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Theorem hoidmvlelem4 41326
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 12486 . . 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 3733 . . . . . . . 8 (𝜑𝑍𝑋)
8 snssi 4472 . . . . . . . 8 (𝑍𝑋 → {𝑍} ⊆ 𝑋)
97, 8syl 17 . . . . . . 7 (𝜑 → {𝑍} ⊆ 𝑋)
105, 9unssd 3938 . . . . . 6 (𝜑 → (𝑌 ∪ {𝑍}) ⊆ 𝑋)
114, 10syl5eqss 3796 . . . . 5 (𝜑𝑊𝑋)
12 ssfi 8335 . . . . 5 ((𝑋 ∈ Fin ∧ 𝑊𝑋) → 𝑊 ∈ Fin)
133, 11, 12syl2anc 565 . . . 4 (𝜑𝑊 ∈ Fin)
14 hoidmvlelem4.a . . . 4 (𝜑𝐴:𝑊⟶ℝ)
15 hoidmvlelem4.b . . . 4 (𝜑𝐵:𝑊⟶ℝ)
162, 13, 14, 15hoidmvcl 41310 . . 3 (𝜑 → (𝐴(𝐿𝑊)𝐵) ∈ (0[,)+∞))
171, 16sseldi 3748 . 2 (𝜑 → (𝐴(𝐿𝑊)𝐵) ∈ ℝ)
18 1red 10256 . . . 4 (𝜑 → 1 ∈ ℝ)
19 hoidmvlelem4.e . . . . 5 (𝜑𝐸 ∈ ℝ+)
2019rpred 12074 . . . 4 (𝜑𝐸 ∈ ℝ)
2118, 20readdcld 10270 . . 3 (𝜑 → (1 + 𝐸) ∈ ℝ)
22 nfv 1994 . . . . 5 𝑗𝜑
23 nnex 11227 . . . . . 6 ℕ ∈ V
2423a1i 11 . . . . 5 (𝜑 → ℕ ∈ V)
25 icossicc 12465 . . . . . 6 (0[,)+∞) ⊆ (0[,]+∞)
2613adantr 466 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → 𝑊 ∈ Fin)
27 hoidmvlelem4.c . . . . . . . . 9 (𝜑𝐶:ℕ⟶(ℝ ↑𝑚 𝑊))
2827ffvelrnda 6502 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ∈ (ℝ ↑𝑚 𝑊))
29 elmapi 8030 . . . . . . . 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 2837 . . . . . . . . . . . . 13 (𝑗 = → (𝑗𝑌𝑌))
33 fveq2 6332 . . . . . . . . . . . . 13 (𝑗 = → (𝑐𝑗) = (𝑐))
3433breq1d 4794 . . . . . . . . . . . . . 14 (𝑗 = → ((𝑐𝑗) ≤ 𝑥 ↔ (𝑐) ≤ 𝑥))
3534, 33ifbieq1d 4246 . . . . . . . . . . . . 13 (𝑗 = → if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥) = if((𝑐) ≤ 𝑥, (𝑐), 𝑥))
3632, 33, 35ifbieq12d 4250 . . . . . . . . . . . 12 (𝑗 = → if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)) = if(𝑌, (𝑐), if((𝑐) ≤ 𝑥, (𝑐), 𝑥)))
3736cbvmptv 4882 . . . . . . . . . . 11 (𝑗𝑊 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥))) = (𝑊 ↦ if(𝑌, (𝑐), if((𝑐) ≤ 𝑥, (𝑐), 𝑥)))
3837mpteq2i 4873 . . . . . . . . . 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 4873 . . . . . . . . 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 2792 . . . . . . . 8 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑊 ↦ if(𝑌, (𝑐), if((𝑐) ≤ 𝑥, (𝑐), 𝑥)))))
41 snidg 4343 . . . . . . . . . . . . 13 (𝑍 ∈ (𝑋𝑌) → 𝑍 ∈ {𝑍})
426, 41syl 17 . . . . . . . . . . . 12 (𝜑𝑍 ∈ {𝑍})
43 elun2 3930 . . . . . . . . . . . 12 (𝑍 ∈ {𝑍} → 𝑍 ∈ (𝑌 ∪ {𝑍}))
4442, 43syl 17 . . . . . . . . . . 11 (𝜑𝑍 ∈ (𝑌 ∪ {𝑍}))
454a1i 11 . . . . . . . . . . . 12 (𝜑𝑊 = (𝑌 ∪ {𝑍}))
4645eqcomd 2776 . . . . . . . . . . 11 (𝜑 → (𝑌 ∪ {𝑍}) = 𝑊)
4744, 46eleqtrd 2851 . . . . . . . . . 10 (𝜑𝑍𝑊)
4815, 47ffvelrnd 6503 . . . . . . . . 9 (𝜑 → (𝐵𝑍) ∈ ℝ)
4948adantr 466 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐵𝑍) ∈ ℝ)
50 hoidmvlelem4.d . . . . . . . . . 10 (𝜑𝐷:ℕ⟶(ℝ ↑𝑚 𝑊))
