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Theorem hoidmvlelem4 40575
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:   𝑥,𝑘   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 12265 . . 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 3579 . . . . . . . 8 (𝜑𝑍𝑋)
8 snssi 4330 . . . . . . . 8 (𝑍𝑋 → {𝑍} ⊆ 𝑋)
97, 8syl 17 . . . . . . 7 (𝜑 → {𝑍} ⊆ 𝑋)
105, 9unssd 3781 . . . . . 6 (𝜑 → (𝑌 ∪ {𝑍}) ⊆ 𝑋)
114, 10syl5eqss 3641 . . . . 5 (𝜑𝑊𝑋)
12 ssfi 8165 . . . . 5 ((𝑋 ∈ Fin ∧ 𝑊𝑋) → 𝑊 ∈ Fin)
133, 11, 12syl2anc 692 . . . 4 (𝜑𝑊 ∈ Fin)
14 hoidmvlelem4.a . . . 4 (𝜑𝐴:𝑊⟶ℝ)
15 hoidmvlelem4.b . . . 4 (𝜑𝐵:𝑊⟶ℝ)
162, 13, 14, 15hoidmvcl 40559 . . 3 (𝜑 → (𝐴(𝐿𝑊)𝐵) ∈ (0[,)+∞))
171, 16sseldi 3593 . 2 (𝜑 → (𝐴(𝐿𝑊)𝐵) ∈ ℝ)
18 1red 10040 . . . 4 (𝜑 → 1 ∈ ℝ)
19 hoidmvlelem4.e . . . . 5 (𝜑𝐸 ∈ ℝ+)
2019rpred 11857 . . . 4 (𝜑𝐸 ∈ ℝ)
2118, 20readdcld 10054 . . 3 (𝜑 → (1 + 𝐸) ∈ ℝ)
22 nfv 1841 . . . . 5 𝑗𝜑
23 nnex 11011 . . . . . 6 ℕ ∈ V
2423a1i 11 . . . . 5 (𝜑 → ℕ ∈ V)
25 icossicc 12245 . . . . . 6 (0[,)+∞) ⊆ (0[,]+∞)
2613adantr 481 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → 𝑊 ∈ Fin)
27 hoidmvlelem4.c . . . . . . . . 9 (𝜑𝐶:ℕ⟶(ℝ ↑𝑚 𝑊))
2827ffvelrnda 6345 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ∈ (ℝ ↑𝑚 𝑊))
29 elmapi 7864 . . . . . . . 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 2687 . . . . . . . . . . . . 13 (𝑗 = → (𝑗𝑌𝑌))
33 fveq2 6178 . . . . . . . . . . . . 13 (𝑗 = → (𝑐𝑗) = (𝑐))
3433breq1d 4654 . . . . . . . . . . . . . 14 (𝑗 = → ((𝑐𝑗) ≤ 𝑥 ↔ (𝑐) ≤ 𝑥))
3534, 33ifbieq1d 4100 . . . . . . . . . . . . 13 (𝑗 = → if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥) = if((𝑐) ≤ 𝑥, (𝑐), 𝑥))
3632, 33, 35ifbieq12d 4104 . . . . . . . . . . . 12 (𝑗 = → if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)) = if(𝑌, (𝑐), if((𝑐) ≤ 𝑥, (𝑐), 𝑥)))
3736cbvmptv 4741 . . . . . . . . . . 11 (𝑗𝑊 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥))) = (𝑊 ↦ if(𝑌, (𝑐), if((𝑐) ≤ 𝑥, (𝑐), 𝑥)))
3837mpteq2i 4732 . . . . . . . . . 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 4732 . . . . . . . . 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 2642 . . . . . . . 8 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑊 ↦ if(𝑌, (𝑐), if((𝑐) ≤ 𝑥, (𝑐), 𝑥)))))
41 snidg 4197 . . . . . . . . . . . . 13 (𝑍 ∈ (𝑋𝑌) → 𝑍 ∈ {𝑍})
426, 41syl 17 . . . . . . . . . . . 12 (𝜑𝑍 ∈ {𝑍})
43 elun2 3773 . . . . . . . . . . . 12 (𝑍 ∈ {𝑍} → 𝑍 ∈ (𝑌 ∪ {𝑍}))
4442, 43syl 17 . . . . . . . . . . 11 (𝜑𝑍 ∈ (𝑌 ∪ {𝑍}))
454a1i 11 . . . . . . . . . . . 12 (𝜑𝑊 = (𝑌 ∪ {𝑍}))
