Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hoidmvlelem4 Structured version   Visualization version   GIF version

Theorem hoidmvlelem4 42746
 Description: The dimensional volume of a multidimensional half-open interval is less than or equal the generalized sum of the dimensional volumes of countable half-open intervals that cover it. Induction step of Lemma 115B of [Fremlin1] p. 29, case nonempty interval and dimension of the space greater than 1. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
Hypotheses
Ref Expression
hoidmvlelem4.l 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))
hoidmvlelem4.x (𝜑𝑋 ∈ Fin)
hoidmvlelem4.y (𝜑𝑌𝑋)
hoidmvlelem4.n (𝜑𝑌 ≠ ∅)
hoidmvlelem4.z (𝜑𝑍 ∈ (𝑋𝑌))
hoidmvlelem4.w 𝑊 = (𝑌 ∪ {𝑍})
hoidmvlelem4.a (𝜑𝐴:𝑊⟶ℝ)
hoidmvlelem4.b (𝜑𝐵:𝑊⟶ℝ)
hoidmvlelem4.k ((𝜑𝑘𝑊) → (𝐴𝑘) < (𝐵𝑘))
hoidmvlelem4.c (𝜑𝐶:ℕ⟶(ℝ ↑m 𝑊))
hoidmvlelem4.d (𝜑𝐷:ℕ⟶(ℝ ↑m 𝑊))
hoidmvlelem4.r (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)
hoidmvlelem4.h 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑m 𝑊) ↦ (𝑗𝑊 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)))))
hoidmvlelem4.14 𝐺 = ((𝐴𝑌)(𝐿𝑌)(𝐵𝑌))
hoidmvlelem4.e (𝜑𝐸 ∈ ℝ+)
hoidmvlelem4.u 𝑈 = {𝑧 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))))}
hoidmvlelem4.s 𝑆 = sup(𝑈, ℝ, < )
hoidmvlelem4.i (𝜑 → ∀𝑒 ∈ (ℝ ↑m 𝑌)∀𝑓 ∈ (ℝ ↑m 𝑌)∀𝑔 ∈ ((ℝ ↑m 𝑌) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑌) ↑m ℕ)(X𝑘𝑌 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑌 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑌)(𝑗))))))
hoidmvlelem4.i2 (𝜑X𝑘𝑊 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑊 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))
Assertion
Ref Expression
hoidmvlelem4 (𝜑 → (𝐴(𝐿𝑊)𝐵) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗))))))
Distinct variable groups:   𝐴,𝑎,𝑏,,𝑗,𝑘,𝑥   𝐴,𝑐,,𝑗,𝑘,𝑥   𝐴,𝑒,𝑓,𝑔,,𝑗,𝑘   𝑧,𝐴,,𝑗   𝐵,𝑎,𝑏,,𝑗,𝑘,𝑥   𝐵,𝑐   𝐵,𝑓,𝑔   𝑧,𝐵   𝐶,𝑎,𝑏,,𝑗,𝑘,𝑥   𝐶,𝑐   𝐶,𝑔   𝑧,𝐶   𝐷,𝑎,𝑏,,𝑗,𝑘,𝑥   𝐷,𝑐   𝐷,𝑔   𝑧,𝐷   𝐸,𝑎,𝑏,,𝑘,𝑥   𝐸,𝑐   𝑧,𝐸   𝐺,𝑎,𝑏,,𝑘,𝑥   𝐺,𝑐   𝑧,𝐺   𝐻,𝑎,𝑏,𝑗,𝑘   𝐻,𝑐   𝑧,𝐻   𝐿,𝑎,𝑏,,𝑗,𝑘,𝑥   𝐿,𝑐   𝑒,𝐿,𝑓,𝑔   𝑧,𝐿   𝑆,𝑎,𝑏,,𝑗,𝑘,𝑥   𝑆,𝑐   𝑆,𝑔   𝑧,𝑆   𝑈,𝑎,𝑏,𝑗,𝑘,𝑥   𝑈,𝑐   𝑧,𝑈   𝑊,𝑎,𝑏,,𝑗,𝑘,𝑥   𝑊,𝑐   𝑧,𝑊   𝑌,𝑎,𝑏,,𝑗,𝑘,𝑥   𝑌,𝑐   𝑒,𝑌,𝑓,𝑔   𝑍,𝑎,𝑏,,𝑗,𝑘,𝑥   𝑍,𝑐   𝑔,𝑍   𝑧,𝑍   𝜑,𝑎,𝑏,,𝑗,𝑘,𝑥   𝜑,𝑐
Allowed substitution hints:   𝜑(𝑧,𝑒,𝑓,𝑔)   𝐵(𝑒)   𝐶(𝑒,𝑓)   𝐷(𝑒,𝑓)   𝑆(𝑒,𝑓)   𝑈(𝑒,𝑓,𝑔,)   𝐸(𝑒,𝑓,𝑔,𝑗)   𝐺(𝑒,𝑓,𝑔,𝑗)   𝐻(𝑥,𝑒,𝑓,𝑔,)   𝑊(𝑒,𝑓,𝑔)   𝑋(𝑥,𝑧,𝑒,𝑓,𝑔,,𝑗,𝑘,𝑎,𝑏,𝑐)   𝑌(𝑧)   𝑍(𝑒,𝑓)

Proof of Theorem hoidmvlelem4
Dummy variables 𝑦 𝑢 𝑖 𝑙 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rge0ssre 12834 . . 3 (0[,)+∞) ⊆ ℝ
2 hoidmvlelem4.l . . . 4 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))
3 hoidmvlelem4.x . . . . 5 (𝜑𝑋 ∈ Fin)
4 hoidmvlelem4.w . . . . . 6 𝑊 = (𝑌 ∪ {𝑍})
5 hoidmvlelem4.y . . . . . . 7 (𝜑𝑌𝑋)
6 hoidmvlelem4.z . . . . . . . . 9 (𝜑𝑍 ∈ (𝑋𝑌))
76eldifad 3952 . . . . . . . 8 (𝜑𝑍𝑋)
8 snssi 4740 . . . . . . . 8 (𝑍𝑋 → {𝑍} ⊆ 𝑋)
97, 8syl 17 . . . . . . 7 (𝜑 → {𝑍} ⊆ 𝑋)
105, 9unssd 4166 . . . . . 6 (𝜑 → (𝑌 ∪ {𝑍}) ⊆ 𝑋)
114, 10eqsstrid 4019 . . . . 5 (𝜑𝑊𝑋)
12 ssfi 8727 . . . . 5 ((𝑋 ∈ Fin ∧ 𝑊𝑋) → 𝑊 ∈ Fin)
133, 11, 12syl2anc 584 . . . 4 (𝜑𝑊 ∈ Fin)
14 hoidmvlelem4.a . . . 4 (𝜑𝐴:𝑊⟶ℝ)
15 hoidmvlelem4.b . . . 4 (𝜑𝐵:𝑊⟶ℝ)
162, 13, 14, 15hoidmvcl 42730 . . 3 (𝜑 → (𝐴(𝐿𝑊)𝐵) ∈ (0[,)+∞))
171, 16sseldi 3969 . 2 (𝜑 → (𝐴(𝐿𝑊)𝐵) ∈ ℝ)
18 1red 10631 . . . 4 (𝜑 → 1 ∈ ℝ)
19 hoidmvlelem4.e . . . . 5 (𝜑𝐸 ∈ ℝ+)
2019rpred 12421 . . . 4 (𝜑𝐸 ∈ ℝ)
2118, 20readdcld 10659 . . 3 (𝜑 → (1 + 𝐸) ∈ ℝ)
22 nfv 1908 . . . . 5 𝑗𝜑
23 nnex 11633 . . . . . 6 ℕ ∈ V
2423a1i 11 . . . . 5 (𝜑 → ℕ ∈ V)
