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 46606
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 4080 . . . . . . . . 9 {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))} ⊆ (𝐴[,]𝐵)
64, 5eqsstri 4030 . . . . . . . 8 𝑈 ⊆ (𝐴[,]𝐵)
76a1i 11 . . . . . . 7 (𝜑𝑈 ⊆ (𝐴[,]𝐵))
82rexrd 11311 . . . . . . . . . . . 12 (𝜑𝐴 ∈ ℝ*)
93rexrd 11311 . . . . . . . . . . . 12 (𝜑𝐵 ∈ ℝ*)
10 hoidmv1lelem1.l . . . . . . . . . . . . 13 (𝜑𝐴 < 𝐵)
112, 3, 10ltled 11409 . . . . . . . . . . . 12 (𝜑𝐴𝐵)
12 lbicc2 13504 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
138, 9, 11, 12syl3anc 1373 . . . . . . . . . . 11 (𝜑𝐴 ∈ (𝐴[,]𝐵))
142recnd 11289 . . . . . . . . . . . . 13 (𝜑𝐴 ∈ ℂ)
1514subidd 11608 . . . . . . . . . . . 12 (𝜑 → (𝐴𝐴) = 0)
16 nfv 1914 . . . . . . . . . . . . 13 𝑗𝜑
17 nnex 12272 . . . . . . . . . . . . . 14 ℕ ∈ V
1817a1i 11 . . . . . . . . . . . . 13 (𝜑 → ℕ ∈ V)
19 volf 25564 . . . . . . . . . . . . . . 15 vol:dom vol⟶(0[,]+∞)
2019a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → vol:dom vol⟶(0[,]+∞))
21 hoidmv1lelem1.c . . . . . . . . . . . . . . . 16 (𝜑𝐶:ℕ⟶ℝ)
2221ffvelcdmda 7104 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ∈ ℝ)
23 hoidmv1lelem1.d . . . . . . . . . . . . . . . . . 18 (𝜑𝐷:ℕ⟶ℝ)
2423ffvelcdmda 7104 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗) ∈ ℝ)
252adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → 𝐴 ∈ ℝ)
2624, 25ifcld 4572 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴) ∈ ℝ)
2726rexrd 11311 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴) ∈ ℝ*)
28 icombl 25599 . . . . . . . . . . . . . . 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, 29ffvelcdmd 7105 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴))) ∈ (0[,]+∞))
3116, 18, 30sge0ge0mpt 46453 . . . . . . . . . . . 12 (𝜑 → 0 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴))))))
3215, 31eqbrtrd 5165 . . . . . . . . . . 11 (𝜑 → (𝐴𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴))))))
3313, 32jca 511 . . . . . . . . . 10 (𝜑 → (𝐴 ∈ (𝐴[,]𝐵) ∧ (𝐴𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴)))))))
34 oveq1 7438 . . . . . . . . . . . 12 (𝑧 = 𝐴 → (𝑧𝐴) = (𝐴𝐴))
35 breq2 5147 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝐴 → ((𝐷𝑗) ≤ 𝑧 ↔ (𝐷𝑗) ≤ 𝐴))
36 id 22 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝐴𝑧 = 𝐴)
3735, 36ifbieq2d 4552 . . . . . . . . . . . . . . . 16 (𝑧 = 𝐴 → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴))
3837oveq2d 7447 . . . . . . . . . . . . . . 15 (𝑧 = 𝐴 → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) = ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴)))
3938fveq2d 6910 . . . . . . . . . . . . . 14 (𝑧 = 𝐴 → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))) = (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴))))
4039mpteq2dv 5244 . . . . . . . . . . . . 13 (𝑧 = 𝐴 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴)))))
4140fveq2d 6910 . . . . . . . . . . . 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 5156 . . . . . . . . . . 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 3692 . . . . . . . . . 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 234 . . . . . . . . 9 (𝜑𝐴 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))})
4544, 4eleqtrrdi 2852 . . . . . . . 8 (𝜑𝐴𝑈)
4645ne0d 4342 . . . . . . 7 (𝜑𝑈 ≠ ∅)
472, 3, 7, 46supicc 13541 . . . . . 6 (𝜑 → sup(𝑈, ℝ, < ) ∈ (𝐴[,]𝐵))
481, 47eqeltrid 2845 . . . . 5 (𝜑𝑆 ∈ (𝐴[,]𝐵))
491a1i 11 . . . . . . 7 (𝜑𝑆 = sup(𝑈, ℝ, < ))
50 nfv 1914 . . . . . . . . 9 𝑧𝜑
512, 3iccssred 13474 . . . . . . . . . . . . 13 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
527, 51sstrd 3994 . . . . . . . . . . . 12 (𝜑𝑈 ⊆ ℝ)
5352sselda 3983 . . . . . . . . . . 11 ((𝜑𝑧𝑈) → 𝑧 ∈ ℝ)
54 nfv 1914 . . . . . . . . . . . . . . . 16 𝑗(𝜑𝑧𝑈)
5517a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑈) → ℕ ∈ V)
5619a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → vol:dom vol⟶(0[,]+∞))
5722adantlr 715 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (𝐶𝑗) ∈ ℝ)
5824adantlr 715 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (𝐷𝑗) ∈ ℝ)
5953adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → 𝑧 ∈ ℝ)
6058, 59ifcld 4572 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ∈ ℝ)
6160rexrd 11311 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ∈ ℝ*)
62 icombl 25599 . . . . . . . . . . . . . . . . . 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, 63ffvelcdmd 7105 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))) ∈ (0[,]+∞))
6554, 55, 64sge0xrclmpt 46443 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ∈ ℝ*)
66 pnfxr 11315 . . . . . . . . . . . . . . . 16 +∞ ∈ ℝ*
6766a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑈) → +∞ ∈ ℝ*)
68 hoidmv1lelem1.r . . . . . . . . . . . . . . . . . 18 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) ∈ ℝ)
