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 41445
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 3847 . . . . . . . . 9 {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))} ⊆ (𝐴[,]𝐵)
64, 5eqsstri 3795 . . . . . . . 8 𝑈 ⊆ (𝐴[,]𝐵)
76a1i 11 . . . . . . 7 (𝜑𝑈 ⊆ (𝐴[,]𝐵))
82rexrd 10343 . . . . . . . . . . . 12 (𝜑𝐴 ∈ ℝ*)
93rexrd 10343 . . . . . . . . . . . 12 (𝜑𝐵 ∈ ℝ*)
10 hoidmv1lelem1.l . . . . . . . . . . . . 13 (𝜑𝐴 < 𝐵)
112, 3, 10ltled 10439 . . . . . . . . . . . 12 (𝜑𝐴𝐵)
12 lbicc2 12492 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
138, 9, 11, 12syl3anc 1490 . . . . . . . . . . 11 (𝜑𝐴 ∈ (𝐴[,]𝐵))
142recnd 10322 . . . . . . . . . . . . 13 (𝜑𝐴 ∈ ℂ)
1514subidd 10634 . . . . . . . . . . . 12 (𝜑 → (𝐴𝐴) = 0)
16 nfv 2009 . . . . . . . . . . . . 13 𝑗𝜑
17 nnex 11281 . . . . . . . . . . . . . 14 ℕ ∈ V
1817a1i 11 . . . . . . . . . . . . 13 (𝜑 → ℕ ∈ V)
19 volf 23587 . . . . . . . . . . . . . . 15 vol:dom vol⟶(0[,]+∞)
2019a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → vol:dom vol⟶(0[,]+∞))
21 hoidmv1lelem1.c . . . . . . . . . . . . . . . 16 (𝜑𝐶:ℕ⟶ℝ)
2221ffvelrnda 6549 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ∈ ℝ)
23 hoidmv1lelem1.d . . . . . . . . . . . . . . . . . 18 (𝜑𝐷:ℕ⟶ℝ)
2423ffvelrnda 6549 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗) ∈ ℝ)
252adantr 472 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → 𝐴 ∈ ℝ)
2624, 25ifcld 4288 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴) ∈ ℝ)
2726rexrd 10343 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴) ∈ ℝ*)
28 icombl 23622 . . . . . . . . . . . . . . 15 (((𝐶𝑗) ∈ ℝ ∧ if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴) ∈ ℝ*) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴)) ∈ dom vol)
2922, 27, 28syl2anc 579 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴)) ∈ dom vol)
3020, 29ffvelrnd 6550 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴))) ∈ (0[,]+∞))
3116, 18, 30sge0ge0mpt 41292 . . . . . . . . . . . 12 (𝜑 → 0 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴))))))
3215, 31eqbrtrd 4831 . . . . . . . . . . 11 (𝜑 → (𝐴𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴))))))
3313, 32jca 507 . . . . . . . . . 10 (𝜑 → (𝐴 ∈ (𝐴[,]𝐵) ∧ (𝐴𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴)))))))
34 oveq1 6849 . . . . . . . . . . . 12 (𝑧 = 𝐴 → (𝑧𝐴) = (𝐴𝐴))
35 breq2 4813 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝐴 → ((𝐷𝑗) ≤ 𝑧 ↔ (𝐷𝑗) ≤ 𝐴))
36 id 22 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝐴𝑧 = 𝐴)
3735, 36ifbieq2d 4268 . . . . . . . . . . . . . . . 16 (𝑧 = 𝐴 → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴))
3837oveq2d 6858 . . . . . . . . . . . . . . 15 (𝑧 = 𝐴 → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) = ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴)))
3938fveq2d 6379 . . . . . . . . . . . . . 14 (𝑧 = 𝐴 → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))) = (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴))))
4039mpteq2dv 4904 . . . . . . . . . . . . 13 (𝑧 = 𝐴 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴)))))
4140fveq2d 6379 . . . . . . . . . . . 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 4822 . . . . . . . . . . 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 3519 . . . . . . . . . 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 225 . . . . . . . . 9 (𝜑𝐴 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))})
4544, 4syl6eleqr 2855 . . . . . . . 8 (𝜑𝐴𝑈)
4645ne0d 4086 . . . . . . 7 (𝜑𝑈 ≠ ∅)
472, 3, 7, 46supicc 12527 . . . . . 6 (𝜑 → sup(𝑈, ℝ, < ) ∈ (𝐴[,]𝐵))
481, 47syl5eqel 2848 . . . . 5 (𝜑𝑆 ∈ (𝐴[,]𝐵))
491a1i 11 . . . . . . 7 (𝜑𝑆 = sup(𝑈, ℝ, < ))
50 nfv 2009 . . . . . . . . 9 𝑧𝜑
512, 3iccssred 40369 . . . . . . . . . . . . 13 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
527, 51sstrd 3771 . . . . . . . . . . . 12 (𝜑𝑈 ⊆ ℝ)
5352sselda 3761 . . . . . . . . . . 11 ((𝜑𝑧𝑈) → 𝑧 ∈ ℝ)
54 nfv 2009 . . . . . . . . . . . . . . . 16 𝑗(𝜑𝑧𝑈)
5517a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑈) → ℕ ∈ V)
5619a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → vol:dom vol⟶(0[,]+∞))