25 icossicc 12814 . . . . . 6 (0[,)+∞) ⊆ (0[,]+∞)
2613adantr 481 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → 𝑊 ∈ Fin)
27 hoidmvlelem4.c . . . . . . . . 9 (𝜑𝐶:ℕ⟶(ℝ ↑m 𝑊))
2827ffvelrnda 6847 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ∈ (ℝ ↑m 𝑊))
29 elmapi 8418 . . . . . . . 8 ((𝐶𝑗) ∈ (ℝ ↑m 𝑊) → (𝐶𝑗):𝑊⟶ℝ)
3028, 29syl 17 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗):𝑊⟶ℝ)
31 hoidmvlelem4.h . . . . . . . . 9 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑m 𝑊) ↦ (𝑗𝑊 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)))))
32 eleq1 2905 . . . . . . . . . . . . 13 (𝑗 = → (𝑗𝑌𝑌))
33 fveq2 6667 . . . . . . . . . . . . 13 (𝑗 = → (𝑐𝑗) = (𝑐))
3433breq1d 5073 . . . . . . . . . . . . . 14 (𝑗 = → ((𝑐𝑗) ≤ 𝑥 ↔ (𝑐) ≤ 𝑥))
3534, 33ifbieq1d 4493 . . . . . . . . . . . . 13 (𝑗 = → if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥) = if((𝑐) ≤ 𝑥, (𝑐), 𝑥))
3632, 33, 35ifbieq12d 4497 . . . . . . . . . . . 12 (𝑗 = → if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)) = if(𝑌, (𝑐), if((𝑐) ≤ 𝑥, (𝑐), 𝑥)))
3736cbvmptv 5166 . . . . . . . . . . 11 (𝑗𝑊 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥))) = (𝑊 ↦ if(𝑌, (𝑐), if((𝑐) ≤ 𝑥, (𝑐), 𝑥)))
3837mpteq2i 5155 . . . . . . . . . 10 (𝑐 ∈ (ℝ ↑m 𝑊) ↦ (𝑗𝑊 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)))) = (𝑐 ∈ (ℝ ↑m 𝑊) ↦ (𝑊 ↦ if(𝑌, (𝑐), if((𝑐) ≤ 𝑥, (𝑐), 𝑥))))
3938mpteq2i 5155 . . . . . . . . 9 (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑m 𝑊) ↦ (𝑗𝑊 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥))))) = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑m 𝑊) ↦ (𝑊 ↦ if(𝑌, (𝑐), if((𝑐) ≤ 𝑥, (𝑐), 𝑥)))))
4031, 39eqtri 2849 . . . . . . . 8 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑m 𝑊) ↦ (𝑊 ↦ if(𝑌, (𝑐), if((𝑐) ≤ 𝑥, (𝑐), 𝑥)))))
41 snidg 4596 . . . . . . . . . . . . 13 (𝑍 ∈ (𝑋𝑌) → 𝑍 ∈ {𝑍})
426, 41syl 17 . . . . . . . . . . . 12 (𝜑𝑍 ∈ {𝑍})
43 elun2 4157 . . . . . . . . . . . 12 (𝑍 ∈ {𝑍} → 𝑍 ∈ (𝑌 ∪ {𝑍}))
4442, 43syl 17 . . . . . . . . . . 11 (𝜑𝑍 ∈ (𝑌 ∪ {𝑍}))
454a1i 11 . . . . . . . . . . . 12 (𝜑𝑊 = (𝑌 ∪ {𝑍}))
4645eqcomd 2832 . . . . . . . . . . 11 (𝜑 → (𝑌 ∪ {𝑍}) = 𝑊)
4744, 46eleqtrd 2920 . . . . . . . . . 10 (𝜑𝑍𝑊)
4815, 47ffvelrnd 6848 . . . . . . . . 9 (𝜑 → (𝐵𝑍) ∈ ℝ)
4948adantr 481 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐵𝑍) ∈ ℝ)
50 hoidmvlelem4.d . . . . . . . . . 10 (𝜑𝐷:ℕ⟶(ℝ ↑m 𝑊))
5150ffvelrnda 6847 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗) ∈ (ℝ ↑m 𝑊))
52 elmapi 8418 . . . . . . . . 9 ((𝐷𝑗) ∈ (ℝ ↑m 𝑊) → (𝐷𝑗):𝑊⟶ℝ)
5351, 52syl 17 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗):𝑊⟶ℝ)
5440, 49, 26, 53hsphoif 42724 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ((𝐻‘(𝐵𝑍))‘(𝐷𝑗)):𝑊⟶ℝ)
552, 26, 30, 54hoidmvcl 42730 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))) ∈ (0[,)+∞))
5625, 55sseldi 3969 . . . . 5 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))) ∈ (0[,]+∞))
5722, 24, 56sge0clmpt 42573 . . . 4 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))))) ∈ (0[,]+∞))
5822, 24, 56sge0xrclmpt 42576 . . . . 5 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))))) ∈ ℝ*)
59 pnfxr 10684 . . . . . 6 +∞ ∈ ℝ*
6059a1i 11 . . . . 5 (𝜑 → +∞ ∈ ℝ*)
61 hoidmvlelem4.r . . . . . . 7 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)
6261rexrd 10680 . . . . . 6 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ*)
632, 26, 30, 53hoidmvcl 42730 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)) ∈ (0[,)+∞))
6425, 63sseldi 3969 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)) ∈ (0[,]+∞))
656eldifbd 3953 . . . . . . . . . 10 (𝜑 → ¬ 𝑍𝑌)