5150ffvelrnda 6502 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗) ∈ (ℝ ↑𝑚 𝑊))
52 elmapi 8030 . . . . . . . . 9 ((𝐷𝑗) ∈ (ℝ ↑𝑚 𝑊) → (𝐷𝑗):𝑊⟶ℝ)
5351, 52syl 17 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗):𝑊⟶ℝ)
5440, 49, 26, 53hsphoif 41304 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ((𝐻‘(𝐵𝑍))‘(𝐷𝑗)):𝑊⟶ℝ)
552, 26, 30, 54hoidmvcl 41310 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))) ∈ (0[,)+∞))
5625, 55sseldi 3748 . . . . 5 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))) ∈ (0[,]+∞))
5722, 24, 56sge0clmpt 41153 . . . 4 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))))) ∈ (0[,]+∞))
5822, 24, 56sge0xrclmpt 41156 . . . . 5 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))))) ∈ ℝ*)
59 pnfxr 10293 . . . . . 6 +∞ ∈ ℝ*
6059a1i 11 . . . . 5 (𝜑 → +∞ ∈ ℝ*)
61 hoidmvlelem4.r . . . . . . 7 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)
6261rexrd 10290 . . . . . 6 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ*)
632, 26, 30, 53hoidmvcl 41310 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)) ∈ (0[,)+∞))
6425, 63sseldi 3748 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)) ∈ (0[,]+∞))
656eldifbd 3734 . . . . . . . . . 10 (𝜑 → ¬ 𝑍𝑌)
6647, 65eldifd 3732 . . . . . . . . 9 (𝜑𝑍 ∈ (𝑊𝑌))
6766adantr 466 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → 𝑍 ∈ (𝑊𝑌))
682, 26, 67, 4, 49, 40, 30, 53hsphoidmvle 41314 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))) ≤ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))
6922, 24, 56, 64, 68sge0lempt 41138 . . . . . 6 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))))) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
7061ltpnfd 12159 . . . . . 6 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) < +∞)
7158, 62, 60, 69, 70xrlelttrd 12195 . . . . 5 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))))) < +∞)
7258, 60, 71xrltned 40083 . . . 4 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))))) ≠ +∞)
73 ge0xrre 40270 . . . 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 565 . . 3 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))))) ∈ ℝ)
7521, 74remulcld 10271 . 2 (𝜑 → ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗)))))) ∈ ℝ)
7621, 61remulcld 10271 . 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 530 . . . . . . . 8 (𝜑 → (𝜑𝑍𝑊))
81 eleq1 2837 . . . . . . . . . . 11 (𝑘 = 𝑍 → (𝑘𝑊𝑍𝑊))
8281anbi2d 606 . . . . . . . . . 10 (𝑘 = 𝑍 → ((𝜑𝑘𝑊) ↔ (𝜑𝑍𝑊)))
83 fveq2 6332 . . . . . . . . . . 11 (𝑘 = 𝑍 → (𝐴𝑘) = (𝐴𝑍))
84 fveq2 6332 . . . . . . . . . . 11 (𝑘 = 𝑍 → (𝐵𝑘) = (𝐵𝑍))
8583, 84breq12d 4797 . . . . . . . . . 10 (𝑘 = 𝑍 → ((𝐴𝑘) < (𝐵𝑘) ↔ (𝐴𝑍) < (𝐵𝑍)))
8682, 85imbi12d 333 . . . . . . . . 9 (𝑘 = 𝑍 → (((𝜑𝑘𝑊) → (𝐴𝑘) < (𝐵𝑘)) ↔ ((𝜑𝑍𝑊) → (𝐴𝑍) < (𝐵𝑍))))
