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Theorem hoidmvlelem5 43225
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. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
Hypotheses
Ref Expression
hoidmvlelem5.l 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))
hoidmvlelem5.f (𝜑𝑋 ∈ Fin)
hoidmvlelem5.y (𝜑𝑌𝑋)
hoidmvlelem5.z (𝜑𝑍 ∈ (𝑋𝑌))
hoidmvlelem5.w 𝑊 = (𝑌 ∪ {𝑍})
hoidmvlelem5.a (𝜑𝐴:𝑊⟶ℝ)
hoidmvlelem5.b (𝜑𝐵:𝑊⟶ℝ)
hoidmvlelem5.c (𝜑𝐶:ℕ⟶(ℝ ↑m 𝑊))
hoidmvlelem5.d (𝜑𝐷:ℕ⟶(ℝ ↑m 𝑊))
hoidmvlelem5.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))))))
hoidmvlelem5.s (𝜑X𝑘𝑊 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑊 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))
hoidmvlelem5.n (𝜑𝑌 ≠ ∅)
Assertion
Ref Expression
hoidmvlelem5 (𝜑 → (𝐴(𝐿𝑊)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
Distinct variable groups:   𝐴,𝑎,𝑏,,𝑗,𝑘,𝑥   𝐴,𝑒,𝑓,𝑔,,𝑗,𝑘   𝐵,𝑎,𝑏,,𝑗,𝑘,𝑥   𝐵,𝑓,𝑔   𝐶,𝑎,𝑏,,𝑗,𝑘,𝑥   𝐶,𝑔   𝐷,𝑎,𝑏,,𝑗,𝑘,𝑥   𝐷,𝑔   𝐿,𝑎,𝑏,,𝑗,𝑘,𝑥   𝑒,𝐿,𝑓,𝑔   𝑊,𝑎,𝑏,,𝑗,𝑘,𝑥   𝑔,𝑊   𝑌,𝑎,𝑏,,𝑗,𝑘,𝑥   𝑒,𝑌,𝑓,𝑔   𝑍,𝑎,𝑏,,𝑗,𝑘,𝑥   𝑔,𝑍   𝜑,𝑎,𝑏,,𝑗,𝑘,𝑥
Allowed substitution hints:   𝜑(𝑒,𝑓,𝑔)   𝐵(𝑒)   𝐶(𝑒,𝑓)   𝐷(𝑒,𝑓)   𝑊(𝑒,𝑓)   𝑋(𝑥,𝑒,𝑓,𝑔,,𝑗,𝑘,𝑎,𝑏)   𝑍(𝑒,𝑓)

Proof of Theorem hoidmvlelem5
Dummy variables 𝑟 𝑠 𝑐 𝑤 𝑧 𝑖 𝑙 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1915 . . . . 5 𝑠𝜑
2 nfre1 3268 . . . . 5 𝑠𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)
31, 2nfan 1900 . . . 4 𝑠(𝜑 ∧ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠))
4 hoidmvlelem5.l . . . 4 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))
5 hoidmvlelem5.w . . . . . 6 𝑊 = (𝑌 ∪ {𝑍})
6 hoidmvlelem5.f . . . . . . . 8 (𝜑𝑋 ∈ Fin)
7 hoidmvlelem5.y . . . . . . . 8 (𝜑𝑌𝑋)
8 ssfi 8726 . . . . . . . 8 ((𝑋 ∈ Fin ∧ 𝑌𝑋) → 𝑌 ∈ Fin)
96, 7, 8syl2anc 587 . . . . . . 7 (𝜑𝑌 ∈ Fin)
10 snfi 8581 . . . . . . . 8 {𝑍} ∈ Fin
1110a1i 11 . . . . . . 7 (𝜑 → {𝑍} ∈ Fin)
12 unfi 8773 . . . . . . 7 ((𝑌 ∈ Fin ∧ {𝑍} ∈ Fin) → (𝑌 ∪ {𝑍}) ∈ Fin)
139, 11, 12syl2anc 587 . . . . . 6 (𝜑 → (𝑌 ∪ {𝑍}) ∈ Fin)
145, 13eqeltrid 2897 . . . . 5 (𝜑𝑊 ∈ Fin)
1514adantr 484 . . . 4 ((𝜑 ∧ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → 𝑊 ∈ Fin)
16 hoidmvlelem5.a . . . . 5 (𝜑𝐴:𝑊⟶ℝ)
1716adantr 484 . . . 4 ((𝜑 ∧ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → 𝐴:𝑊⟶ℝ)
18 hoidmvlelem5.b . . . . 5 (𝜑𝐵:𝑊⟶ℝ)
1918adantr 484 . . . 4 ((𝜑 ∧ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → 𝐵:𝑊⟶ℝ)
20 simpr 488 . . . 4 ((𝜑 ∧ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠))
213, 4, 15, 17, 19, 20hoidmvval0 43213 . . 3 ((𝜑 ∧ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → (𝐴(𝐿𝑊)𝐵) = 0)
22 nnex 11635 . . . . . 6 ℕ ∈ V
2322a1i 11 . . . . 5 (𝜑 → ℕ ∈ V)
24 icossicc 12818 . . . . . . 7 (0[,)+∞) ⊆ (0[,]+∞)
2514adantr 484 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → 𝑊 ∈ Fin)
26 hoidmvlelem5.c . . . . . . . . . 10 (𝜑𝐶:ℕ⟶(ℝ ↑m 𝑊))
2726ffvelrnda 6832 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ∈ (ℝ ↑m 𝑊))
28 elmapi 8415 . . . . . . . . 9 ((𝐶𝑗) ∈ (ℝ ↑m 𝑊) → (𝐶𝑗):𝑊⟶ℝ)
2927, 28syl 17 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗):𝑊⟶ℝ)
30 hoidmvlelem5.d . . . . . . . . . 10 (𝜑𝐷:ℕ⟶(ℝ ↑m 𝑊))
3130ffvelrnda 6832 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗) ∈ (ℝ ↑m 𝑊))
32 elmapi 8415 . . . . . . . . 9 ((𝐷𝑗) ∈ (ℝ ↑m 𝑊) → (𝐷𝑗):𝑊⟶ℝ)