6647, 65eldifd 3951 . . . . . . . . 9 (𝜑𝑍 ∈ (𝑊𝑌))
6766adantr 481 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → 𝑍 ∈ (𝑊𝑌))
682, 26, 67, 4, 49, 40, 30, 53hsphoidmvle 42734 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))) ≤ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))
6922, 24, 56, 64, 68sge0lempt 42558 . . . . . 6 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))))) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
7061ltpnfd 12506 . . . . . 6 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) < +∞)
7158, 62, 60, 69, 70xrlelttrd 12543 . . . . 5 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))))) < +∞)
7258, 60, 71xrltned 41490 . . . 4 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))))) ≠ +∞)
73 ge0xrre 41672 . . . 4 (((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))))) ∈ (0[,]+∞) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))))) ≠ +∞) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))))) ∈ ℝ)
7457, 72, 73syl2anc 584 . . 3 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))))) ∈ ℝ)
7521, 74remulcld 10660 . 2 (𝜑 → ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗)))))) ∈ ℝ)
7621, 61remulcld 10660 . 2 (𝜑 → ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗))))) ∈ ℝ)
77 hoidmvlelem4.14 . . . . . . 7 𝐺 = ((𝐴𝑌)(𝐿𝑌)(𝐵𝑌))
78 hoidmvlelem4.u . . . . . . 7 𝑈 = {𝑧 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))))}
79 hoidmvlelem4.s . . . . . . 7 𝑆 = sup(𝑈, ℝ, < )
8047ancli 549 . . . . . . . 8 (𝜑 → (𝜑𝑍𝑊))
81 eleq1 2905 . . . . . . . . . . 11 (𝑘 = 𝑍 → (𝑘𝑊𝑍𝑊))
8281anbi2d 628 . . . . . . . . . 10 (𝑘 = 𝑍 → ((𝜑𝑘𝑊) ↔ (𝜑𝑍𝑊)))
83 fveq2 6667 . . . . . . . . . . 11 (𝑘 = 𝑍 → (𝐴𝑘) = (𝐴𝑍))
84 fveq2 6667 . . . . . . . . . . 11 (𝑘 = 𝑍 → (𝐵𝑘) = (𝐵𝑍))
8583, 84breq12d 5076 . . . . . . . . . 10 (𝑘 = 𝑍 → ((𝐴𝑘) < (𝐵𝑘) ↔ (𝐴𝑍) < (𝐵𝑍)))
8682, 85imbi12d 346 . . . . . . . . 9 (𝑘 = 𝑍 → (((𝜑𝑘𝑊) → (𝐴𝑘) < (𝐵𝑘)) ↔ ((𝜑𝑍𝑊) → (𝐴𝑍) < (𝐵𝑍))))
87 hoidmvlelem4.k . . . . . . . . 9 ((𝜑𝑘𝑊) → (𝐴𝑘) < (𝐵𝑘))
8886, 87vtoclg 3573 . . . . . . . 8 (𝑍𝑊 → ((𝜑𝑍𝑊) → (𝐴𝑍) < (𝐵𝑍)))
8947, 80, 88sylc 65 . . . . . . 7 (𝜑 → (𝐴𝑍) < (𝐵𝑍))
902, 3, 5, 6, 4, 14, 15, 27, 50, 61, 31, 77, 19, 78, 79, 89hoidmvlelem1 42743 . . . . . 6 (𝜑𝑆𝑈)
9148rexrd 10680 . . . . . . . 8 (𝜑 → (𝐵𝑍) ∈ ℝ*)
92 iccssxr 12809 . . . . . . . . 9 ((𝐴𝑍)[,](𝐵𝑍)) ⊆ ℝ*
93 ssrab2 4060 . . . . . . . . . . 11 {𝑧 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))))} ⊆ ((𝐴𝑍)[,](𝐵𝑍))
9478, 93eqsstri 4005 . . . . . . . . . 10 𝑈 ⊆ ((𝐴𝑍)[,](𝐵𝑍))
9594, 90sseldi 3969 . . . . . . . . 9 (𝜑𝑆 ∈ ((𝐴𝑍)[,](𝐵𝑍)))
9692, 95sseldi 3969 . . . . . . . 8 (𝜑𝑆 ∈ ℝ*)
97 simpl 483 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝑆) → 𝜑)
98 simpr 485 . . . . . . . . . . 11 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝑆) → ¬ (𝐵𝑍) ≤ 𝑆)
9914, 47ffvelrnd 6848 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴𝑍) ∈ ℝ)
10099, 48iccssred 41645 . . . . . . . . . . . . . 14 (𝜑 → ((𝐴𝑍)[,](𝐵𝑍)) ⊆ ℝ)
101100, 95sseldd 3972 . . . . . . . . . . . . 13 (𝜑𝑆 ∈ ℝ)
102101adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝑆) → 𝑆 ∈ ℝ)
10397, 48syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝑆) → (𝐵𝑍) ∈ ℝ)
104102, 103ltnled 10776 . . . . . . . . . . 11 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝑆) → (𝑆 < (𝐵𝑍) ↔ ¬ (𝐵𝑍) ≤ 𝑆))
10598, 104mpbird 258 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝑆) → 𝑆 < (𝐵𝑍))
1063adantr 481 . . . . . . . . . . 11 ((𝜑𝑆 < (𝐵𝑍)) → 𝑋 ∈ Fin)
1075adantr 481 . . . . . . . . . . 11 ((𝜑𝑆 < (𝐵𝑍)) → 𝑌𝑋)
1086adantr 481 . . . . . . . . . . 11 ((𝜑𝑆 < (𝐵𝑍)) → 𝑍 ∈ (𝑋𝑌))