87 hoidmvlelem4.k . . . . . . . . 9 ((𝜑𝑘𝑊) → (𝐴𝑘) < (𝐵𝑘))
8886, 87vtoclg 3415 . . . . . . . 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 41323 . . . . . 6 (𝜑𝑆𝑈)
9148rexrd 10290 . . . . . . . 8 (𝜑 → (𝐵𝑍) ∈ ℝ*)
92 iccssxr 12460 . . . . . . . . 9 ((𝐴𝑍)[,](𝐵𝑍)) ⊆ ℝ*
93 ssrab2 3834 . . . . . . . . . . 11 {𝑧 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))))} ⊆ ((𝐴𝑍)[,](𝐵𝑍))
9478, 93eqsstri 3782 . . . . . . . . . 10 𝑈 ⊆ ((𝐴𝑍)[,](𝐵𝑍))
9594, 90sseldi 3748 . . . . . . . . 9 (𝜑𝑆 ∈ ((𝐴𝑍)[,](𝐵𝑍)))
9692, 95sseldi 3748 . . . . . . . 8 (𝜑𝑆 ∈ ℝ*)
97 simpl 468 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝑆) → 𝜑)
98 simpr 471 . . . . . . . . . . 11 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝑆) → ¬ (𝐵𝑍) ≤ 𝑆)
9914, 47ffvelrnd 6503 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴𝑍) ∈ ℝ)
10099, 48iccssred 40242 . . . . . . . . . . . . . 14 (𝜑 → ((𝐴𝑍)[,](𝐵𝑍)) ⊆ ℝ)
101100, 95sseldd 3751 . . . . . . . . . . . . 13 (𝜑𝑆 ∈ ℝ)
102101adantr 466 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝑆) → 𝑆 ∈ ℝ)
10397, 48syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝑆) → (𝐵𝑍) ∈ ℝ)
104102, 103ltnled 10385 . . . . . . . . . . 11 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝑆) → (𝑆 < (𝐵𝑍) ↔ ¬ (𝐵𝑍) ≤ 𝑆))
10598, 104mpbird 247 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝑆) → 𝑆 < (𝐵𝑍))
1063adantr 466 . . . . . . . . . . 11 ((𝜑𝑆 < (𝐵𝑍)) → 𝑋 ∈ Fin)
1075adantr 466 . . . . . . . . . . 11 ((𝜑𝑆 < (𝐵𝑍)) → 𝑌𝑋)
1086adantr 466 . . . . . . . . . . 11 ((𝜑𝑆 < (𝐵𝑍)) → 𝑍 ∈ (𝑋𝑌))
10914adantr 466 . . . . . . . . . . 11 ((𝜑𝑆 < (𝐵𝑍)) → 𝐴:𝑊⟶ℝ)
11015adantr 466 . . . . . . . . . . 11 ((𝜑𝑆 < (𝐵𝑍)) → 𝐵:𝑊⟶ℝ)
11187adantlr 686 . . . . . . . . . . 11 (((𝜑𝑆 < (𝐵𝑍)) ∧ 𝑘𝑊) → (𝐴𝑘) < (𝐵𝑘))
112 eqid 2770 . . . . . . . . . . 11 (𝑦𝑌 ↦ 0) = (𝑦𝑌 ↦ 0)
11327adantr 466 . . . . . . . . . . 11 ((𝜑𝑆 < (𝐵𝑍)) → 𝐶:ℕ⟶(ℝ ↑𝑚 𝑊))
114 fveq2 6332 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑗 → (𝐶𝑖) = (𝐶𝑗))
115114fveq1d 6334 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → ((𝐶𝑖)‘𝑍) = ((𝐶𝑗)‘𝑍))
116 fveq2 6332 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑗 → (𝐷𝑖) = (𝐷𝑗))
117116fveq1d 6334 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → ((𝐷𝑖)‘𝑍) = ((𝐷𝑗)‘𝑍))
118115, 117oveq12d 6810 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)) = (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)))
119118eleq2d 2835 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)) ↔ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))))
120114reseq1d 5533 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → ((𝐶𝑖) ↾ 𝑌) = ((𝐶𝑗) ↾ 𝑌))
121119, 120ifbieq1d 4246 . . . . . . . . . . . 12 (𝑖 = 𝑗 → if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐶𝑖) ↾ 𝑌), (𝑦𝑌 ↦ 0)) = if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), (𝑦𝑌 ↦ 0)))