4645eqcomd 2626 . . . . . . . . . . 11 (𝜑 → (𝑌 ∪ {𝑍}) = 𝑊)
4744, 46eleqtrd 2701 . . . . . . . . . 10 (𝜑𝑍𝑊)
4815, 47ffvelrnd 6346 . . . . . . . . 9 (𝜑 → (𝐵𝑍) ∈ ℝ)
4948adantr 481 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐵𝑍) ∈ ℝ)
50 hoidmvlelem4.d . . . . . . . . . 10 (𝜑𝐷:ℕ⟶(ℝ ↑𝑚 𝑊))
5150ffvelrnda 6345 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗) ∈ (ℝ ↑𝑚 𝑊))
52 elmapi 7864 . . . . . . . . 9 ((𝐷𝑗) ∈ (ℝ ↑𝑚 𝑊) → (𝐷𝑗):𝑊⟶ℝ)
5351, 52syl 17 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗):𝑊⟶ℝ)
5440, 49, 26, 53hsphoif 40553 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ((𝐻‘(𝐵𝑍))‘(𝐷𝑗)):𝑊⟶ℝ)
552, 26, 30, 54hoidmvcl 40559 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))) ∈ (0[,)+∞))
5625, 55sseldi 3593 . . . . 5 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))) ∈ (0[,]+∞))
5722, 24, 56sge0clmpt 40405 . . . 4 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))))) ∈ (0[,]+∞))
5822, 24, 56sge0xrclmpt 40408 . . . . 5 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))))) ∈ ℝ*)
59 pnfxr 10077 . . . . . 6 +∞ ∈ ℝ*
6059a1i 11 . . . . 5 (𝜑 → +∞ ∈ ℝ*)
61 hoidmvlelem4.r . . . . . . 7 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)
6261rexrd 10074 . . . . . 6 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ*)
632, 26, 30, 53hoidmvcl 40559 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)) ∈ (0[,)+∞))
6425, 63sseldi 3593 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)) ∈ (0[,]+∞))
656eldifbd 3580 . . . . . . . . . 10 (𝜑 → ¬ 𝑍𝑌)
6647, 65eldifd 3578 . . . . . . . . 9 (𝜑𝑍 ∈ (𝑊𝑌))
6766adantr 481 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → 𝑍 ∈ (𝑊𝑌))
682, 26, 67, 4, 49, 40, 30, 53hsphoidmvle 40563 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))) ≤ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))
6922, 24, 56, 64, 68sge0lempt 40390 . . . . . 6 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))))) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
7061ltpnfd 11940 . . . . . 6 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) < +∞)
7158, 62, 60, 69, 70xrlelttrd 11976 . . . . 5 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))))) < +∞)
7258, 60, 71xrltned 39386 . . . 4 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))))) ≠ +∞)
73 ge0xrre 39561 . . . 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 692 . . 3 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))))) ∈ ℝ)
7521, 74remulcld 10055 . 2 (𝜑 → ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗)))))) ∈ ℝ)