3331, 32syl 17 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗):𝑊⟶ℝ)
344, 25, 29, 33hoidmvcl 43208 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)) ∈ (0[,)+∞))
3524, 34sseldi 3916 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)) ∈ (0[,]+∞))
3635fmpttd 6860 . . . . 5 (𝜑 → (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗))):ℕ⟶(0[,]+∞))
3723, 36sge0ge0 43010 . . . 4 (𝜑 → 0 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
3837adantr 484 . . 3 ((𝜑 ∧ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → 0 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
3921, 38eqbrtrd 5055 . 2 ((𝜑 ∧ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → (𝐴(𝐿𝑊)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
40 icossxr 12814 . . . . . . 7 (0[,)+∞) ⊆ ℝ*
414, 14, 16, 18hoidmvcl 43208 . . . . . . 7 (𝜑 → (𝐴(𝐿𝑊)𝐵) ∈ (0[,)+∞))
4240, 41sseldi 3916 . . . . . 6 (𝜑 → (𝐴(𝐿𝑊)𝐵) ∈ ℝ*)
4342adantr 484 . . . . 5 ((𝜑 ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (𝐴(𝐿𝑊)𝐵) ∈ ℝ*)
4423, 36sge0xrcl 43011 . . . . . 6 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ*)
4544adantr 484 . . . . 5 ((𝜑 ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ*)
46 rge0ssre 12838 . . . . . . . . 9 (0[,)+∞) ⊆ ℝ
4746, 41sseldi 3916 . . . . . . . 8 (𝜑 → (𝐴(𝐿𝑊)𝐵) ∈ ℝ)
48 ltpnf 12507 . . . . . . . 8 ((𝐴(𝐿𝑊)𝐵) ∈ ℝ → (𝐴(𝐿𝑊)𝐵) < +∞)
4947, 48syl 17 . . . . . . 7 (𝜑 → (𝐴(𝐿𝑊)𝐵) < +∞)
5049adantr 484 . . . . . 6 ((𝜑 ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (𝐴(𝐿𝑊)𝐵) < +∞)
51 id 22 . . . . . . . 8 ((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞ → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞)
5251eqcomd 2807 . . . . . . 7 ((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞ → +∞ = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
5352adantl 485 . . . . . 6 ((𝜑 ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → +∞ = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
5450, 53breqtrd 5059 . . . . 5 ((𝜑 ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (𝐴(𝐿𝑊)𝐵) < (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
5543, 45, 54xrltled 12535 . . . 4 ((𝜑 ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (𝐴(𝐿𝑊)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
5655adantlr 714 . . 3 (((𝜑 ∧ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (𝐴(𝐿𝑊)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
57 simpll 766 . . . 4 (((𝜑 ∧ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → 𝜑)
58 simpr 488 . . . . . 6 ((𝜑 ∧ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠))
5916ffvelrnda 6832 . . . . . . . . . 10 ((𝜑𝑠𝑊) → (𝐴𝑠) ∈ ℝ)
6018ffvelrnda 6832 . . . . . . . . . 10 ((𝜑𝑠𝑊) → (𝐵𝑠) ∈ ℝ)
6159, 60ltnled 10780 . . . . . . . . 9 ((𝜑𝑠𝑊) → ((𝐴𝑠) < (𝐵𝑠) ↔ ¬ (𝐵𝑠) ≤ (𝐴𝑠)))
6261ralbidva 3164 . . . . . . . 8 (𝜑 → (∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠) ↔ ∀𝑠𝑊 ¬ (𝐵𝑠) ≤ (𝐴𝑠)))
63 ralnex 3202 . . . . . . . . 9 (∀𝑠𝑊 ¬ (𝐵𝑠) ≤ (𝐴𝑠) ↔ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠))
6463a1i 11 . . . . . . . 8 (𝜑 → (∀𝑠𝑊 ¬ (𝐵𝑠) ≤ (𝐴𝑠) ↔ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)))
6562, 64bitrd 282 . . . . . . 7 (𝜑 → (∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠) ↔ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)))