6968rexrd 11311 . . . . . . . . . . . . . . . . 17 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) ∈ ℝ*)
7069adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) ∈ ℝ*)
7124rexrd 11311 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗) ∈ ℝ*)
72 icombl 25599 . . . . . . . . . . . . . . . . . . . 20 (((𝐶𝑗) ∈ ℝ ∧ (𝐷𝑗) ∈ ℝ*) → ((𝐶𝑗)[,)(𝐷𝑗)) ∈ dom vol)
7322, 71, 72syl2anc 584 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)[,)(𝐷𝑗)) ∈ dom vol)
7420, 73ffvelcdmd 7105 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)(𝐷𝑗))) ∈ (0[,]+∞))
7574adantlr 715 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)(𝐷𝑗))) ∈ (0[,]+∞))
7673adantlr 715 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶𝑗)[,)(𝐷𝑗)) ∈ dom vol)
7722rexrd 11311 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ∈ ℝ*)
7877adantlr 715 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (𝐶𝑗) ∈ ℝ*)
7971adantlr 715 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (𝐷𝑗) ∈ ℝ*)
8022leidd 11829 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ≤ (𝐶𝑗))
8180adantlr 715 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (𝐶𝑗) ≤ (𝐶𝑗))
82 min1 13231 . . . . . . . . . . . . . . . . . . . 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 13457 . . . . . . . . . . . . . . . . . . 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 839 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ⊆ ((𝐶𝑗)[,)(𝐷𝑗)))
86 volss 25568 . . . . . . . . . . . . . . . . . 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 1373 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))) ≤ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))
8854, 55, 64, 75, 87sge0lempt 46425 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))))
8968ltpnfd 13163 . . . . . . . . . . . . . . . . 17 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) < +∞)
9089adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) < +∞)
9165, 70, 67, 88, 90xrlelttrd 13202 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) < +∞)
9265, 67, 91xrltned 45368 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ≠ +∞)
9392neneqd 2945 . . . . . . . . . . . . 13 ((𝜑𝑧𝑈) → ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) = +∞)
94 eqid 2737 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))
9564, 94fmptd 7134 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑈) → (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))):ℕ⟶(0[,]+∞))
9655, 95sge0repnf 46401 . . . . . . . . . . . . 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 257 . . . . . . . . . . . 12 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ∈ ℝ)
982adantr 480 . . . . . . . . . . . 12 ((𝜑𝑧𝑈) → 𝐴 ∈ ℝ)
9997, 98readdcld 11290 . . . . . . . . . . 11 ((𝜑𝑧𝑈) → ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) + 𝐴) ∈ ℝ)
10051, 48sseldd 3984 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑆 ∈ ℝ)
101100adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ ℕ) → 𝑆 ∈ ℝ)
10224, 101ifcld 4572 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ∈ ℝ)
103102rexrd 11311 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ∈ ℝ*)
104 icombl 25599 . . . . . . . . . . . . . . . . . . 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, 105ffvelcdmd 7105 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))) ∈ (0[,]+∞))
10716, 18, 106sge0xrclmpt 46443 . . . . . . . . . . . . . . . 16 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) ∈ ℝ*)
10866a1i 11 . . . . . . . . . . . . . . . 16 (𝜑 → +∞ ∈ ℝ*)
109 min1 13231 . . . . . . . . . . . . . . . . . . . . 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 13457 . . . . . . . . . . . . . . . . . . . 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 839 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)) ⊆ ((𝐶𝑗)[,)(𝐷𝑗)))
113 volss 25568 . . . . . . . . . . . . . . . . . . 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 1373 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))) ≤ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))
11516, 18, 106, 74, 114sge0lempt 46425 . . . . . . . . . . . . . . . . 17 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))))
116107, 69, 108, 115, 89xrlelttrd 13202 . . . . . . . . . . . . . . . 16 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) < +∞)
117107, 108, 116xrltned 45368 . . . . . . . . . . . . . . 15 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) ≠ +∞)
118117neneqd 2945 . . . . . . . . . . . . . 14 (𝜑 → ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) = +∞)
119 eqid 2737 . . . . . . . . . . . . . . . 16 (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))
120106, 119fmptd 7134 . . . . . . . . . . . . . . 15 (𝜑 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))):ℕ⟶(0[,]+∞))