122121cbvmptv 4882 . . . . . . . . . . 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 466 . . . . . . . . . . 11 ((𝜑𝑆 < (𝐵𝑍)) → 𝐷:ℕ⟶(ℝ ↑𝑚 𝑊))
124116reseq1d 5533 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → ((𝐷𝑖) ↾ 𝑌) = ((𝐷𝑗) ↾ 𝑌))
125119, 124ifbieq1d 4246 . . . . . . . . . . . 12 (𝑖 = 𝑗 → if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐷𝑖) ↾ 𝑌), (𝑦𝑌 ↦ 0)) = if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), (𝑦𝑌 ↦ 0)))
126125cbvmptv 4882 . . . . . . . . . . 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 466 . . . . . . . . . . 11 ((𝜑𝑆 < (𝐵𝑍)) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)
12819adantr 466 . . . . . . . . . . 11 ((𝜑𝑆 < (𝐵𝑍)) → 𝐸 ∈ ℝ+)
12990adantr 466 . . . . . . . . . . 11 ((𝜑𝑆 < (𝐵𝑍)) → 𝑆𝑈)
130 simpr 471 . . . . . . . . . . 11 ((𝜑𝑆 < (𝐵𝑍)) → 𝑆 < (𝐵𝑍))
131 biid 251 . . . . . . . . . . . . . . . . 17 (𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)) ↔ 𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)))
132 eqidd 2771 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑦 → 0 = 0)
133132cbvmptv 4882 . . . . . . . . . . . . . . . . 17 (𝑤𝑌 ↦ 0) = (𝑦𝑌 ↦ 0)
134131, 133ifbieq2i 4247 . . . . . . . . . . . . . . . 16 if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐶𝑖) ↾ 𝑌), (𝑤𝑌 ↦ 0)) = if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐶𝑖) ↾ 𝑌), (𝑦𝑌 ↦ 0))
135134mpteq2i 4873 . . . . . . . . . . . . . . 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 6338 . . . . . . . . . . . . 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 4247 . . . . . . . . . . . . . . . 16 if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐷𝑖) ↾ 𝑌), (𝑤𝑌 ↦ 0)) = if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐷𝑖) ↾ 𝑌), (𝑦𝑌 ↦ 0))
140139mpteq2i 4873 . . . . . . . . . . . . . . 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 6338 . . . . . . . . . . . . 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 6810 . . . . . . . . . . . 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 4882 . . . . . . . . . . 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 466 . . . . . . . . . . 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 466 . . . . . . . . . . 11 ((𝜑𝑆 < (𝐵𝑍)) → X𝑘𝑊 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑊 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))
149 eqid 2770 . . . . . . . . . . . 12 (𝑥X𝑦𝑌 ((𝐴𝑦)[,)(𝐵𝑦)) ↦ (𝑦𝑊 ↦ if(𝑦𝑌, (𝑥𝑦), 𝑆))) = (𝑥X𝑦𝑌 ((𝐴𝑦)[,)(𝐵𝑦)) ↦ (𝑦𝑊 ↦ if(𝑦𝑌, (𝑥𝑦), 𝑆)))
150 fveq2 6332 . . . . . . . . . . . . . . 15 (𝑦 = 𝑘 → (𝐴𝑦) = (𝐴𝑘))
151 fveq2 6332 . . . . . . . . . . . . . . 15 (𝑦 = 𝑘 → (𝐵𝑦) = (𝐵𝑘))
152150, 151oveq12d 6810 . . . . . . . . . . . . . 14 (𝑦 = 𝑘 → ((𝐴𝑦)[,)(𝐵𝑦)) = ((𝐴𝑘)[,)(𝐵𝑘)))