6665adantr 484 . . . . . 6 ((𝜑 ∧ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → (∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠) ↔ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)))
6758, 66mpbird 260 . . . . 5 ((𝜑 ∧ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠))
6867adantr 484 . . . 4 (((𝜑 ∧ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠))
69 simpr 488 . . . . . 6 ((𝜑 ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞)
7022a1i 11 . . . . . . 7 ((𝜑 ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → ℕ ∈ V)
7136adantr 484 . . . . . . 7 ((𝜑 ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗))):ℕ⟶(0[,]+∞))
7270, 71sge0repnf 43012 . . . . . 6 ((𝜑 ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → ((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ ↔ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞))
7369, 72mpbird 260 . . . . 5 ((𝜑 ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)
7473adantlr 714 . . . 4 (((𝜑 ∧ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)
75 simpll 766 . . . . . . 7 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → (𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)))
76 fveq2 6649 . . . . . . . . . . . . 13 (𝑗 = 𝑖 → (𝐶𝑗) = (𝐶𝑖))
77 fveq2 6649 . . . . . . . . . . . . 13 (𝑗 = 𝑖 → (𝐷𝑗) = (𝐷𝑖))
7876, 77oveq12d 7157 . . . . . . . . . . . 12 (𝑗 = 𝑖 → ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)) = ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))
7978cbvmptv 5136 . . . . . . . . . . 11 (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗))) = (𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))
8079fveq2i 6652 . . . . . . . . . 10 ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖))))
8180eleq1i 2883 . . . . . . . . 9 ((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ ↔ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ)
8281biimpi 219 . . . . . . . 8 ((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ → (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ)
8382ad2antlr 726 . . . . . . 7 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ)
84 simpr 488 . . . . . . 7 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝑟 ∈ ℝ+)
856ad3antrrr 729 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝑋 ∈ Fin)
867ad3antrrr 729 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝑌𝑋)
87 hoidmvlelem5.n . . . . . . . . 9 (𝜑𝑌 ≠ ∅)
8887ad3antrrr 729 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝑌 ≠ ∅)
89 hoidmvlelem5.z . . . . . . . . 9 (𝜑𝑍 ∈ (𝑋𝑌))
9089ad3antrrr 729 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝑍 ∈ (𝑋𝑌))
9116ad3antrrr 729 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝐴:𝑊⟶ℝ)
9218ad3antrrr 729 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝐵:𝑊⟶ℝ)
93 fveq2 6649 . . . . . . . . . . . . . 14 (𝑠 = 𝑘 → (𝐴𝑠) = (𝐴𝑘))
94 fveq2 6649 . . . . . . . . . . . . . 14 (𝑠 = 𝑘 → (𝐵𝑠) = (𝐵𝑘))
9593, 94breq12d 5046 . . . . . . . . . . . . 13 (𝑠 = 𝑘 → ((𝐴𝑠) < (𝐵𝑠) ↔ (𝐴𝑘) < (𝐵𝑘)))
9695cbvralvw 3399 . . . . . . . . . . . 12 (∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠) ↔ ∀𝑘𝑊 (𝐴𝑘) < (𝐵𝑘))
9796biimpi 219 . . . . . . . . . . 11 (∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠) → ∀𝑘𝑊 (𝐴𝑘) < (𝐵𝑘))