5722adantlr 706 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (𝐶𝑗) ∈ ℝ)
5824adantlr 706 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (𝐷𝑗) ∈ ℝ)
5953adantr 472 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → 𝑧 ∈ ℝ)
6058, 59ifcld 4288 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ∈ ℝ)
6160rexrd 10343 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ∈ ℝ*)
62 icombl 23622 . . . . . . . . . . . . . . . . . 18 (((𝐶𝑗) ∈ ℝ ∧ if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ∈ ℝ*) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ∈ dom vol)
6357, 61, 62syl2anc 579 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ∈ dom vol)
6456, 63ffvelrnd 6550 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))) ∈ (0[,]+∞))
6554, 55, 64sge0xrclmpt 41282 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ∈ ℝ*)
66 pnfxr 10346 . . . . . . . . . . . . . . . 16 +∞ ∈ ℝ*
6766a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑈) → +∞ ∈ ℝ*)
68 hoidmv1lelem1.r . . . . . . . . . . . . . . . . . 18 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) ∈ ℝ)
6968rexrd 10343 . . . . . . . . . . . . . . . . 17 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) ∈ ℝ*)
7069adantr 472 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) ∈ ℝ*)
7124rexrd 10343 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗) ∈ ℝ*)
72 icombl 23622 . . . . . . . . . . . . . . . . . . . 20 (((𝐶𝑗) ∈ ℝ ∧ (𝐷𝑗) ∈ ℝ*) → ((𝐶𝑗)[,)(𝐷𝑗)) ∈ dom vol)
7322, 71, 72syl2anc 579 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)[,)(𝐷𝑗)) ∈ dom vol)
7420, 73ffvelrnd 6550 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)(𝐷𝑗))) ∈ (0[,]+∞))
7574adantlr 706 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)(𝐷𝑗))) ∈ (0[,]+∞))
7673adantlr 706 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶𝑗)[,)(𝐷𝑗)) ∈ dom vol)
7722rexrd 10343 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ∈ ℝ*)
7877adantlr 706 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (𝐶𝑗) ∈ ℝ*)
7971adantlr 706 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (𝐷𝑗) ∈ ℝ*)
8022leidd 10848 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ≤ (𝐶𝑗))
8180adantlr 706 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (𝐶𝑗) ≤ (𝐶𝑗))
82 min1 12222 . . . . . . . . . . . . . . . . . . . 20 (((𝐷𝑗) ∈ ℝ ∧ 𝑧 ∈ ℝ) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ (𝐷𝑗))
8358, 59, 82syl2anc 579 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ (𝐷𝑗))
84 icossico 12445 . . . . . . . . . . . . . . . . . . 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 867 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ⊆ ((𝐶𝑗)[,)(𝐷𝑗)))
86 volss 23591 . . . . . . . . . . . . . . . . . 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 1490 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))) ≤ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))
8854, 55, 64, 75, 87sge0lempt 41264 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))))
8968ltpnfd 12155 . . . . . . . . . . . . . . . . 17 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) < +∞)
9089adantr 472 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) < +∞)
9165, 70, 67, 88, 90xrlelttrd 12193 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) < +∞)
9265, 67, 91xrltned 40211 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ≠ +∞)
9392neneqd 2942 . . . . . . . . . . . . 13 ((𝜑𝑧𝑈) → ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) = +∞)
94 eqid 2765 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))
9564, 94fmptd 6574 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑈) → (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))):ℕ⟶(0[,]+∞))
9655, 95sge0repnf 41240 . . . . . . . . . . . . 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 248 . . . . . . . . . . . 12 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ∈ ℝ)
982adantr 472 . . . . . . . . . . . 12 ((𝜑𝑧𝑈) → 𝐴 ∈ ℝ)
9997, 98readdcld 10323 . . . . . . . . . . 11 ((𝜑𝑧𝑈) → ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) + 𝐴) ∈ ℝ)
10051, 48sseldd 3762 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑆 ∈ ℝ)