10914adantr 481 . . . . . . . . . . 11 ((𝜑𝑆 < (𝐵𝑍)) → 𝐴:𝑊⟶ℝ)
11015adantr 481 . . . . . . . . . . 11 ((𝜑𝑆 < (𝐵𝑍)) → 𝐵:𝑊⟶ℝ)
11187adantlr 711 . . . . . . . . . . 11 (((𝜑𝑆 < (𝐵𝑍)) ∧ 𝑘𝑊) → (𝐴𝑘) < (𝐵𝑘))
112 eqid 2826 . . . . . . . . . . 11 (𝑦𝑌 ↦ 0) = (𝑦𝑌 ↦ 0)
11327adantr 481 . . . . . . . . . . 11 ((𝜑𝑆 < (𝐵𝑍)) → 𝐶:ℕ⟶(ℝ ↑m 𝑊))
114 fveq2 6667 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑗 → (𝐶𝑖) = (𝐶𝑗))
115114fveq1d 6669 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → ((𝐶𝑖)‘𝑍) = ((𝐶𝑗)‘𝑍))
116 fveq2 6667 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑗 → (𝐷𝑖) = (𝐷𝑗))
117116fveq1d 6669 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → ((𝐷𝑖)‘𝑍) = ((𝐷𝑗)‘𝑍))
118115, 117oveq12d 7166 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)) = (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)))
119118eleq2d 2903 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)) ↔ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))))
120114reseq1d 5851 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → ((𝐶𝑖) ↾ 𝑌) = ((𝐶𝑗) ↾ 𝑌))
121119, 120ifbieq1d 4493 . . . . . . . . . . . 12 (𝑖 = 𝑗 → if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐶𝑖) ↾ 𝑌), (𝑦𝑌 ↦ 0)) = if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), (𝑦𝑌 ↦ 0)))
122121cbvmptv 5166 . . . . . . . . . . 11 (𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐶𝑖) ↾ 𝑌), (𝑦𝑌 ↦ 0))) = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), (𝑦𝑌 ↦ 0)))
12350adantr 481 . . . . . . . . . . 11 ((𝜑𝑆 < (𝐵𝑍)) → 𝐷:ℕ⟶(ℝ ↑m 𝑊))
124116reseq1d 5851 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → ((𝐷𝑖) ↾ 𝑌) = ((𝐷𝑗) ↾ 𝑌))
125119, 124ifbieq1d 4493 . . . . . . . . . . . 12 (𝑖 = 𝑗 → if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐷𝑖) ↾ 𝑌), (𝑦𝑌 ↦ 0)) = if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), (𝑦𝑌 ↦ 0)))
126125cbvmptv 5166 . . . . . . . . . . 11 (𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐷𝑖) ↾ 𝑌), (𝑦𝑌 ↦ 0))) = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), (𝑦𝑌 ↦ 0)))
12761adantr 481 . . . . . . . . . . 11 ((𝜑𝑆 < (𝐵𝑍)) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)
12819adantr 481 . . . . . . . . . . 11 ((𝜑𝑆 < (𝐵𝑍)) → 𝐸 ∈ ℝ+)
12990adantr 481 . . . . . . . . . . 11 ((𝜑𝑆 < (𝐵𝑍)) → 𝑆𝑈)
130 simpr 485 . . . . . . . . . . 11 ((𝜑𝑆 < (𝐵𝑍)) → 𝑆 < (𝐵𝑍))
131 biid 262 . . . . . . . . . . . . . . . . 17 (𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)) ↔ 𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)))
132 eqidd 2827 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑦 → 0 = 0)
133132cbvmptv 5166 . . . . . . . . . . . . . . . . 17 (𝑤𝑌 ↦ 0) = (𝑦𝑌 ↦ 0)
134131, 133ifbieq2i 4494 . . . . . . . . . . . . . . . 16 if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐶𝑖) ↾ 𝑌), (𝑤𝑌 ↦ 0)) = if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐶𝑖) ↾ 𝑌), (𝑦𝑌 ↦ 0))
135134mpteq2i 5155 . . . . . . . . . . . . . . 15 (𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐶𝑖) ↾ 𝑌), (𝑤𝑌 ↦ 0))) = (𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐶𝑖) ↾ 𝑌), (𝑦𝑌 ↦ 0)))
136135a1i 11 . . . . . . . . . . . . . 14 (𝑙 = 𝑗 → (𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐶𝑖) ↾ 𝑌), (𝑤𝑌 ↦ 0))) = (𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐶𝑖) ↾ 𝑌), (𝑦𝑌 ↦ 0))))
137 id 22 . . . . . . . . . . . . . 14 (𝑙 = 𝑗𝑙 = 𝑗)
138136, 137fveq12d 6674 . . . . . . . . . . . . 13 (𝑙 = 𝑗 → ((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐶𝑖) ↾ 𝑌), (𝑤𝑌 ↦ 0)))‘𝑙) = ((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐶𝑖) ↾ 𝑌), (𝑦𝑌 ↦ 0)))‘𝑗))