9897adantr 484 . . . . . . . . . 10 ((∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠) ∧ 𝑘𝑊) → ∀𝑘𝑊 (𝐴𝑘) < (𝐵𝑘))
99 simpr 488 . . . . . . . . . 10 ((∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠) ∧ 𝑘𝑊) → 𝑘𝑊)
100 rspa 3174 . . . . . . . . . 10 ((∀𝑘𝑊 (𝐴𝑘) < (𝐵𝑘) ∧ 𝑘𝑊) → (𝐴𝑘) < (𝐵𝑘))
10198, 99, 100syl2anc 587 . . . . . . . . 9 ((∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠) ∧ 𝑘𝑊) → (𝐴𝑘) < (𝐵𝑘))
102101ad5ant25 761 . . . . . . . 8 (((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) ∧ 𝑘𝑊) → (𝐴𝑘) < (𝐵𝑘))
10326ad3antrrr 729 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝐶:ℕ⟶(ℝ ↑m 𝑊))
10430ad3antrrr 729 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝐷:ℕ⟶(ℝ ↑m 𝑊))
10581biimpri 231 . . . . . . . . 9 ((Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)
106105ad2antlr 726 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)
107 fveq1 6648 . . . . . . . . . . . . 13 (𝑑 = 𝑐 → (𝑑𝑖) = (𝑐𝑖))
108107breq1d 5043 . . . . . . . . . . . . . 14 (𝑑 = 𝑐 → ((𝑑𝑖) ≤ 𝑥 ↔ (𝑐𝑖) ≤ 𝑥))
109108, 107ifbieq1d 4451 . . . . . . . . . . . . 13 (𝑑 = 𝑐 → if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥) = if((𝑐𝑖) ≤ 𝑥, (𝑐𝑖), 𝑥))
110107, 109ifeq12d 4448 . . . . . . . . . . . 12 (𝑑 = 𝑐 → if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥)) = if(𝑖𝑌, (𝑐𝑖), if((𝑐𝑖) ≤ 𝑥, (𝑐𝑖), 𝑥)))
111110mpteq2dv 5129 . . . . . . . . . . 11 (𝑑 = 𝑐 → (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥))) = (𝑖𝑊 ↦ if(𝑖𝑌, (𝑐𝑖), if((𝑐𝑖) ≤ 𝑥, (𝑐𝑖), 𝑥))))
112 eleq1w 2875 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → (𝑖𝑌𝑗𝑌))
113 fveq2 6649 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → (𝑐𝑖) = (𝑐𝑗))
114113breq1d 5043 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → ((𝑐𝑖) ≤ 𝑥 ↔ (𝑐𝑗) ≤ 𝑥))
115114, 113ifbieq1d 4451 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → if((𝑐𝑖) ≤ 𝑥, (𝑐𝑖), 𝑥) = if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥))
116112, 113, 115ifbieq12d 4455 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → if(𝑖𝑌, (𝑐𝑖), if((𝑐𝑖) ≤ 𝑥, (𝑐𝑖), 𝑥)) = if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)))
117116cbvmptv 5136 . . . . . . . . . . . 12 (𝑖𝑊 ↦ if(𝑖𝑌, (𝑐𝑖), if((𝑐𝑖) ≤ 𝑥, (𝑐𝑖), 𝑥))) = (𝑗𝑊 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)))
118117a1i 11 . . . . . . . . . . 11 (𝑑 = 𝑐 → (𝑖𝑊 ↦ if(𝑖𝑌, (𝑐𝑖), if((𝑐𝑖) ≤ 𝑥, (𝑐𝑖), 𝑥))) = (𝑗𝑊 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥))))
119111, 118eqtrd 2836 . . . . . . . . . 10 (𝑑 = 𝑐 → (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥))) = (𝑗𝑊 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥))))
120119cbvmptv 5136 . . . . . . . . 9 (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥)))) = (𝑐 ∈ (ℝ ↑m 𝑊) ↦ (𝑗𝑊 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥))))
121120mpteq2i 5125 . . . . . . . 8 (𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥))))) = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑m 𝑊) ↦ (𝑗𝑊 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)))))
122 eqid 2801 . . . . . . . 8 ((𝐴𝑌)(𝐿𝑌)(𝐵𝑌)) = ((𝐴𝑌)(𝐿𝑌)(𝐵𝑌))
123 simpr 488 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝑟 ∈ ℝ+)
124 oveq1 7146 . . . . . . . . . . 11 (𝑤 = 𝑧 → (𝑤 − (𝐴𝑍)) = (𝑧 − (𝐴𝑍)))
125124oveq2d 7155 . . . . . . . . . 10 (𝑤 = 𝑧 → (((𝐴𝑌)(𝐿𝑌)(𝐵𝑌)) · (𝑤 − (𝐴𝑍))) = (((𝐴𝑌)(𝐿𝑌)(𝐵𝑌)) · (𝑧 − (𝐴𝑍))))