7621, 61remulcld 10055 . 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 573 . . . . . . . 8 (𝜑 → (𝜑𝑍𝑊))
81 eleq1 2687 . . . . . . . . . . 11 (𝑘 = 𝑍 → (𝑘𝑊𝑍𝑊))
8281anbi2d 739 . . . . . . . . . 10 (𝑘 = 𝑍 → ((𝜑𝑘𝑊) ↔ (𝜑𝑍𝑊)))
83 fveq2 6178 . . . . . . . . . . 11 (𝑘 = 𝑍 → (𝐴𝑘) = (𝐴𝑍))
84 fveq2 6178 . . . . . . . . . . 11 (𝑘 = 𝑍 → (𝐵𝑘) = (𝐵𝑍))
8583, 84breq12d 4657 . . . . . . . . . 10 (𝑘 = 𝑍 → ((𝐴𝑘) < (𝐵𝑘) ↔ (𝐴𝑍) < (𝐵𝑍)))
8682, 85imbi12d 334 . . . . . . . . 9 (𝑘 = 𝑍 → (((𝜑𝑘𝑊) → (𝐴𝑘) < (𝐵𝑘)) ↔ ((𝜑𝑍𝑊) → (𝐴𝑍) < (𝐵𝑍))))
87 hoidmvlelem4.k . . . . . . . . 9 ((𝜑𝑘𝑊) → (𝐴𝑘) < (𝐵𝑘))
8886, 87vtoclg 3261 . . . . . . . 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 40572 . . . . . 6 (𝜑𝑆𝑈)
9148rexrd 10074 . . . . . . . 8 (𝜑 → (𝐵𝑍) ∈ ℝ*)
92 iccssxr 12241 . . . . . . . . 9 ((𝐴𝑍)[,](𝐵𝑍)) ⊆ ℝ*
93 ssrab2 3679 . . . . . . . . . . 11 {𝑧 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))))} ⊆ ((𝐴𝑍)[,](𝐵𝑍))
9478, 93eqsstri 3627 . . . . . . . . . 10 𝑈 ⊆ ((𝐴𝑍)[,](𝐵𝑍))
9594, 90sseldi 3593 . . . . . . . . 9 (𝜑𝑆 ∈ ((𝐴𝑍)[,](𝐵𝑍)))
9692, 95sseldi 3593 . . . . . . . 8 (𝜑𝑆 ∈ ℝ*)
97 simpl 473 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝑆) → 𝜑)
98 simpr 477 . . . . . . . . . . 11 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝑆) → ¬ (𝐵𝑍) ≤ 𝑆)
9914, 47ffvelrnd 6346 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴𝑍) ∈ ℝ)
10099, 48iccssred 39530 . . . . . . . . . . . . . 14 (𝜑 → ((𝐴𝑍)[,](𝐵𝑍)) ⊆ ℝ)
101100, 95sseldd 3596 . . . . . . . . . . . . 13 (𝜑𝑆 ∈ ℝ)
102101adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝑆) → 𝑆 ∈ ℝ)
10397, 48syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝑆) → (𝐵𝑍) ∈ ℝ)
104102, 103ltnled 10169 . . . . . . . . . . 11 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝑆) → (𝑆 < (𝐵𝑍) ↔ ¬ (𝐵𝑍) ≤ 𝑆))
10598, 104mpbird 247 . . . . . . . . . 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 750 . . . . . . . . . . 11 (((𝜑𝑆 < (𝐵𝑍)) ∧ 𝑘𝑊) → (𝐴𝑘) < (𝐵𝑘))
112 eqid 2620 . . . . . . . . . . 11 (𝑦𝑌 ↦ 0) = (𝑦𝑌 ↦ 0)
11327adantr 481 . . . . . . . . . . 11 ((𝜑𝑆 < (𝐵𝑍)) → 𝐶:ℕ⟶(ℝ ↑𝑚 𝑊))
114 fveq2 6178 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑗 → (𝐶𝑖) = (𝐶𝑗))
115114fveq1d 6180 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → ((𝐶𝑖)‘𝑍) = ((𝐶𝑗)‘𝑍))
116 fveq2 6178 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑗 → (𝐷𝑖) = (𝐷𝑗))
117116fveq1d 6180 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → ((𝐷𝑖)‘𝑍) = ((𝐷𝑗)‘𝑍))
118115, 117oveq12d 6653 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)) = (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)))