139131, 133ifbieq2i 4494 . . . . . . . . . . . . . . . 16 if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐷𝑖) ↾ 𝑌), (𝑤𝑌 ↦ 0)) = if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐷𝑖) ↾ 𝑌), (𝑦𝑌 ↦ 0))
140139mpteq2i 5155 . . . . . . . . . . . . . . 15 (𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐷𝑖) ↾ 𝑌), (𝑤𝑌 ↦ 0))) = (𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐷𝑖) ↾ 𝑌), (𝑦𝑌 ↦ 0)))
141140a1i 11 . . . . . . . . . . . . . 14 (𝑙 = 𝑗 → (𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐷𝑖) ↾ 𝑌), (𝑤𝑌 ↦ 0))) = (𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐷𝑖) ↾ 𝑌), (𝑦𝑌 ↦ 0))))
142141, 137fveq12d 6674 . . . . . . . . . . . . 13 (𝑙 = 𝑗 → ((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐷𝑖) ↾ 𝑌), (𝑤𝑌 ↦ 0)))‘𝑙) = ((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐷𝑖) ↾ 𝑌), (𝑦𝑌 ↦ 0)))‘𝑗))
143138, 142oveq12d 7166 . . . . . . . . . . . 12 (𝑙 = 𝑗 → (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐶𝑖) ↾ 𝑌), (𝑤𝑌 ↦ 0)))‘𝑙)(𝐿𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐷𝑖) ↾ 𝑌), (𝑤𝑌 ↦ 0)))‘𝑙)) = (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐶𝑖) ↾ 𝑌), (𝑦𝑌 ↦ 0)))‘𝑗)(𝐿𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐷𝑖) ↾ 𝑌), (𝑦𝑌 ↦ 0)))‘𝑗)))
144143cbvmptv 5166 . . . . . . . . . . 11 (𝑙 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐶𝑖) ↾ 𝑌), (𝑤𝑌 ↦ 0)))‘𝑙)(𝐿𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐷𝑖) ↾ 𝑌), (𝑤𝑌 ↦ 0)))‘𝑙))) = (𝑗 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐶𝑖) ↾ 𝑌), (𝑦𝑌 ↦ 0)))‘𝑗)(𝐿𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐷𝑖) ↾ 𝑌), (𝑦𝑌 ↦ 0)))‘𝑗)))
145 hoidmvlelem4.i . . . . . . . . . . . 12 (𝜑 → ∀𝑒 ∈ (ℝ ↑m 𝑌)∀𝑓 ∈ (ℝ ↑m 𝑌)∀𝑔 ∈ ((ℝ ↑m 𝑌) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑌) ↑m ℕ)(X𝑘𝑌 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑌 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑌)(𝑗))))))
146145adantr 481 . . . . . . . . . . 11 ((𝜑𝑆 < (𝐵𝑍)) → ∀𝑒 ∈ (ℝ ↑m 𝑌)∀𝑓 ∈ (ℝ ↑m 𝑌)∀𝑔 ∈ ((ℝ ↑m 𝑌) ↑m ℕ)∀ ∈ ((ℝ ↑m 𝑌) ↑m ℕ)(X𝑘𝑌 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑌 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑌)(𝑗))))))
147 hoidmvlelem4.i2 . . . . . . . . . . . 12 (𝜑X𝑘𝑊 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑊 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))
148147adantr 481 . . . . . . . . . . 11 ((𝜑𝑆 < (𝐵𝑍)) → X𝑘𝑊 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑊 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))
149 eqid 2826 . . . . . . . . . . . 12 (𝑥X𝑦𝑌 ((𝐴𝑦)[,)(𝐵𝑦)) ↦ (𝑦𝑊 ↦ if(𝑦𝑌, (𝑥𝑦), 𝑆))) = (𝑥X𝑦𝑌 ((𝐴𝑦)[,)(𝐵𝑦)) ↦ (𝑦𝑊 ↦ if(𝑦𝑌, (𝑥𝑦), 𝑆)))
150 fveq2 6667 . . . . . . . . . . . . . . 15 (𝑦 = 𝑘 → (𝐴𝑦) = (𝐴𝑘))
151 fveq2 6667 . . . . . . . . . . . . . . 15 (𝑦 = 𝑘 → (𝐵𝑦) = (𝐵𝑘))
152150, 151oveq12d 7166 . . . . . . . . . . . . . 14 (𝑦 = 𝑘 → ((𝐴𝑦)[,)(𝐵𝑦)) = ((𝐴𝑘)[,)(𝐵𝑘)))
153152cbvixpv 8468 . . . . . . . . . . . . 13 X𝑦𝑌 ((𝐴𝑦)[,)(𝐵𝑦)) = X𝑘𝑌 ((𝐴𝑘)[,)(𝐵𝑘))
154 eleq1 2905 . . . . . . . . . . . . . . 15 (𝑦 = 𝑘 → (𝑦𝑌𝑘𝑌))
155 fveq2 6667 . . . . . . . . . . . . . . 15 (𝑦 = 𝑘 → (𝑥𝑦) = (𝑥𝑘))
156154, 155ifbieq1d 4493 . . . . . . . . . . . . . 14 (𝑦 = 𝑘 → if(𝑦𝑌, (𝑥𝑦), 𝑆) = if(𝑘𝑌, (𝑥𝑘), 𝑆))
157156cbvmptv 5166 . . . . . . . . . . . . 13 (𝑦𝑊 ↦ if(𝑦𝑌, (𝑥𝑦), 𝑆)) = (𝑘𝑊 ↦ if(𝑘𝑌, (𝑥𝑘), 𝑆))
158153, 157mpteq12i 5156 . . . . . . . . . . . 12 (𝑥X𝑦𝑌 ((𝐴𝑦)[,)(𝐵𝑦)) ↦ (𝑦𝑊 ↦ if(𝑦𝑌, (𝑥𝑦), 𝑆))) = (𝑥X𝑘𝑌 ((𝐴𝑘)[,)(𝐵𝑘)) ↦ (𝑘𝑊 ↦ if(𝑘𝑌, (𝑥𝑘), 𝑆)))
159149, 158eqtri 2849 . . . . . . . . . . 11 (𝑥X𝑦𝑌 ((𝐴𝑦)[,)(𝐵𝑦)) ↦ (𝑦𝑊 ↦ if(𝑦𝑌, (𝑥𝑦), 𝑆))) = (𝑥X𝑘𝑌 ((𝐴𝑘)[,)(𝐵𝑘)) ↦ (𝑘𝑊 ↦ if(𝑘𝑌, (𝑥𝑘), 𝑆)))