153152cbvixpv 8079 . . . . . . . . . . . . 13 X𝑦𝑌 ((𝐴𝑦)[,)(𝐵𝑦)) = X𝑘𝑌 ((𝐴𝑘)[,)(𝐵𝑘))
154 eleq1 2837 . . . . . . . . . . . . . . 15 (𝑦 = 𝑘 → (𝑦𝑌𝑘𝑌))
155 fveq2 6332 . . . . . . . . . . . . . . 15 (𝑦 = 𝑘 → (𝑥𝑦) = (𝑥𝑘))
156154, 155ifbieq1d 4246 . . . . . . . . . . . . . 14 (𝑦 = 𝑘 → if(𝑦𝑌, (𝑥𝑦), 𝑆) = if(𝑘𝑌, (𝑥𝑘), 𝑆))
157156cbvmptv 4882 . . . . . . . . . . . . 13 (𝑦𝑊 ↦ if(𝑦𝑌, (𝑥𝑦), 𝑆)) = (𝑘𝑊 ↦ if(𝑘𝑌, (𝑥𝑘), 𝑆))
158153, 157mpteq12i 4874 . . . . . . . . . . . 12 (𝑥X𝑦𝑌 ((𝐴𝑦)[,)(𝐵𝑦)) ↦ (𝑦𝑊 ↦ if(𝑦𝑌, (𝑥𝑦), 𝑆))) = (𝑥X𝑘𝑌 ((𝐴𝑘)[,)(𝐵𝑘)) ↦ (𝑘𝑊 ↦ if(𝑘𝑌, (𝑥𝑘), 𝑆)))
159149, 158eqtri 2792 . . . . . . . . . . 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 41325 . . . . . . . . . 10 ((𝜑𝑆 < (𝐵𝑍)) → ∃𝑢𝑈 𝑆 < 𝑢)
16197, 105, 160syl2anc 565 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝑆) → ∃𝑢𝑈 𝑆 < 𝑢)
16294a1i 11 . . . . . . . . . . . . . . . . . 18 (𝜑𝑈 ⊆ ((𝐴𝑍)[,](𝐵𝑍)))
163162, 100sstrd 3760 . . . . . . . . . . . . . . . . 17 (𝜑𝑈 ⊆ ℝ)
164163adantr 466 . . . . . . . . . . . . . . . 16 ((𝜑𝑢𝑈) → 𝑈 ⊆ ℝ)
165 ne0i 4067 . . . . . . . . . . . . . . . . 17 (𝑢𝑈𝑈 ≠ ∅)
166165adantl 467 . . . . . . . . . . . . . . . 16 ((𝜑𝑢𝑈) → 𝑈 ≠ ∅)
16799rexrd 10290 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐴𝑍) ∈ ℝ*)
168167adantr 466 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑢𝑈) → (𝐴𝑍) ∈ ℝ*)
16991adantr 466 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑢𝑈) → (𝐵𝑍) ∈ ℝ*)
170162sselda 3750 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑢𝑈) → 𝑢 ∈ ((𝐴𝑍)[,](𝐵𝑍)))
171 iccleub 12433 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑍) ∈ ℝ* ∧ (𝐵𝑍) ∈ ℝ*𝑢 ∈ ((𝐴𝑍)[,](𝐵𝑍))) → 𝑢 ≤ (𝐵𝑍))
172168, 169, 170, 171syl3anc 1475 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑢𝑈) → 𝑢 ≤ (𝐵𝑍))
173172ralrimiva 3114 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑢𝑈 𝑢 ≤ (𝐵𝑍))
174 breq2 4788 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (𝐵𝑍) → (𝑢𝑦𝑢 ≤ (𝐵𝑍)))
175174ralbidv 3134 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (𝐵𝑍) → (∀𝑢𝑈 𝑢𝑦 ↔ ∀𝑢𝑈 𝑢 ≤ (𝐵𝑍)))
176175rspcev 3458 . . . . . . . . . . . . . . . . . 18 (((𝐵𝑍) ∈ ℝ ∧ ∀𝑢𝑈 𝑢 ≤ (𝐵𝑍)) → ∃𝑦 ∈ ℝ ∀𝑢𝑈 𝑢𝑦)
17748, 173, 176syl2anc 565 . . . . . . . . . . . . . . . . 17 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑢𝑈 𝑢𝑦)
178177adantr 466 . . . . . . . . . . . . . . . 16 ((𝜑𝑢𝑈) → ∃𝑦 ∈ ℝ ∀𝑢𝑈 𝑢𝑦)
179 simpr 471 . . . . . . . . . . . . . . . 16 ((𝜑𝑢𝑈) → 𝑢𝑈)
180 suprub 11185 . . . . . . . . . . . . . . . 16 (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑢𝑈 𝑢𝑦) ∧ 𝑢𝑈) → 𝑢 ≤ sup(𝑈, ℝ, < ))
181164, 166, 178, 179, 180syl31anc 1478 . . . . . . . . . . . . . . 15 ((𝜑𝑢𝑈) → 𝑢 ≤ sup(𝑈, ℝ, < ))
182181, 79syl6breqr 4826 . . . . . . . . . . . . . 14 ((𝜑𝑢𝑈) → 𝑢𝑆)
183182ralrimiva 3114 . . . . . . . . . . . . 13 (𝜑 → ∀𝑢𝑈 𝑢𝑆)