119118eleq2d 2685 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)) ↔ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))))
120114reseq1d 5384 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → ((𝐶𝑖) ↾ 𝑌) = ((𝐶𝑗) ↾ 𝑌))
121119, 120ifbieq1d 4100 . . . . . . . . . . . 12 (𝑖 = 𝑗 → if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐶𝑖) ↾ 𝑌), (𝑦𝑌 ↦ 0)) = if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), (𝑦𝑌 ↦ 0)))
122121cbvmptv 4741 . . . . . . . . . . 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 5384 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → ((𝐷𝑖) ↾ 𝑌) = ((𝐷𝑗) ↾ 𝑌))
125119, 124ifbieq1d 4100 . . . . . . . . . . . 12 (𝑖 = 𝑗 → if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐷𝑖) ↾ 𝑌), (𝑦𝑌 ↦ 0)) = if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), (𝑦𝑌 ↦ 0)))
126125cbvmptv 4741 . . . . . . . . . . 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 477 . . . . . . . . . . 11 ((𝜑𝑆 < (𝐵𝑍)) → 𝑆 < (𝐵𝑍))
131 biid 251 . . . . . . . . . . . . . . . . 17 (𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)) ↔ 𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)))
132 eqidd 2621 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑦 → 0 = 0)
133132cbvmptv 4741 . . . . . . . . . . . . . . . . 17 (𝑤𝑌 ↦ 0) = (𝑦𝑌 ↦ 0)
134131, 133ifbieq2i 4101 . . . . . . . . . . . . . . . 16 if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐶𝑖) ↾ 𝑌), (𝑤𝑌 ↦ 0)) = if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐶𝑖) ↾ 𝑌), (𝑦𝑌 ↦ 0))
135134mpteq2i 4732 . . . . . . . . . . . . . . 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 6184 . . . . . . . . . . . . 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 4101 . . . . . . . . . . . . . . . 16 if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐷𝑖) ↾ 𝑌), (𝑤𝑌 ↦ 0)) = if(𝑆 ∈ (((𝐶𝑖)‘𝑍)[,)((𝐷𝑖)‘𝑍)), ((𝐷𝑖) ↾ 𝑌), (𝑦𝑌 ↦ 0))
140139mpteq2i 4732 . . . . . . . . . . . . . . 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 6184 . . . . . . . . . . . . 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 6653 . . . . . . . . . . . 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 4741 . . . . . . . . . . 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 2620 . . . . . . . . . . . 12 (𝑥X𝑦𝑌 ((𝐴𝑦)[,)(𝐵𝑦)) ↦ (𝑦𝑊 ↦ if(𝑦𝑌, (𝑥𝑦), 𝑆))) = (𝑥X𝑦𝑌 ((𝐴𝑦)[,)(𝐵𝑦)) ↦ (𝑦𝑊 ↦ if(𝑦𝑌, (𝑥𝑦), 𝑆)))
150 fveq2 6178 . . . . . . . . . . . . . . 15 (𝑦 = 𝑘 → (𝐴𝑦) = (𝐴𝑘))
151 fveq2 6178 . . . . . . . . . . . . . . 15 (𝑦 = 𝑘 → (𝐵𝑦) = (𝐵𝑘))