1602, 106, 107, 108, 4, 109, 110, 111, 112, 113, 122, 123, 126, 127, 31, 77, 128, 78, 129, 130, 144, 146, 148, 159hoidmvlelem3 42745 . . . . . . . . . 10 ((𝜑𝑆 < (𝐵𝑍)) → ∃𝑢𝑈 𝑆 < 𝑢)
16197, 105, 160syl2anc 584 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝑆) → ∃𝑢𝑈 𝑆 < 𝑢)
16294a1i 11 . . . . . . . . . . . . . . . . . 18 (𝜑𝑈 ⊆ ((𝐴𝑍)[,](𝐵𝑍)))
163162, 100sstrd 3981 . . . . . . . . . . . . . . . . 17 (𝜑𝑈 ⊆ ℝ)
164163adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑢𝑈) → 𝑈 ⊆ ℝ)
165 ne0i 4304 . . . . . . . . . . . . . . . . 17 (𝑢𝑈𝑈 ≠ ∅)
166165adantl 482 . . . . . . . . . . . . . . . 16 ((𝜑𝑢𝑈) → 𝑈 ≠ ∅)
16799rexrd 10680 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐴𝑍) ∈ ℝ*)
168167adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑢𝑈) → (𝐴𝑍) ∈ ℝ*)
16991adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑢𝑈) → (𝐵𝑍) ∈ ℝ*)
170162sselda 3971 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑢𝑈) → 𝑢 ∈ ((𝐴𝑍)[,](𝐵𝑍)))
171 iccleub 12782 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑍) ∈ ℝ* ∧ (𝐵𝑍) ∈ ℝ*𝑢 ∈ ((𝐴𝑍)[,](𝐵𝑍))) → 𝑢 ≤ (𝐵𝑍))
172168, 169, 170, 171syl3anc 1365 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑢𝑈) → 𝑢 ≤ (𝐵𝑍))
173172ralrimiva 3187 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑢𝑈 𝑢 ≤ (𝐵𝑍))
174 brralrspcev 5123 . . . . . . . . . . . . . . . . . 18 (((𝐵𝑍) ∈ ℝ ∧ ∀𝑢𝑈 𝑢 ≤ (𝐵𝑍)) → ∃𝑦 ∈ ℝ ∀𝑢𝑈 𝑢𝑦)
17548, 173, 174syl2anc 584 . . . . . . . . . . . . . . . . 17 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑢𝑈 𝑢𝑦)
176175adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑢𝑈) → ∃𝑦 ∈ ℝ ∀𝑢𝑈 𝑢𝑦)
177 simpr 485 . . . . . . . . . . . . . . . 16 ((𝜑𝑢𝑈) → 𝑢𝑈)
178 suprub 11591 . . . . . . . . . . . . . . . 16 (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑢𝑈 𝑢𝑦) ∧ 𝑢𝑈) → 𝑢 ≤ sup(𝑈, ℝ, < ))
179164, 166, 176, 177, 178syl31anc 1367 . . . . . . . . . . . . . . 15 ((𝜑𝑢𝑈) → 𝑢 ≤ sup(𝑈, ℝ, < ))
180179, 79breqtrrdi 5105 . . . . . . . . . . . . . 14 ((𝜑𝑢𝑈) → 𝑢𝑆)
181180ralrimiva 3187 . . . . . . . . . . . . 13 (𝜑 → ∀𝑢𝑈 𝑢𝑆)
182164, 177sseldd 3972 . . . . . . . . . . . . . . 15 ((𝜑𝑢𝑈) → 𝑢 ∈ ℝ)
183101adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑢𝑈) → 𝑆 ∈ ℝ)
184182, 183lenltd 10775 . . . . . . . . . . . . . 14 ((𝜑𝑢𝑈) → (𝑢𝑆 ↔ ¬ 𝑆 < 𝑢))
185184ralbidva 3201 . . . . . . . . . . . . 13 (𝜑 → (∀𝑢𝑈 𝑢𝑆 ↔ ∀𝑢𝑈 ¬ 𝑆 < 𝑢))
186181, 185mpbid 233 . . . . . . . . . . . 12 (𝜑 → ∀𝑢𝑈 ¬ 𝑆 < 𝑢)
187 ralnex 3241 . . . . . . . . . . . 12 (∀𝑢𝑈 ¬ 𝑆 < 𝑢 ↔ ¬ ∃𝑢𝑈 𝑆 < 𝑢)
188186, 187sylib 219 . . . . . . . . . . 11 (𝜑 → ¬ ∃𝑢𝑈 𝑆 < 𝑢)
189188adantr 481 . . . . . . . . . 10 ((𝜑𝑆 < (𝐵𝑍)) → ¬ ∃𝑢𝑈 𝑆 < 𝑢)
19097, 105, 189syl2anc 584 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝑆) → ¬ ∃𝑢𝑈 𝑆 < 𝑢)
191161, 190condan 814 . . . . . . . 8 (𝜑 → (𝐵𝑍) ≤ 𝑆)
192 iccleub 12782 . . . . . . . . 9 (((𝐴𝑍) ∈ ℝ* ∧ (𝐵𝑍) ∈ ℝ*𝑆 ∈ ((𝐴𝑍)[,](𝐵𝑍))) → 𝑆 ≤ (𝐵𝑍))
193167, 91, 95, 192syl3anc 1365 . . . . . . . 8 (𝜑𝑆 ≤ (𝐵𝑍))
19491, 96, 191, 193xrletrid 12538 . . . . . . 7 (𝜑 → (𝐵𝑍) = 𝑆)
19578eqcomi 2835 . . . . . . . 8 {𝑧 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))))} = 𝑈
196195a1i 11 . . . . . . 7 (𝜑 → {𝑧 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))))} = 𝑈)
197194, 196eleq12d 2912 . . . . . 6 (𝜑 → ((𝐵𝑍) ∈ {𝑧 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))))} ↔ 𝑆𝑈))
19890, 197mpbird 258 . . . . 5 (𝜑 → (𝐵𝑍) ∈ {𝑧 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))))})
199 oveq1 7155 . . . . . . . 8 (𝑧 = (𝐵𝑍) → (𝑧 − (𝐴𝑍)) = ((𝐵𝑍) − (𝐴𝑍)))
200199oveq2d 7164 . . . . . . 7 (𝑧 = (𝐵𝑍) → (𝐺 · (𝑧 − (𝐴𝑍))) = (𝐺 · ((𝐵𝑍) − (𝐴𝑍))))
201 fveq2 6667 . . . . . . . . . . . 12 (𝑧 = (𝐵𝑍) → (𝐻𝑧) = (𝐻‘(𝐵𝑍)))