12118, 120sge0repnf 46401 . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . 13 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) ∈ ℝ)
123122, 2readdcld 11290 . . . . . . . . . . . 12 (𝜑 → ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴) ∈ ℝ)
124123adantr 480 . . . . . . . . . . 11 ((𝜑𝑧𝑈) → ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴) ∈ ℝ)
1254eleq2i 2833 . . . . . . . . . . . . . . . 16 (𝑧𝑈𝑧 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))})
126125biimpi 216 . . . . . . . . . . . . . . 15 (𝑧𝑈𝑧 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))})
127126adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑈) → 𝑧 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))})
128 rabid 3458 . . . . . . . . . . . . . 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 218 . . . . . . . . . . . . 13 ((𝜑𝑧𝑈) → (𝑧 ∈ (𝐴[,]𝐵) ∧ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))))
130129simprd 495 . . . . . . . . . . . 12 ((𝜑𝑧𝑈) → (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))))
13153, 98, 97lesubaddd 11860 . . . . . . . . . . . 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 232 . . . . . . . . . . 11 ((𝜑𝑧𝑈) → 𝑧 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) + 𝐴))
133122adantr 480 . . . . . . . . . . . 12 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) ∈ ℝ)
134106adantlr 715 . . . . . . . . . . . . 13 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))) ∈ (0[,]+∞))
135105adantlr 715 . . . . . . . . . . . . . 14 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)) ∈ dom vol)
136103adantlr 715 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ∈ ℝ*)
13760adantr 480 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ∈ ℝ)
138 eqidd 2738 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → (𝐷𝑗) = (𝐷𝑗))
139 iftrue 4531 . . . . . . . . . . . . . . . . . . 19 ((𝐷𝑗) ≤ 𝑧 → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = (𝐷𝑗))
140139adantl 481 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = (𝐷𝑗))
14158adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → (𝐷𝑗) ∈ ℝ)
14259adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → 𝑧 ∈ ℝ)
143100ad3antrrr 730 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → 𝑆 ∈ ℝ)
144 simpr 484 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → (𝐷𝑗) ≤ 𝑧)
14552adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑧𝑈) → 𝑈 ⊆ ℝ)
14646adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑧𝑈) → 𝑈 ≠ ∅)
1472, 3jca 511 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
148 iccsupr 13482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑈 ⊆ (𝐴[,]𝐵) ∧ 𝐴𝑈) → (𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥))
149147, 7, 45, 148syl3anc 1373 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥))
150149simp3d 1145 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥)
151150adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑧𝑈) → ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥)
152127, 125sylibr 234 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑧𝑈) → 𝑧𝑈)
153 suprub 12229 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥) ∧ 𝑧𝑈) → 𝑧 ≤ sup(𝑈, ℝ, < ))
154145, 146, 151, 152, 153syl31anc 1375 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑧𝑈) → 𝑧 ≤ sup(𝑈, ℝ, < ))
155154, 1breqtrrdi 5185 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑧𝑈) → 𝑧𝑆)
156155ad2antrr 726 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → 𝑧𝑆)
157141, 142, 143, 144, 156letrd 11418 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → (𝐷𝑗) ≤ 𝑆)
158157iftrued 4533 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) = (𝐷𝑗))
159138, 140, 1583eqtr4d 2787 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
160137, 159eqled 11364 . . . . . . . . . . . . . . . 16 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
16159adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → 𝑧 ∈ ℝ)
16258adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → (𝐷𝑗) ∈ ℝ)
163 simpr 484 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → ¬ (𝐷𝑗) ≤ 𝑧)
164161, 162ltnled 11408 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → (𝑧 < (𝐷𝑗) ↔ ¬ (𝐷𝑗) ≤ 𝑧))
165163, 164mpbird 257 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → 𝑧 < (𝐷𝑗))
166161, 162, 165ltled 11409 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → 𝑧 ≤ (𝐷𝑗))
167166adantr 480 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ (𝐷𝑗) ≤ 𝑆) → 𝑧 ≤ (𝐷𝑗))
168 iffalse 4534 . . . . . . . . . . . . . . . . . . . 20 (¬ (𝐷𝑗) ≤ 𝑧 → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = 𝑧)
169168ad2antlr 727 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ (𝐷𝑗) ≤ 𝑆) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = 𝑧)
170 iftrue 4531 . . . . . . . . . . . . . . . . . . . 20 ((𝐷𝑗) ≤ 𝑆 → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) = (𝐷𝑗))