152150, 151oveq12d 6653 . . . . . . . . . . . . . 14 (𝑦 = 𝑘 → ((𝐴𝑦)[,)(𝐵𝑦)) = ((𝐴𝑘)[,)(𝐵𝑘)))
153152cbvixpv 7911 . . . . . . . . . . . . 13 X𝑦𝑌 ((𝐴𝑦)[,)(𝐵𝑦)) = X𝑘𝑌 ((𝐴𝑘)[,)(𝐵𝑘))
154 eleq1 2687 . . . . . . . . . . . . . . 15 (𝑦 = 𝑘 → (𝑦𝑌𝑘𝑌))
155 fveq2 6178 . . . . . . . . . . . . . . 15 (𝑦 = 𝑘 → (𝑥𝑦) = (𝑥𝑘))
156154, 155ifbieq1d 4100 . . . . . . . . . . . . . 14 (𝑦 = 𝑘 → if(𝑦𝑌, (𝑥𝑦), 𝑆) = if(𝑘𝑌, (𝑥𝑘), 𝑆))
157156cbvmptv 4741 . . . . . . . . . . . . 13 (𝑦𝑊 ↦ if(𝑦𝑌, (𝑥𝑦), 𝑆)) = (𝑘𝑊 ↦ if(𝑘𝑌, (𝑥𝑘), 𝑆))
158153, 157mpteq12i 4733 . . . . . . . . . . . 12 (𝑥X𝑦𝑌 ((𝐴𝑦)[,)(𝐵𝑦)) ↦ (𝑦𝑊 ↦ if(𝑦𝑌, (𝑥𝑦), 𝑆))) = (𝑥X𝑘𝑌 ((𝐴𝑘)[,)(𝐵𝑘)) ↦ (𝑘𝑊 ↦ if(𝑘𝑌, (𝑥𝑘), 𝑆)))
159149, 158eqtri 2642 . . . . . . . . . . 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 40574 . . . . . . . . . 10 ((𝜑𝑆 < (𝐵𝑍)) → ∃𝑢𝑈 𝑆 < 𝑢)
16197, 105, 160syl2anc 692 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝑆) → ∃𝑢𝑈 𝑆 < 𝑢)
16294a1i 11 . . . . . . . . . . . . . . . . . 18 (𝜑𝑈 ⊆ ((𝐴𝑍)[,](𝐵𝑍)))
163162, 100sstrd 3605 . . . . . . . . . . . . . . . . 17 (𝜑𝑈 ⊆ ℝ)
164163adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑢𝑈) → 𝑈 ⊆ ℝ)
165 ne0i 3913 . . . . . . . . . . . . . . . . 17 (𝑢𝑈𝑈 ≠ ∅)
166165adantl 482 . . . . . . . . . . . . . . . 16 ((𝜑𝑢𝑈) → 𝑈 ≠ ∅)
16799rexrd 10074 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐴𝑍) ∈ ℝ*)
168167adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑢𝑈) → (𝐴𝑍) ∈ ℝ*)
16991adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑢𝑈) → (𝐵𝑍) ∈ ℝ*)
170162sselda 3595 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑢𝑈) → 𝑢 ∈ ((𝐴𝑍)[,](𝐵𝑍)))
171 iccleub 12214 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑍) ∈ ℝ* ∧ (𝐵𝑍) ∈ ℝ*𝑢 ∈ ((𝐴𝑍)[,](𝐵𝑍))) → 𝑢 ≤ (𝐵𝑍))
172168, 169, 170, 171syl3anc 1324 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑢𝑈) → 𝑢 ≤ (𝐵𝑍))
173172ralrimiva 2963 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑢𝑈 𝑢 ≤ (𝐵𝑍))
174 breq2 4648 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (𝐵𝑍) → (𝑢𝑦𝑢 ≤ (𝐵𝑍)))
175174ralbidv 2983 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (𝐵𝑍) → (∀𝑢𝑈 𝑢𝑦 ↔ ∀𝑢𝑈 𝑢 ≤ (𝐵𝑍)))
176175rspcev 3304 . . . . . . . . . . . . . . . . . 18 (((𝐵𝑍) ∈ ℝ ∧ ∀𝑢𝑈 𝑢 ≤ (𝐵𝑍)) → ∃𝑦 ∈ ℝ ∀𝑢𝑈 𝑢𝑦)
17748, 173, 176syl2anc 692 . . . . . . . . . . . . . . . . 17 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑢𝑈 𝑢𝑦)
178177adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑢𝑈) → ∃𝑦 ∈ ℝ ∀𝑢𝑈 𝑢𝑦)
179 simpr 477 . . . . . . . . . . . . . . . 16 ((𝜑𝑢𝑈) → 𝑢𝑈)
180 suprub 10969 . . . . . . . . . . . . . . . 16 (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑢𝑈 𝑢𝑦) ∧ 𝑢𝑈) → 𝑢 ≤ sup(𝑈, ℝ, < ))