202201fveq1d 6669 . . . . . . . . . . 11 (𝑧 = (𝐵𝑍) → ((𝐻𝑧)‘(𝐷𝑗)) = ((𝐻‘(𝐵𝑍))‘(𝐷𝑗)))
203202oveq2d 7164 . . . . . . . . . 10 (𝑧 = (𝐵𝑍) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))) = ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))))
204203mpteq2dv 5159 . . . . . . . . 9 (𝑧 = (𝐵𝑍) → (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗)))) = (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗)))))
205204fveq2d 6671 . . . . . . . 8 (𝑧 = (𝐵𝑍) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))))))
206205oveq2d 7164 . . . . . . 7 (𝑧 = (𝐵𝑍) → ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗)))))) = ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗)))))))
207200, 206breq12d 5076 . . . . . 6 (𝑧 = (𝐵𝑍) → ((𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗)))))) ↔ (𝐺 · ((𝐵𝑍) − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))))))))
208207elrab 3684 . . . . 5 ((𝐵𝑍) ∈ {𝑧 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))))} ↔ ((𝐵𝑍) ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∧ (𝐺 · ((𝐵𝑍) − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))))))))
209198, 208sylib 219 . . . 4 (𝜑 → ((𝐵𝑍) ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∧ (𝐺 · ((𝐵𝑍) − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))))))))
210209simprd 496 . . 3 (𝜑 → (𝐺 · ((𝐵𝑍) − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗)))))))
2113, 5ssfid 8730 . . . . . 6 (𝜑𝑌 ∈ Fin)
212 eqid 2826 . . . . . 6 𝑘𝑌 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = ∏𝑘𝑌 (vol‘((𝐴𝑘)[,)(𝐵𝑘)))
2132, 211, 6, 65, 4, 14, 15, 212hoiprodp1 42736 . . . . 5 (𝜑 → (𝐴(𝐿𝑊)𝐵) = (∏𝑘𝑌 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) · (vol‘((𝐴𝑍)[,)(𝐵𝑍)))))
214 eqidd 2827 . . . . . . 7 (𝜑 → ∏𝑘𝑌 ((𝐵𝑘) − (𝐴𝑘)) = ∏𝑘𝑌 ((𝐵𝑘) − (𝐴𝑘)))
21514adantr 481 . . . . . . . . . 10 ((𝜑𝑘𝑌) → 𝐴:𝑊⟶ℝ)
216 ssun1 4152 . . . . . . . . . . . 12 𝑌 ⊆ (𝑌 ∪ {𝑍})
2174eqcomi 2835 . . . . . . . . . . . 12 (𝑌 ∪ {𝑍}) = 𝑊
218216, 217sseqtri 4007 . . . . . . . . . . 11 𝑌𝑊
219 simpr 485 . . . . . . . . . . 11 ((𝜑𝑘𝑌) → 𝑘𝑌)
220218, 219sseldi 3969 . . . . . . . . . 10 ((𝜑𝑘𝑌) → 𝑘𝑊)
221215, 220ffvelrnd 6848 . . . . . . . . 9 ((𝜑𝑘𝑌) → (𝐴𝑘) ∈ ℝ)
22215adantr 481 . . . . . . . . . 10 ((𝜑𝑘𝑌) → 𝐵:𝑊⟶ℝ)
223222, 220ffvelrnd 6848 . . . . . . . . 9 ((𝜑𝑘𝑌) → (𝐵𝑘) ∈ ℝ)
224220, 87syldan 591 . . . . . . . . 9 ((𝜑𝑘𝑌) → (𝐴𝑘) < (𝐵𝑘))
225221, 223, 224volicon0 42723 . . . . . . . 8 ((𝜑𝑘𝑌) → (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = ((𝐵𝑘) − (𝐴𝑘)))
226225prodeq2dv 15267 . . . . . . 7 (𝜑 → ∏𝑘𝑌 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = ∏𝑘𝑌 ((𝐵𝑘) − (𝐴𝑘)))
22777a1i 11 . . . . . . . 8 (𝜑𝐺 = ((𝐴𝑌)(𝐿𝑌)(𝐵𝑌)))
228 hoidmvlelem4.n . . . . . . . . 9 (𝜑𝑌 ≠ ∅)
229218a1i 11 . . . . . . . . . 10 (𝜑𝑌𝑊)
23014, 229fssresd 6542 . . . . . . . . 9 (𝜑 → (𝐴𝑌):𝑌⟶ℝ)
23115, 229fssresd 6542 . . . . . . . . 9 (𝜑 → (𝐵𝑌):𝑌⟶ℝ)
2322, 211, 228, 230, 231hoidmvn0val 42732 . . . . . . . 8 (𝜑 → ((𝐴𝑌)(𝐿𝑌)(𝐵𝑌)) = ∏𝑘𝑌 (vol‘(((𝐴𝑌)‘𝑘)[,)((𝐵𝑌)‘𝑘))))
233 fvres 6686 . . . . . . . . . . . . 13 (𝑘𝑌 → ((𝐴𝑌)‘𝑘) = (𝐴𝑘))
234 fvres 6686 . . . . . . . . . . . . 13 (𝑘𝑌 → ((𝐵𝑌)‘𝑘) = (𝐵𝑘))
235233, 234oveq12d 7166 . . . . . . . . . . . 12 (𝑘𝑌 → (((𝐴𝑌)‘𝑘)[,)((𝐵𝑌)‘𝑘)) = ((𝐴𝑘)[,)(𝐵𝑘)))
236235fveq2d 6671 . . . . . . . . . . 11 (𝑘𝑌 → (vol‘(((𝐴𝑌)‘𝑘)[,)((𝐵𝑌)‘𝑘))) = (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
237236adantl 482 . . . . . . . . . 10 ((𝜑𝑘𝑌) → (vol‘(((𝐴𝑌)‘𝑘)[,)((𝐵𝑌)‘𝑘))) = (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
238 volico 42134 . . . . . . . . . . 11 (((𝐴𝑘) ∈ ℝ ∧ (𝐵𝑘) ∈ ℝ) → (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = if((𝐴𝑘) < (𝐵𝑘), ((𝐵𝑘) − (𝐴𝑘)), 0))
