . . . . . . . . . 11 (𝑧 = 𝑆 → ((𝐷𝑗) ≤ 𝑧 ↔ (𝐷𝑗) ≤ 𝑆))
200 id 22 . . . . . . . . . . 11 (𝑧 = 𝑆𝑧 = 𝑆)
201199, 200ifbieq2d 4406 . . . . . . . . . 10 (𝑧 = 𝑆 → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
202201oveq2d 7032 . . . . . . . . 9 (𝑧 = 𝑆 → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) = ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))
203202fveq2d 6542 . . . . . . . 8 (𝑧 = 𝑆 → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))) = (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))
204203mpteq2dv 5056 . . . . . . 7 (𝑧 = 𝑆 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))))
205204fveq2d 6542 . . . . . 6 (𝑧 = 𝑆 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) = (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))))
206198, 205breq12d 4975 . . . . 5 (𝑧 = 𝑆 → ((𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ↔ (𝑆𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))))))
207206elrab 3618 . . . 4 (𝑆 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))} ↔ (𝑆 ∈ (𝐴[,]𝐵) ∧ (𝑆𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))))))
208197, 207sylibr 235 . . 3 (𝜑𝑆 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))})
209208, 4syl6eleqr 2894 . 2 (𝜑𝑆𝑈)
210209, 45, 1503jca 1121 1 (𝜑 → (𝑆𝑈𝐴𝑈 ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1080   = wceq 1522  wcel 2081  wne 2984  wral 3105  wrex 3106  {crab 3109  Vcvv 3437  wss 3859  c0 4211  ifcif 4381   class class class wbr 4962  cmpt 5041  dom cdm 5443  wf 6221  cfv 6225  (class class class)co 7016  supcsup 8750  cr 10382  0cc0 10383   + caddc 10386  +∞cpnf 10518  *cxr 10520   < clt 10521  cle 10522  cmin 10717  cn 11486  [,)cico 12590  [,]cicc 12591  volcvol 23747  Σ^csumge0 42186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-inf2 8950  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460  ax-pre-sup 10461
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-of 7267  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-2o 7954  df-oadd 7957  df-er 8139  df-map 8258  df-pm 8259  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-sup 8752  df-inf 8753  df-oi 8820  df-dju 9176  df-card 9214  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-div 11146  df-nn 11487  df-2 11548  df-3 11549  df-n0 11746  df-z 11830  df-uz 12094  df-q 12198  df-rp 12240  df-xadd 12358  df-ioo 12592  df-ico 12594  df-icc 12595  df-fz 12743  df-fzo 12884  df-fl 13012  df-seq 13220  df-exp 13280  df-hash 13541  df-cj 14292  df-re 14293  df-im 14294  df-sqrt 14428  df-abs 14429  df-clim 14679  df-rlim 14680  df-sum 14877  df-xmet 20220  df-met 20221  df-ovol 23748  df-vol 23749  df-sumge0 42187
This theorem is referenced by:  hoidmv1lelem3  42417
  Copyright terms: Public domain W3C validator