181164, 166, 178, 179, 180syl31anc 1327 . . . . . . . . . . . . . . 15 ((𝜑𝑢𝑈) → 𝑢 ≤ sup(𝑈, ℝ, < ))
182181, 79syl6breqr 4686 . . . . . . . . . . . . . 14 ((𝜑𝑢𝑈) → 𝑢𝑆)
183182ralrimiva 2963 . . . . . . . . . . . . 13 (𝜑 → ∀𝑢𝑈 𝑢𝑆)
184164, 179sseldd 3596 . . . . . . . . . . . . . . 15 ((𝜑𝑢𝑈) → 𝑢 ∈ ℝ)
185101adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑢𝑈) → 𝑆 ∈ ℝ)
186184, 185lenltd 10168 . . . . . . . . . . . . . 14 ((𝜑𝑢𝑈) → (𝑢𝑆 ↔ ¬ 𝑆 < 𝑢))
187186ralbidva 2982 . . . . . . . . . . . . 13 (𝜑 → (∀𝑢𝑈 𝑢𝑆 ↔ ∀𝑢𝑈 ¬ 𝑆 < 𝑢))
188183, 187mpbid 222 . . . . . . . . . . . 12 (𝜑 → ∀𝑢𝑈 ¬ 𝑆 < 𝑢)
189 ralnex 2989 . . . . . . . . . . . 12 (∀𝑢𝑈 ¬ 𝑆 < 𝑢 ↔ ¬ ∃𝑢𝑈 𝑆 < 𝑢)
190188, 189sylib 208 . . . . . . . . . . 11 (𝜑 → ¬ ∃𝑢𝑈 𝑆 < 𝑢)
191190adantr 481 . . . . . . . . . 10 ((𝜑𝑆 < (𝐵𝑍)) → ¬ ∃𝑢𝑈 𝑆 < 𝑢)
19297, 105, 191syl2anc 692 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝐵𝑍) ≤ 𝑆) → ¬ ∃𝑢𝑈 𝑆 < 𝑢)
193161, 192condan 834 . . . . . . . 8 (𝜑 → (𝐵𝑍) ≤ 𝑆)
194 iccleub 12214 . . . . . . . . 9 (((𝐴𝑍) ∈ ℝ* ∧ (𝐵𝑍) ∈ ℝ*𝑆 ∈ ((𝐴𝑍)[,](𝐵𝑍))) → 𝑆 ≤ (𝐵𝑍))
195167, 91, 95, 194syl3anc 1324 . . . . . . . 8 (𝜑𝑆 ≤ (𝐵𝑍))
19691, 96, 193, 195xrletrid 11971 . . . . . . 7 (𝜑 → (𝐵𝑍) = 𝑆)
19778eqcomi 2629 . . . . . . . 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 2693 . . . . . 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 6642 . . . . . . . 8 (𝑧 = (𝐵𝑍) → (𝑧 − (𝐴𝑍)) = ((𝐵𝑍) − (𝐴𝑍)))
202201oveq2d 6651 . . . . . . 7 (𝑧 = (𝐵𝑍) → (𝐺 · (𝑧 − (𝐴𝑍))) = (𝐺 · ((𝐵𝑍) − (𝐴𝑍))))
203 fveq2 6178 . . . . . . . . . . . 12 (𝑧 = (𝐵𝑍) → (𝐻𝑧) = (𝐻‘(𝐵𝑍)))
204203fveq1d 6180 . . . . . . . . . . 11 (𝑧 = (𝐵𝑍) → ((𝐻𝑧)‘(𝐷𝑗)) = ((𝐻‘(𝐵𝑍))‘(𝐷𝑗)))
205204oveq2d 6651 . . . . . . . . . 10 (𝑧 = (𝐵𝑍) → ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))) = ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))))
206205mpteq2dv 4736 . . . . . . . . 9 (𝑧 = (𝐵𝑍) → (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗)))) = (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗)))))
207206fveq2d 6182 . . . . . . . 8 (𝑧 = (𝐵𝑍) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗))))))
208207oveq2d 6651 . . . . . . 7 (𝑧 = (𝐵𝑍) → ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗)))))) = ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗)))))))
209202, 208breq12d 4657 . . . . . 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 3357 . . . . 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 479 . . 3 (𝜑 → (𝐺 · ((𝐵𝑍) − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻‘(𝐵𝑍))‘(𝐷𝑗)))))))