239221, 223, 238syl2anc 584 . . . . . . . . . 10 ((𝜑𝑘𝑌) → (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = if((𝐴𝑘) < (𝐵𝑘), ((𝐵𝑘) − (𝐴𝑘)), 0))
240239, 225eqtr3d 2863 . . . . . . . . . 10 ((𝜑𝑘𝑌) → if((𝐴𝑘) < (𝐵𝑘), ((𝐵𝑘) − (𝐴𝑘)), 0) = ((𝐵𝑘) − (𝐴𝑘)))
241237, 239, 2403eqtrd 2865 . . . . . . . . 9 ((𝜑𝑘𝑌) → (vol‘(((𝐴𝑌)‘𝑘)[,)((𝐵𝑌)‘𝑘))) = ((𝐵𝑘) − (𝐴𝑘)))
242241prodeq2dv 15267 . . . . . . . 8 (𝜑 → ∏𝑘𝑌 (vol‘(((𝐴𝑌)‘𝑘)[,)((𝐵𝑌)‘𝑘))) = ∏𝑘𝑌 ((𝐵𝑘) − (𝐴𝑘)))
243227, 232, 2423eqtrd 2865 . . . . . . 7 (𝜑𝐺 = ∏𝑘𝑌 ((𝐵𝑘) − (𝐴𝑘)))
244214, 226, 2433eqtr4d 2871 . . . . . 6 (𝜑 → ∏𝑘𝑌 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = 𝐺)
24599, 48, 89volicon0 42723 . . . . . 6 (𝜑 → (vol‘((𝐴𝑍)[,)(𝐵𝑍))) = ((𝐵𝑍) − (𝐴𝑍)))
246244, 245oveq12d 7166 . . . . 5 (𝜑 → (∏𝑘𝑌 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) · (vol‘((𝐴𝑍)[,)(𝐵𝑍)))) = (𝐺 · ((𝐵𝑍) − (𝐴𝑍))))
247213, 246eqtrd 2861 . . . 4 (𝜑 → (𝐴(𝐿𝑊)𝐵) = (𝐺 · ((𝐵𝑍) − (𝐴𝑍))))
248247breq1d 5073 . . 3 (𝜑 → ((𝐴(𝐿𝑊)𝐵) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗)))))) ↔ (𝐺 · ((𝐵𝑍) − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))))))))
249210, 248mpbird 258 . 2 (𝜑 → (𝐴(𝐿𝑊)𝐵) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗)))))))
250 0le1 11152 . . . . 5 0 ≤ 1
251250a1i 11 . . . 4 (𝜑 → 0 ≤ 1)
25219rpge0d 12425 . . . 4 (𝜑 → 0 ≤ 𝐸)
25318, 20, 251, 252addge0d 11205 . . 3 (𝜑 → 0 ≤ (1 + 𝐸))
25474, 61, 21, 253, 69lemul2ad 11569 . 2 (𝜑 → ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗)))))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗))))))
25517, 75, 76, 249, 254letrd 10786 1 (𝜑 → (𝐴(𝐿𝑊)𝐵) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗))))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1530   ∈ wcel 2107   ≠ wne 3021  ∀wral 3143  ∃wrex 3144  {crab 3147  Vcvv 3500   ∖ cdif 3937   ∪ cun 3938   ⊆ wss 3940  ∅c0 4295  ifcif 4470  {csn 4564  ∪ ciun 4917   class class class wbr 5063   ↦ cmpt 5143   ↾ cres 5556  ⟶wf 6348  ‘cfv 6352  (class class class)co 7148   ∈ cmpo 7150   ↑m cmap 8396  Xcixp 8450  Fincfn 8498  supcsup 8893  ℝcr 10525  0cc0 10526  1c1 10527   + caddc 10529   · cmul 10531  +∞cpnf 10661  ℝ*cxr 10663   < clt 10664   ≤ cle 10665   − cmin 10859  ℕcn 11627  ℝ+crp 12379  [,)cico 12730  [,]cicc 12731  ∏cprod 15249  volcvol 23979  Σ^csumge0 42510 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-inf2 9093  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-isom 6361  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-of 7399  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-2o 8094  df-oadd 8097  df-er 8279  df-map 8398  df-pm 8399  df-ixp 8451  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-fi 8864  df-sup 8895  df-inf 8896  df-oi 8963  df-dju 9319  df-card 9357  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11628  df-2 11689  df-3 11690  df-n0 11887  df-z 11971  df-uz 12233  df-q 12338  df-rp 12380  df-xneg 12497  df-xadd 12498  df-xmul 12499  df-ioo 12732  df-ico 12734  df-icc 12735  df-fz 12883  df-fzo 13024  df-fl 13152  df-seq 13360  df-exp 13420  df-hash 13681  df-cj 14448  df-re 14449  df-im 14450  df-sqrt 14584  df-abs 14585  df-clim 14835  df-rlim 14836  df-sum 15033  df-prod 15250  df-rest 16686  df-topgen 16707  df-psmet 20453  df-xmet 20454  df-met 20455  df-bl 20456  df-mopn 20457  df-top 21418  df-topon 21435  df-bases 21470  df-cmp 21911  df-ovol 23980  df-vol 23981  df-sumge0 42511 This theorem is referenced by:  hoidmvlelem5  42747
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