101100adantr 472 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ ℕ) → 𝑆 ∈ ℝ)
10224, 101ifcld 4288 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ∈ ℝ)
103102rexrd 10343 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ∈ ℝ*)
104 icombl 23622 . . . . . . . . . . . . . . . . . . 19 (((𝐶𝑗) ∈ ℝ ∧ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ∈ ℝ*) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)) ∈ dom vol)
10522, 103, 104syl2anc 579 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)) ∈ dom vol)
10620, 105ffvelrnd 6550 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))) ∈ (0[,]+∞))
10716, 18, 106sge0xrclmpt 41282 . . . . . . . . . . . . . . . 16 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) ∈ ℝ*)
10866a1i 11 . . . . . . . . . . . . . . . 16 (𝜑 → +∞ ∈ ℝ*)
109 min1 12222 . . . . . . . . . . . . . . . . . . . . 21 (((𝐷𝑗) ∈ ℝ ∧ 𝑆 ∈ ℝ) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ≤ (𝐷𝑗))
11024, 101, 109syl2anc 579 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ≤ (𝐷𝑗))
111 icossico 12445 . . . . . . . . . . . . . . . . . . . 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 867 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)) ⊆ ((𝐶𝑗)[,)(𝐷𝑗)))
113 volss 23591 . . . . . . . . . . . . . . . . . . 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 1490 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))) ≤ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))
11516, 18, 106, 74, 114sge0lempt 41264 . . . . . . . . . . . . . . . . 17 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))))
116107, 69, 108, 115, 89xrlelttrd 12193 . . . . . . . . . . . . . . . 16 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) < +∞)
117107, 108, 116xrltned 40211 . . . . . . . . . . . . . . 15 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) ≠ +∞)
118117neneqd 2942 . . . . . . . . . . . . . 14 (𝜑 → ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) = +∞)
119 eqid 2765 . . . . . . . . . . . . . . . 16 (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))
120106, 119fmptd 6574 . . . . . . . . . . . . . . 15 (𝜑 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))):ℕ⟶(0[,]+∞))
12118, 120sge0repnf 41240 . . . . . . . . . . . . . 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 248 . . . . . . . . . . . . 13 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) ∈ ℝ)
123122, 2readdcld 10323 . . . . . . . . . . . 12 (𝜑 → ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴) ∈ ℝ)
124123adantr 472 . . . . . . . . . . 11 ((𝜑𝑧𝑈) → ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴) ∈ ℝ)
1254eleq2i 2836 . . . . . . . . . . . . . . . 16 (𝑧𝑈𝑧 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))})
126125biimpi 207 . . . . . . . . . . . . . . 15 (𝑧𝑈𝑧 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))})
127126adantl 473 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑈) → 𝑧 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))})
128 rabid 3263 . . . . . . . . . . . . . 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 209 . . . . . . . . . . . . 13 ((𝜑𝑧𝑈) → (𝑧 ∈ (𝐴[,]𝐵) ∧ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))))
130129simprd 489 . . . . . . . . . . . 12 ((𝜑𝑧𝑈) → (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))))
13153, 98, 97lesubaddd 10878 . . . . . . . . . . . 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 223 . . . . . . . . . . 11 ((𝜑𝑧𝑈) → 𝑧 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) + 𝐴))
133122adantr 472 . . . . . . . . . . . 12 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) ∈ ℝ)
134106adantlr 706 . . . . . . . . . . . . 13 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))) ∈ (0[,]+∞))
135105adantlr 706 . . . . . . . . . . . . . 14 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)) ∈ dom vol)
136103adantlr 706 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ∈ ℝ*)
13760adantr 472 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ∈ ℝ)
138 eqidd 2766 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → (𝐷𝑗) = (𝐷𝑗))
139 iftrue 4249 . . . . . . . . . . . . . . . . . . 19 ((𝐷𝑗) ≤ 𝑧 → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = (𝐷𝑗))