184164, 179sseldd 3751 . . . . . . . . . . . . . . 15 ((𝜑𝑢𝑈) → 𝑢 ∈ ℝ)
185101adantr 466 . . . . . . . . . . . . . . 15 ((𝜑𝑢𝑈) → 𝑆 ∈ ℝ)
186184, 185lenltd 10384 . . . . . . . . . . . . . 14 ((𝜑𝑢𝑈) → (𝑢𝑆 ↔ ¬ 𝑆 < 𝑢))
187186ralbidva 3133 . . . . . . . . . . . . 13 (𝜑 → (∀𝑢𝑈 𝑢𝑆 ↔ ∀𝑢𝑈 ¬ 𝑆 < 𝑢))
188183, 187mpbid 222 . . . . . . . . . . . 12 (𝜑 → ∀𝑢𝑈 ¬ 𝑆 < 𝑢)
189 ralnex 3140 . . . . . . . . . . . 12 (∀𝑢𝑈 ¬ 𝑆 < 𝑢 ↔ ¬ ∃𝑢𝑈 𝑆 < 𝑢)
190188, 189sylib 208 . . . . . . . . . . 11 (𝜑 → ¬ ∃𝑢𝑈 𝑆 < 𝑢)
191190adantr 466 . . . . . . . . . 10 ((𝜑𝑆 < (𝐵𝑍)) → ¬ ∃𝑢𝑈 𝑆 < 𝑢)
19297, 105, 191syl2anc 565 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝑆) → ¬ ∃𝑢𝑈 𝑆 < 𝑢)
193161, 192condan 801 . . . . . . . 8 (𝜑 → (𝐵𝑍) ≤ 𝑆)
194 iccleub 12433 . . . . . . . . 9 (((𝐴𝑍) ∈ ℝ* ∧ (𝐵𝑍) ∈ ℝ*𝑆 ∈ ((𝐴𝑍)[,](𝐵𝑍))) → 𝑆 ≤ (𝐵𝑍))
195167, 91, 95, 194syl3anc 1475 . . . . . . . 8 (𝜑𝑆 ≤ (𝐵𝑍))
19691, 96, 193, 195xrletrid 12190 . . . . . . 7 (𝜑 → (𝐵𝑍) = 𝑆)
19778eqcomi 2779 . . . . . . . 8 {𝑧 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))))} = 𝑈
198197a1i 11 . . . . . . 7 (𝜑 → {𝑧 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))))} = 𝑈)
199196, 198eleq12d 2843 . . . . . 6 (𝜑 → ((𝐵𝑍) ∈ {𝑧 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))))} ↔ 𝑆𝑈))
20090, 199mpbird 247 . . . . 5 (𝜑 → (𝐵𝑍) ∈ {𝑧 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))))})
201 oveq1 6799 . . . . . . . 8 (𝑧 = (𝐵𝑍) → (𝑧 − (𝐴𝑍)) = ((𝐵𝑍) − (𝐴𝑍)))
202201oveq2d 6808 . . . . . . 7 (𝑧 = (𝐵𝑍) → (𝐺 · (𝑧 − (𝐴𝑍))) = (𝐺 · ((𝐵𝑍) − (𝐴𝑍))))
203 fveq2 6332 . . . . . . . . . . . 12 (𝑧 = (𝐵𝑍) → (𝐻𝑧) = (𝐻‘(𝐵𝑍)))
204203fveq1d 6334 . . . . . . . . . . 11 (𝑧 = (𝐵𝑍) → ((𝐻𝑧)‘(𝐷𝑗)) = ((𝐻‘(𝐵𝑍))‘(𝐷𝑗)))
205204oveq2d 6808 . . . . . . . . . 10 (𝑧 = (𝐵𝑍) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))) = ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))))
206205mpteq2dv 4877 . . . . . . . . 9 (𝑧 = (𝐵𝑍) → (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗)))) = (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗)))))
207206fveq2d 6336 . . . . . . . 8 (𝑧 = (𝐵𝑍) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))))))
208207oveq2d 6808 . . . . . . 7 (𝑧 = (𝐵𝑍) → ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗)))))) = ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗)))))))
209202, 208breq12d 4797 . . . . . 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))))))))
210209elrab 3513 . . . . 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))))))))
211200, 210sylib 208 . . . 4 (𝜑 → ((𝐵𝑍) ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∧ (𝐺 · ((𝐵𝑍) − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))))))))
212211simprd 477 . . 3 (𝜑 → (𝐺 · ((𝐵𝑍) − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗)))))))
2133, 5ssfid 8338 . . . . . 6 (𝜑𝑌 ∈ Fin)