126 breq2 5037 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = 𝑥 → ((𝑑𝑖) ≤ 𝑤 ↔ (𝑑𝑖) ≤ 𝑥))
127 eqidd 2802 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = 𝑥 → (𝑑𝑖) = (𝑑𝑖))
128 id 22 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = 𝑥𝑤 = 𝑥)
129126, 127, 128ifbieq12d 4455 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = 𝑥 → if((𝑑𝑖) ≤ 𝑤, (𝑑𝑖), 𝑤) = if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥))
130129ifeq2d 4447 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑥 → if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑤, (𝑑𝑖), 𝑤)) = if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥)))
131130mpteq2dv 5129 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑥 → (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑤, (𝑑𝑖), 𝑤))) = (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥))))
132131mpteq2dv 5129 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑥 → (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑤, (𝑑𝑖), 𝑤)))) = (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥)))))
133132cbvmptv 5136 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑤, (𝑑𝑖), 𝑤))))) = (𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥)))))
134133a1i 11 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑧 → (𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑤, (𝑑𝑖), 𝑤))))) = (𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥))))))
135 id 22 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑧𝑤 = 𝑧)
136134, 135fveq12d 6656 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑧 → ((𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑤, (𝑑𝑖), 𝑤)))))‘𝑤) = ((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥)))))‘𝑧))
137136fveq1d 6651 . . . . . . . . . . . . . . 15 (𝑤 = 𝑧 → (((𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑤, (𝑑𝑖), 𝑤)))))‘𝑤)‘(𝐷𝑙)) = (((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥)))))‘𝑧)‘(𝐷𝑙)))
138137oveq2d 7155 . . . . . . . . . . . . . 14 (𝑤 = 𝑧 → ((𝐶𝑙)(𝐿𝑊)(((𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑤, (𝑑𝑖), 𝑤)))))‘𝑤)‘(𝐷𝑙))) = ((𝐶𝑙)(𝐿𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥)))))‘𝑧)‘(𝐷𝑙))))
139138mpteq2dv 5129 . . . . . . . . . . . . 13 (𝑤 = 𝑧 → (𝑙 ∈ ℕ ↦ ((𝐶𝑙)(𝐿𝑊)(((𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑤, (𝑑𝑖), 𝑤)))))‘𝑤)‘(𝐷𝑙)))) = (𝑙 ∈ ℕ ↦ ((𝐶𝑙)(𝐿𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥)))))‘𝑧)‘(𝐷𝑙)))))
140 fveq2 6649 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑗 → (𝐶𝑙) = (𝐶𝑗))
141 2fveq3 6654 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑗 → (((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥)))))‘𝑧)‘(𝐷𝑙)) = (((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥)))))‘𝑧)‘(𝐷𝑗)))
142140, 141oveq12d 7157 . . . . . . . . . . . . . . 15 (𝑙 = 𝑗 → ((𝐶𝑙)(𝐿𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥)))))‘𝑧)‘(𝐷𝑙))) = ((𝐶𝑗)(𝐿𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥)))))‘𝑧)‘(𝐷𝑗))))
143142cbvmptv 5136 . . . . . . . . . . . . . 14 (𝑙 ∈ ℕ ↦ ((𝐶𝑙)(𝐿𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥)))))‘𝑧)‘(𝐷𝑙)))) = (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥)))))‘𝑧)‘(𝐷𝑗))))