140139adantl 473 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = (𝐷𝑗))
14158adantr 472 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → (𝐷𝑗) ∈ ℝ)
14259adantr 472 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → 𝑧 ∈ ℝ)
143100ad3antrrr 721 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → 𝑆 ∈ ℝ)
144 simpr 477 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → (𝐷𝑗) ≤ 𝑧)
14552adantr 472 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑧𝑈) → 𝑈 ⊆ ℝ)
14646adantr 472 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑧𝑈) → 𝑈 ≠ ∅)
1472, 3jca 507 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
148 iccsupr 12469 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑈 ⊆ (𝐴[,]𝐵) ∧ 𝐴𝑈) → (𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥))
149147, 7, 45, 148syl3anc 1490 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥))
150149simp3d 1174 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥)
151150adantr 472 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑧𝑈) → ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥)
152127, 125sylibr 225 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑧𝑈) → 𝑧𝑈)
153 suprub 11238 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥) ∧ 𝑧𝑈) → 𝑧 ≤ sup(𝑈, ℝ, < ))
154145, 146, 151, 152, 153syl31anc 1492 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑧𝑈) → 𝑧 ≤ sup(𝑈, ℝ, < ))
155154, 1syl6breqr 4851 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑧𝑈) → 𝑧𝑆)
156155ad2antrr 717 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → 𝑧𝑆)
157141, 142, 143, 144, 156letrd 10448 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → (𝐷𝑗) ≤ 𝑆)
158157iftrued 4251 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) = (𝐷𝑗))
159138, 140, 1583eqtr4d 2809 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
160137, 159eqled 10394 . . . . . . . . . . . . . . . 16 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
16159adantr 472 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → 𝑧 ∈ ℝ)
16258adantr 472 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → (𝐷𝑗) ∈ ℝ)
163 simpr 477 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → ¬ (𝐷𝑗) ≤ 𝑧)
164161, 162ltnled 10438 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → (𝑧 < (𝐷𝑗) ↔ ¬ (𝐷𝑗) ≤ 𝑧))
165163, 164mpbird 248 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → 𝑧 < (𝐷𝑗))
166161, 162, 165ltled 10439 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → 𝑧 ≤ (𝐷𝑗))
167166adantr 472 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ (𝐷𝑗) ≤ 𝑆) → 𝑧 ≤ (𝐷𝑗))
168 iffalse 4252 . . . . . . . . . . . . . . . . . . . 20 (¬ (𝐷𝑗) ≤ 𝑧 → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = 𝑧)
169168ad2antlr 718 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ (𝐷𝑗) ≤ 𝑆) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = 𝑧)
170 iftrue 4249 . . . . . . . . . . . . . . . . . . . 20 ((𝐷𝑗) ≤ 𝑆 → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) = (𝐷𝑗))
171170adantl 473 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ (𝐷𝑗) ≤ 𝑆) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) = (𝐷𝑗))
172169, 171breq12d 4822 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ (𝐷𝑗) ≤ 𝑆) → (if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ↔ 𝑧 ≤ (𝐷𝑗)))
173167, 172mpbird 248 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ (𝐷𝑗) ≤ 𝑆) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
174155ad3antrrr 721 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ ¬ (𝐷𝑗) ≤ 𝑆) → 𝑧𝑆)
175168ad2antlr 718 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ ¬ (𝐷𝑗) ≤ 𝑆) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = 𝑧)
176 iffalse 4252 . . . . . . . . . . . . . . . . . . . 20 (¬ (𝐷𝑗) ≤ 𝑆 → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) = 𝑆)
177176adantl 473 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ ¬ (𝐷𝑗) ≤ 𝑆) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) = 𝑆)
178175, 177breq12d 4822 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ ¬ (𝐷𝑗) ≤ 𝑆) → (if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ↔ 𝑧𝑆))