214 eqid 2770 . . . . . 6 𝑘𝑌 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = ∏𝑘𝑌 (vol‘((𝐴𝑘)[,)(𝐵𝑘)))
2152, 213, 6, 65, 4, 14, 15, 214hoiprodp1 41316 . . . . 5 (𝜑 → (𝐴(𝐿𝑊)𝐵) = (∏𝑘𝑌 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) · (vol‘((𝐴𝑍)[,)(𝐵𝑍)))))
216 eqidd 2771 . . . . . . 7 (𝜑 → ∏𝑘𝑌 ((𝐵𝑘) − (𝐴𝑘)) = ∏𝑘𝑌 ((𝐵𝑘) − (𝐴𝑘)))
21714adantr 466 . . . . . . . . . 10 ((𝜑𝑘𝑌) → 𝐴:𝑊⟶ℝ)
218 ssun1 3925 . . . . . . . . . . . 12 𝑌 ⊆ (𝑌 ∪ {𝑍})
2194eqcomi 2779 . . . . . . . . . . . 12 (𝑌 ∪ {𝑍}) = 𝑊
220218, 219sseqtri 3784 . . . . . . . . . . 11 𝑌𝑊
221 simpr 471 . . . . . . . . . . 11 ((𝜑𝑘𝑌) → 𝑘𝑌)
222220, 221sseldi 3748 . . . . . . . . . 10 ((𝜑𝑘𝑌) → 𝑘𝑊)
223217, 222ffvelrnd 6503 . . . . . . . . 9 ((𝜑𝑘𝑌) → (𝐴𝑘) ∈ ℝ)
22415adantr 466 . . . . . . . . . 10 ((𝜑𝑘𝑌) → 𝐵:𝑊⟶ℝ)
225224, 222ffvelrnd 6503 . . . . . . . . 9 ((𝜑𝑘𝑌) → (𝐵𝑘) ∈ ℝ)
226222, 87syldan 571 . . . . . . . . 9 ((𝜑𝑘𝑌) → (𝐴𝑘) < (𝐵𝑘))
227223, 225, 226volicon0 41303 . . . . . . . 8 ((𝜑𝑘𝑌) → (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = ((𝐵𝑘) − (𝐴𝑘)))
228227prodeq2dv 14859 . . . . . . 7 (𝜑 → ∏𝑘𝑌 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = ∏𝑘𝑌 ((𝐵𝑘) − (𝐴𝑘)))
22977a1i 11 . . . . . . . 8 (𝜑𝐺 = ((𝐴𝑌)(𝐿𝑌)(𝐵𝑌)))
230 hoidmvlelem4.n . . . . . . . . 9 (𝜑𝑌 ≠ ∅)
231220a1i 11 . . . . . . . . . 10 (𝜑𝑌𝑊)
23214, 231fssresd 6211 . . . . . . . . 9 (𝜑 → (𝐴𝑌):𝑌⟶ℝ)
23315, 231fssresd 6211 . . . . . . . . 9 (𝜑 → (𝐵𝑌):𝑌⟶ℝ)
2342, 213, 230, 232, 233hoidmvn0val 41312 . . . . . . . 8 (𝜑 → ((𝐴𝑌)(𝐿𝑌)(𝐵𝑌)) = ∏𝑘𝑌 (vol‘(((𝐴𝑌)‘𝑘)[,)((𝐵𝑌)‘𝑘))))
235 fvres 6348 . . . . . . . . . . . . 13 (𝑘𝑌 → ((𝐴𝑌)‘𝑘) = (𝐴𝑘))
236 fvres 6348 . . . . . . . . . . . . 13 (𝑘𝑌 → ((𝐵𝑌)‘𝑘) = (𝐵𝑘))
237235, 236oveq12d 6810 . . . . . . . . . . . 12 (𝑘𝑌 → (((𝐴𝑌)‘𝑘)[,)((𝐵𝑌)‘𝑘)) = ((𝐴𝑘)[,)(𝐵𝑘)))
238237fveq2d 6336 . . . . . . . . . . 11 (𝑘𝑌 → (vol‘(((𝐴𝑌)‘𝑘)[,)((𝐵𝑌)‘𝑘))) = (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
239238adantl 467 . . . . . . . . . 10 ((𝜑𝑘𝑌) → (vol‘(((𝐴𝑌)‘𝑘)[,)((𝐵𝑌)‘𝑘))) = (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
240 volico 40711 . . . . . . . . . . 11 (((𝐴𝑘) ∈ ℝ ∧ (𝐵𝑘) ∈ ℝ) → (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = if((𝐴𝑘) < (𝐵𝑘), ((𝐵𝑘) − (𝐴𝑘)), 0))
241223, 225, 240syl2anc 565 . . . . . . . . . 10 ((𝜑𝑘𝑌) → (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = if((𝐴𝑘) < (𝐵𝑘), ((𝐵𝑘) − (𝐴𝑘)), 0))
242241, 227eqtr3d 2806 . . . . . . . . . 10 ((𝜑𝑘𝑌) → if((𝐴𝑘) < (𝐵𝑘), ((𝐵𝑘) − (𝐴𝑘)), 0) = ((𝐵𝑘) − (𝐴𝑘)))
243239, 241, 2423eqtrd 2808 . . . . . . . . 9 ((𝜑𝑘𝑌) → (vol‘(((𝐴𝑌)‘𝑘)[,)((𝐵𝑌)‘𝑘))) = ((𝐵𝑘) − (𝐴𝑘)))
244243prodeq2dv 14859 . . . . . . . 8 (𝜑 → ∏𝑘𝑌 (vol‘(((𝐴𝑌)‘𝑘)[,)((𝐵𝑌)‘𝑘))) = ∏𝑘𝑌 ((𝐵𝑘) − (𝐴𝑘)))
245229, 234, 2443eqtrd 2808 . . . . . . 7 (𝜑𝐺 = ∏𝑘𝑌 ((𝐵𝑘) − (𝐴𝑘)))