171170adantl 481 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ (𝐷𝑗) ≤ 𝑆) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) = (𝐷𝑗))
172169, 171breq12d 5156 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ (𝐷𝑗) ≤ 𝑆) → (if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ↔ 𝑧 ≤ (𝐷𝑗)))
173167, 172mpbird 257 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ (𝐷𝑗) ≤ 𝑆) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
174155ad3antrrr 730 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ ¬ (𝐷𝑗) ≤ 𝑆) → 𝑧𝑆)
175168ad2antlr 727 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ ¬ (𝐷𝑗) ≤ 𝑆) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = 𝑧)
176 iffalse 4534 . . . . . . . . . . . . . . . . . . . 20 (¬ (𝐷𝑗) ≤ 𝑆 → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) = 𝑆)
177176adantl 481 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ ¬ (𝐷𝑗) ≤ 𝑆) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) = 𝑆)
178175, 177breq12d 5156 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ ¬ (𝐷𝑗) ≤ 𝑆) → (if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ↔ 𝑧𝑆))
179174, 178mpbird 257 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ ¬ (𝐷𝑗) ≤ 𝑆) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
180173, 179pm2.61dan 813 . . . . . . . . . . . . . . . 16 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
181160, 180pm2.61dan 813 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
182 icossico 13457 . . . . . . . . . . . . . . 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 839 . . . . . . . . . . . . . 14 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ⊆ ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))
184 volss 25568 . . . . . . . . . . . . . 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 1373 . . . . . . . . . . . . 13 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))) ≤ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))
18654, 55, 64, 134, 185sge0lempt 46425 . . . . . . . . . . . 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 11877 . . . . . . . . . . 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 11418 . . . . . . . . . 10 ((𝜑𝑧𝑈) → 𝑧 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴))
189188ex 412 . . . . . . . . 9 (𝜑 → (𝑧𝑈𝑧 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴)))
19050, 189ralrimi 3257 . . . . . . . 8 (𝜑 → ∀𝑧𝑈 𝑧 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴))
191 suprleub 12234 . . . . . . . . 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 1375 . . . . . . . 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 257 . . . . . . 7 (𝜑 → sup(𝑈, ℝ, < ) ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴))
19449, 193eqbrtrd 5165 . . . . . 6 (𝜑𝑆 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴))
195100, 2, 122lesubaddd 11860 . . . . . 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 257 . . . . 5 (𝜑 → (𝑆𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))))
19748, 196jca 511 . . . 4 (𝜑 → (𝑆 ∈ (𝐴[,]𝐵) ∧ (𝑆𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))))))
198 oveq1 7438 . . . . . 6 (𝑧 = 𝑆 → (𝑧𝐴) = (𝑆𝐴))
199 breq2 5147 . . . . . . . . . . 11 (𝑧 = 𝑆 → ((𝐷𝑗) ≤ 𝑧 ↔ (𝐷𝑗) ≤ 𝑆))
200 id 22 . . . . . . . . . . 11 (𝑧 = 𝑆𝑧 = 𝑆)
201199, 200ifbieq2d 4552 . . . . . . . . . 10 (𝑧 = 𝑆 → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
202201oveq2d 7447 . . . . . . . . 9 (𝑧 = 𝑆 → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) = ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))
203202fveq2d 6910 . . . . . . . 8 (𝑧 = 𝑆 → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))) = (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))
204203mpteq2dv 5244 . . . . . . 7 (𝑧 = 𝑆 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))))
205204fveq2d 6910 . . . . . 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 5156 . . . . 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 3692 . . . 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 234 . . 3 (𝜑𝑆 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))})
209208, 4eleqtrrdi 2852 . 2 (𝜑𝑆𝑈)
210209, 45, 1503jca 1129 1 (𝜑 → (𝑆𝑈𝐴𝑈 ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  {crab 3436  Vcvv 3480  wss 3951  c0 4333  ifcif 4525   class class class wbr 5143  cmpt 5225  dom cdm 5685  wf 6557  cfv 6561  (class class class)co 7431  supcsup 9480  cr 11154  0cc0 11155   + caddc 11158  +∞cpnf 11292  *cxr 11294   < clt 11295  cle 11296  cmin 11492  cn 12266  [,)cico 13389  [,]cicc 13390  volcvol 25498  Σ^csumge0 46377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xadd 13155  df-ioo 13391  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-rlim 15525  df-sum 15723  df-xmet 21357  df-met 21358  df-ovol 25499  df-vol 25500  df-sumge0 46378
This theorem is referenced by:  hoidmv1lelem3  46608
  Copyright terms: Public domain W3C validator