2133, 5ssfid 8168 . . . . . 6 (𝜑𝑌 ∈ Fin)
214 eqid 2620 . . . . . 6 𝑘𝑌 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = ∏𝑘𝑌 (vol‘((𝐴𝑘)[,)(𝐵𝑘)))
2152, 213, 6, 65, 4, 14, 15, 214hoiprodp1 40565 . . . . 5 (𝜑 → (𝐴(𝐿𝑊)𝐵) = (∏𝑘𝑌 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) · (vol‘((𝐴𝑍)[,)(𝐵𝑍)))))
216 eqidd 2621 . . . . . . 7 (𝜑 → ∏𝑘𝑌 ((𝐵𝑘) − (𝐴𝑘)) = ∏𝑘𝑌 ((𝐵𝑘) − (𝐴𝑘)))
21714adantr 481 . . . . . . . . . 10 ((𝜑𝑘𝑌) → 𝐴:𝑊⟶ℝ)
218 ssun1 3768 . . . . . . . . . . . 12 𝑌 ⊆ (𝑌 ∪ {𝑍})
2194eqcomi 2629 . . . . . . . . . . . 12 (𝑌 ∪ {𝑍}) = 𝑊
220218, 219sseqtri 3629 . . . . . . . . . . 11 𝑌𝑊
221 simpr 477 . . . . . . . . . . 11 ((𝜑𝑘𝑌) → 𝑘𝑌)
222220, 221sseldi 3593 . . . . . . . . . 10 ((𝜑𝑘𝑌) → 𝑘𝑊)
223217, 222ffvelrnd 6346 . . . . . . . . 9 ((𝜑𝑘𝑌) → (𝐴𝑘) ∈ ℝ)
22415adantr 481 . . . . . . . . . 10 ((𝜑𝑘𝑌) → 𝐵:𝑊⟶ℝ)
225224, 222ffvelrnd 6346 . . . . . . . . 9 ((𝜑𝑘𝑌) → (𝐵𝑘) ∈ ℝ)
226222, 87syldan 487 . . . . . . . . 9 ((𝜑𝑘𝑌) → (𝐴𝑘) < (𝐵𝑘))
227223, 225, 226volicon0 40552 . . . . . . . 8 ((𝜑𝑘𝑌) → (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = ((𝐵𝑘) − (𝐴𝑘)))
228227prodeq2dv 14634 . . . . . . 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 6058 . . . . . . . . 9 (𝜑 → (𝐴𝑌):𝑌⟶ℝ)
23315, 231fssresd 6058 . . . . . . . . 9 (𝜑 → (𝐵𝑌):𝑌⟶ℝ)
2342, 213, 230, 232, 233hoidmvn0val 40561 . . . . . . . 8 (𝜑 → ((𝐴𝑌)(𝐿𝑌)(𝐵𝑌)) = ∏𝑘𝑌 (vol‘(((𝐴𝑌)‘𝑘)[,)((𝐵𝑌)‘𝑘))))
235 fvres 6194 . . . . . . . . . . . . 13 (𝑘𝑌 → ((𝐴𝑌)‘𝑘) = (𝐴𝑘))
236 fvres 6194 . . . . . . . . . . . . 13 (𝑘𝑌 → ((𝐵𝑌)‘𝑘) = (𝐵𝑘))
237235, 236oveq12d 6653 . . . . . . . . . . . 12 (𝑘𝑌 → (((𝐴𝑌)‘𝑘)[,)((𝐵𝑌)‘𝑘)) = ((𝐴𝑘)[,)(𝐵𝑘)))
238237fveq2d 6182 . . . . . . . . . . 11 (𝑘𝑌 → (vol‘(((𝐴𝑌)‘𝑘)[,)((𝐵𝑌)‘𝑘))) = (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
239238adantl 482 . . . . . . . . . 10 ((𝜑𝑘𝑌) → (vol‘(((𝐴𝑌)‘𝑘)[,)((𝐵𝑌)‘𝑘))) = (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
240 volico 39963 . . . . . . . . . . 11 (((𝐴𝑘) ∈ ℝ ∧ (𝐵𝑘) ∈ ℝ) → (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = if((𝐴𝑘) < (𝐵𝑘), ((𝐵𝑘) − (𝐴𝑘)), 0))
241223, 225, 240syl2anc 692 . . . . . . . . . 10 ((𝜑𝑘𝑌) → (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = if((𝐴𝑘) < (𝐵𝑘), ((𝐵𝑘) − (𝐴𝑘)), 0))
242241, 227eqtr3d 2656 . . . . . . . . . 10 ((𝜑𝑘𝑌) → if((𝐴𝑘) < (𝐵𝑘), ((𝐵𝑘) − (𝐴𝑘)), 0) = ((𝐵𝑘) − (𝐴𝑘)))
243239, 241, 2423eqtrd 2658 . . . . . . . . 9 ((𝜑𝑘𝑌) → (vol‘(((𝐴𝑌)‘𝑘)[,)((𝐵𝑌)‘𝑘))) = ((𝐵𝑘) − (𝐴𝑘)))
244243prodeq2dv 14634 . . . . . . . 8 (𝜑 → ∏𝑘𝑌 (vol‘(((𝐴𝑌)‘𝑘)[,)((𝐵𝑌)‘𝑘))) = ∏𝑘𝑌 ((𝐵𝑘) − (𝐴𝑘)))