246216, 228, 2453eqtr4d 2814 . . . . . 6 (𝜑 → ∏𝑘𝑌 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = 𝐺)
24799, 48, 89volicon0 41303 . . . . . 6 (𝜑 → (vol‘((𝐴𝑍)[,)(𝐵𝑍))) = ((𝐵𝑍) − (𝐴𝑍)))
248246, 247oveq12d 6810 . . . . 5 (𝜑 → (∏𝑘𝑌 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) · (vol‘((𝐴𝑍)[,)(𝐵𝑍)))) = (𝐺 · ((𝐵𝑍) − (𝐴𝑍))))
249215, 248eqtrd 2804 . . . 4 (𝜑 → (𝐴(𝐿𝑊)𝐵) = (𝐺 · ((𝐵𝑍) − (𝐴𝑍))))
250249breq1d 4794 . . 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))))))))
251212, 250mpbird 247 . 2 (𝜑 → (𝐴(𝐿𝑊)𝐵) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗)))))))
252 0le1 10752 . . . . 5 0 ≤ 1
253252a1i 11 . . . 4 (𝜑 → 0 ≤ 1)
25419rpge0d 12078 . . . 4 (𝜑 → 0 ≤ 𝐸)
25518, 20, 253, 254addge0d 10804 . . 3 (𝜑 → 0 ≤ (1 + 𝐸))
25674, 61, 21, 255, 69lemul2ad 11165 . 2 (𝜑 → ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗)))))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗))))))
25717, 75, 76, 251, 256letrd 10395 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 382   = wceq 1630  wcel 2144  wne 2942  wral 3060  wrex 3061  {crab 3064  Vcvv 3349  cdif 3718  cun 3719  wss 3721  c0 4061  ifcif 4223  {csn 4314   ciun 4652   class class class wbr 4784  cmpt 4861  cres 5251  wf 6027  cfv 6031  (class class class)co 6792  cmpt2 6794  𝑚 cmap 8008  Xcixp 8061  Fincfn 8108  supcsup 8501  cr 10136  0cc0 10137  1c1 10138   + caddc 10140   · cmul 10142  +∞cpnf 10272  *cxr 10274   < clt 10275  cle 10276  cmin 10467  cn 11221  +crp 12034  [,)cico 12381  [,]cicc 12382  cprod 14841  volcvol 23450  Σ^csumge0 41090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-inf2 8701  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214  ax-pre-sup 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-of 7043  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-2o 7713  df-oadd 7716  df-er 7895  df-map 8010  df-pm 8011  df-ixp 8062  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-fi 8472  df-sup 8503  df-inf 8504  df-oi 8570  df-card 8964  df-cda 9191  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-div 10886  df-nn 11222  df-2 11280  df-3 11281  df-n0 11494  df-z 11579  df-uz 11888  df-q 11991  df-rp 12035  df-xneg 12150  df-xadd 12151  df-xmul 12152  df-ioo 12383  df-ico 12385  df-icc 12386  df-fz 12533  df-fzo 12673  df-fl 12800  df-seq 13008  df-exp 13067  df-hash 13321  df-cj 14046  df-re 14047  df-im 14048  df-sqrt 14182  df-abs 14183  df-clim 14426  df-rlim 14427  df-sum 14624  df-prod 14842  df-rest 16290  df-topgen 16311  df-psmet 19952  df-xmet 19953  df-met 19954  df-bl 19955  df-mopn 19956  df-top 20918  df-topon 20935  df-bases 20970  df-cmp 21410  df-ovol 23451  df-vol 23452  df-sumge0 41091
This theorem is referenced by:  hoidmvlelem5  41327
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