179174, 178mpbird 248 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ ¬ (𝐷𝑗) ≤ 𝑆) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
180173, 179pm2.61dan 847 . . . . . . . . . . . . . . . 16 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
181160, 180pm2.61dan 847 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
182 icossico 12445 . . . . . . . . . . . . . . 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 867 . . . . . . . . . . . . . 14 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ⊆ ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))
184 volss 23591 . . . . . . . . . . . . . 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 1490 . . . . . . . . . . . . 13 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))) ≤ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))
18654, 55, 64, 134, 185sge0lempt 41264 . . . . . . . . . . . 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 10895 . . . . . . . . . . 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 10448 . . . . . . . . . 10 ((𝜑𝑧𝑈) → 𝑧 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴))
189188ex 401 . . . . . . . . 9 (𝜑 → (𝑧𝑈𝑧 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴)))
19050, 189ralrimi 3104 . . . . . . . 8 (𝜑 → ∀𝑧𝑈 𝑧 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴))
191 suprleub 11243 . . . . . . . . 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 1492 . . . . . . . 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 248 . . . . . . 7 (𝜑 → sup(𝑈, ℝ, < ) ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴))
19449, 193eqbrtrd 4831 . . . . . 6 (𝜑𝑆 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴))
195100, 2, 122lesubaddd 10878 . . . . . 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 248 . . . . 5 (𝜑 → (𝑆𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))))
19748, 196jca 507 . . . 4 (𝜑 → (𝑆 ∈ (𝐴[,]𝐵) ∧ (𝑆𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))))))
198 oveq1 6849 . . . . . 6 (𝑧 = 𝑆 → (𝑧𝐴) = (𝑆𝐴))
199 breq2 4813 . . . . . . . . . . 11 (𝑧 = 𝑆 → ((𝐷𝑗) ≤ 𝑧 ↔ (𝐷𝑗) ≤ 𝑆))
200 id 22 . . . . . . . . . . 11 (𝑧 = 𝑆𝑧 = 𝑆)
201199, 200ifbieq2d 4268 . . . . . . . . . 10 (𝑧 = 𝑆 → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
202201oveq2d 6858 . . . . . . . . 9 (𝑧 = 𝑆 → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) = ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))
203202fveq2d 6379 . . . . . . . 8 (𝑧 = 𝑆 → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))) = (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))
204203mpteq2dv 4904 . . . . . . 7 (𝑧 = 𝑆 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))))
205204fveq2d 6379 . . . . . 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 4822 . . . . 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 3519 . . . 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 225 . . 3 (𝜑𝑆 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))})
209208, 4syl6eleqr 2855 . 2 (𝜑𝑆𝑈)
210209, 45, 1503jca 1158 1 (𝜑 → (𝑆𝑈𝐴𝑈 ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wral 3055  wrex 3056  {crab 3059  Vcvv 3350  wss 3732  c0 4079  ifcif 4243   class class class wbr 4809  cmpt 4888  dom cdm 5277  wf 6064  cfv 6068  (class class class)co 6842  supcsup 8553  cr 10188  0cc0 10189   + caddc 10192  +∞cpnf 10325  *cxr 10327   < clt 10328  cle 10329  cmin 10520  cn 11274  [,)cico 12379  [,]cicc 12380  volcvol 23521  Σ^csumge0 41216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-map 8062  df-pm 8063  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-sup 8555  df-inf 8556  df-oi 8622  df-card 9016  df-cda 9243  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-n0 11539  df-z 11625  df-uz 11887  df-q 11990  df-rp 12029  df-xadd 12147  df-ioo 12381  df-ico 12383  df-icc 12384  df-fz 12534  df-fzo 12674  df-fl 12801  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14124  df-re 14125  df-im 14126  df-sqrt 14260  df-abs 14261  df-clim 14504  df-rlim 14505  df-sum 14702  df-xmet 20012  df-met 20013  df-ovol 23522  df-vol 23523  df-sumge0 41217
This theorem is referenced by:  hoidmv1lelem3  41447
  Copyright terms: Public domain W3C validator