144143a1i 11 . . . . . . . . . . . . 13 (𝑤 = 𝑧 → (𝑙 ∈ ℕ ↦ ((𝐶𝑙)(𝐿𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥)))))‘𝑧)‘(𝐷𝑙)))) = (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥)))))‘𝑧)‘(𝐷𝑗)))))
145139, 144eqtrd 2836 . . . . . . . . . . . 12 (𝑤 = 𝑧 → (𝑙 ∈ ℕ ↦ ((𝐶𝑙)(𝐿𝑊)(((𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑤, (𝑑𝑖), 𝑤)))))‘𝑤)‘(𝐷𝑙)))) = (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥)))))‘𝑧)‘(𝐷𝑗)))))
146145fveq2d 6653 . . . . . . . . . . 11 (𝑤 = 𝑧 → (Σ^‘(𝑙 ∈ ℕ ↦ ((𝐶𝑙)(𝐿𝑊)(((𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑤, (𝑑𝑖), 𝑤)))))‘𝑤)‘(𝐷𝑙))))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥)))))‘𝑧)‘(𝐷𝑗))))))
147146oveq2d 7155 . . . . . . . . . 10 (𝑤 = 𝑧 → ((1 + 𝑟) · (Σ^‘(𝑙 ∈ ℕ ↦ ((𝐶𝑙)(𝐿𝑊)(((𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑤, (𝑑𝑖), 𝑤)))))‘𝑤)‘(𝐷𝑙)))))) = ((1 + 𝑟) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥)))))‘𝑧)‘(𝐷𝑗)))))))
148125, 147breq12d 5046 . . . . . . . . 9 (𝑤 = 𝑧 → ((((𝐴𝑌)(𝐿𝑌)(𝐵𝑌)) · (𝑤 − (𝐴𝑍))) ≤ ((1 + 𝑟) · (Σ^‘(𝑙 ∈ ℕ ↦ ((𝐶𝑙)(𝐿𝑊)(((𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑤, (𝑑𝑖), 𝑤)))))‘𝑤)‘(𝐷𝑙)))))) ↔ (((𝐴𝑌)(𝐿𝑌)(𝐵𝑌)) · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝑟) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥)))))‘𝑧)‘(𝐷𝑗))))))))
149148cbvrabv 3442 . . . . . . . 8 {𝑤 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (((𝐴𝑌)(𝐿𝑌)(𝐵𝑌)) · (𝑤 − (𝐴𝑍))) ≤ ((1 + 𝑟) · (Σ^‘(𝑙 ∈ ℕ ↦ ((𝐶𝑙)(𝐿𝑊)(((𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑤, (𝑑𝑖), 𝑤)))))‘𝑤)‘(𝐷𝑙))))))} = {𝑧 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (((𝐴𝑌)(𝐿𝑌)(𝐵𝑌)) · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝑟) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥)))))‘𝑧)‘(𝐷𝑗))))))}
150 eqid 2801 . . . . . . . 8 sup({𝑤 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (((𝐴𝑌)(𝐿𝑌)(𝐵𝑌)) · (𝑤 − (𝐴𝑍))) ≤ ((1 + 𝑟) · (Σ^‘(𝑙 ∈ ℕ ↦ ((𝐶𝑙)(𝐿𝑊)(((𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑤, (𝑑𝑖), 𝑤)))))‘𝑤)‘(𝐷𝑙))))))}, ℝ, < ) = sup({𝑤 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (((𝐴𝑌)(𝐿𝑌)(𝐵𝑌)) · (𝑤 − (𝐴𝑍))) ≤ ((1 + 𝑟) · (Σ^‘(𝑙 ∈ ℕ ↦ ((𝐶𝑙)(𝐿𝑊)(((𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑤, (𝑑𝑖), 𝑤)))))‘𝑤)‘(𝐷𝑙))))))}, ℝ, < )
151 hoidmvlelem5.i . . . . . . . . 9 (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))))))
152151ad3antrrr 729 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → ∀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))))))
153 hoidmvlelem5.s . . . . . . . . 9 (𝜑X𝑘𝑊 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑊 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))
154153ad3antrrr 729 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → X𝑘𝑊 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑊 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))
1554, 85, 86, 88, 90, 5, 91, 92, 102, 103, 104, 106, 121, 122, 123, 149, 150, 152, 154hoidmvlelem4 43224 . . . . . . 7 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → (𝐴(𝐿𝑊)𝐵) ≤ ((1 + 𝑟) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗))))))
15675, 83, 84, 155syl21anc 836 . . . . . 6 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → (𝐴(𝐿𝑊)𝐵) ≤ ((1 + 𝑟) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗))))))
157156ralrimiva 3152 . . . . 5 (((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) → ∀𝑟 ∈ ℝ+ (𝐴(𝐿𝑊)𝐵) ≤ ((1 + 𝑟) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗))))))