245229, 234, 2443eqtrd 2658 . . . . . . 7 (𝜑𝐺 = ∏𝑘𝑌 ((𝐵𝑘) − (𝐴𝑘)))
246216, 228, 2453eqtr4d 2664 . . . . . 6 (𝜑 → ∏𝑘𝑌 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) = 𝐺)
24799, 48, 89volicon0 40552 . . . . . 6 (𝜑 → (vol‘((𝐴𝑍)[,)(𝐵𝑍))) = ((𝐵𝑍) − (𝐴𝑍)))
248246, 247oveq12d 6653 . . . . 5 (𝜑 → (∏𝑘𝑌 (vol‘((𝐴𝑘)[,)(𝐵𝑘))) · (vol‘((𝐴𝑍)[,)(𝐵𝑍)))) = (𝐺 · ((𝐵𝑍) − (𝐴𝑍))))
249215, 248eqtrd 2654 . . . 4 (𝜑 → (𝐴(𝐿𝑊)𝐵) = (𝐺 · ((𝐵𝑍) − (𝐴𝑍))))
250249breq1d 4654 . . 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 10536 . . . . 5 0 ≤ 1
253252a1i 11 . . . 4 (𝜑 → 0 ≤ 1)
25419rpge0d 11861 . . . 4 (𝜑 → 0 ≤ 𝐸)
25518, 20, 253, 254addge0d 10588 . . 3 (𝜑 → 0 ≤ (1 + 𝐸))
25674, 61, 21, 255, 69lemul2ad 10949 . 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 10179 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 384   = wceq 1481  wcel 1988  wne 2791  wral 2909  wrex 2910  {crab 2913  Vcvv 3195  cdif 3564  cun 3565  wss 3567  c0 3907  ifcif 4077  {csn 4168   ciun 4511   class class class wbr 4644  cmpt 4720  cres 5106  wf 5872  cfv 5876  (class class class)co 6635  cmpt2 6637  𝑚 cmap 7842  Xcixp 7893  Fincfn 7940  supcsup 8331  cr 9920  0cc0 9921  1c1 9922   + caddc 9924   · cmul 9926  +∞cpnf 10056  *cxr 10058   < clt 10059  cle 10060  cmin 10251  cn 11005  +crp 11817  [,)cico 12162  [,]cicc 12163  cprod 14616  volcvol 23213  Σ^csumge0 40342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998  ax-pre-sup 9999
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-of 6882  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-2o 7546  df-oadd 7549  df-er 7727  df-map 7844  df-pm 7845  df-ixp 7894  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-fi 8302  df-sup 8333  df-inf 8334  df-oi 8400  df-card 8750  df-cda 8975  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-div 10670  df-nn 11006  df-2 11064  df-3 11065  df-n0 11278  df-z 11363  df-uz 11673  df-q 11774  df-rp 11818  df-xneg 11931  df-xadd 11932  df-xmul 11933  df-ioo 12164  df-ico 12166  df-icc 12167  df-fz 12312  df-fzo 12450  df-fl 12576  df-seq 12785  df-exp 12844  df-hash 13101  df-cj 13820  df-re 13821  df-im 13822  df-sqrt 13956  df-abs 13957  df-clim 14200  df-rlim 14201  df-sum 14398  df-prod 14617  df-rest 16064  df-topgen 16085  df-psmet 19719  df-xmet 19720  df-met 19721  df-bl 19722  df-mopn 19723  df-top 20680  df-topon 20697  df-bases 20731  df-cmp 21171  df-ovol 23214  df-vol 23215  df-sumge0 40343
This theorem is referenced by:  hoidmvlelem5  40576
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