158 nfv 1915 . . . . . 6 𝑟((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)
15942ad2antrr 725 . . . . . 6 (((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) → (𝐴(𝐿𝑊)𝐵) ∈ ℝ*)
160 0xr 10681 . . . . . . . 8 0 ∈ ℝ*
161160a1i 11 . . . . . . 7 (((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) → 0 ∈ ℝ*)
162 pnfxr 10688 . . . . . . . 8 +∞ ∈ ℝ*
163162a1i 11 . . . . . . 7 (((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) → +∞ ∈ ℝ*)
16444ad2antrr 725 . . . . . . 7 (((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ*)
16537ad2antrr 725 . . . . . . 7 (((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) → 0 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
166 ltpnf 12507 . . . . . . . 8 ((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) < +∞)
167166adantl 485 . . . . . . 7 (((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) < +∞)
168161, 163, 164, 165, 167elicod 12779 . . . . . 6 (((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ (0[,)+∞))
169158, 159, 168xralrple2 41973 . . . . 5 (((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) → ((𝐴(𝐿𝑊)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ↔ ∀𝑟 ∈ ℝ+ (𝐴(𝐿𝑊)𝐵) ≤ ((1 + 𝑟) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))))
170157, 169mpbird 260 . . . 4 (((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) → (𝐴(𝐿𝑊)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
17157, 68, 74, 170syl21anc 836 . . 3 (((𝜑 ∧ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (𝐴(𝐿𝑊)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
17256, 171pm2.61dan 812 . 2 ((𝜑 ∧ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → (𝐴(𝐿𝑊)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
17339, 172pm2.61dan 812 1 (𝜑 → (𝐴(𝐿𝑊)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  wne 2990  wral 3109  wrex 3110  {crab 3113  Vcvv 3444  cdif 3881  cun 3882  wss 3884  c0 4246  ifcif 4428  {csn 4528   ciun 4884   class class class wbr 5033  cmpt 5113  cres 5525  wf 6324  cfv 6328  (class class class)co 7139  cmpo 7141  m cmap 8393  Xcixp 8448  Fincfn 8496  supcsup 8892  cr 10529  0cc0 10530  1c1 10531   + caddc 10533   · cmul 10535  +∞cpnf 10665  *cxr 10667   < clt 10668  cle 10669  cmin 10863  cn 11629  +crp 12381  [,)cico 12732  [,]cicc 12733  cprod 15254  volcvol 24070  Σ^csumge0 42988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-ixp 8449  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fi 8863  df-sup 8894  df-inf 8895  df-oi 8962  df-dju 9318  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-q 12341  df-rp 12382  df-xneg 12499  df-xadd 12500  df-xmul 12501  df-ioo 12734  df-ico 12736  df-icc 12737  df-fz 12890  df-fzo 13033  df-fl 13161  df-seq 13369  df-exp 13430  df-hash 13691  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-clim 14840  df-rlim 14841  df-sum 15038  df-prod 15255  df-rest 16691  df-topgen 16712  df-psmet 20086  df-xmet 20087  df-met 20088  df-bl 20089  df-mopn 20090  df-top 21502  df-topon 21519  df-bases 21554  df-cmp 21995  df-ovol 24071  df-vol 24072  df-sumge0 42989
This theorem is referenced by:  hoidmvle  43226
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