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Theorem hoidmvlelem5 44137
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 1917 . . . . 5 𝑠𝜑
2 nfre1 3239 . . . . 5 𝑠𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)
31, 2nfan 1902 . . . 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 8956 . . . . . . . 8 ((𝑋 ∈ Fin ∧ 𝑌𝑋) → 𝑌 ∈ Fin)
96, 7, 8syl2anc 584 . . . . . . 7 (𝜑𝑌 ∈ Fin)
10 snfi 8834 . . . . . . . 8 {𝑍} ∈ Fin
1110a1i 11 . . . . . . 7 (𝜑 → {𝑍} ∈ Fin)
12 unfi 8955 . . . . . . 7 ((𝑌 ∈ Fin ∧ {𝑍} ∈ Fin) → (𝑌 ∪ {𝑍}) ∈ Fin)
139, 11, 12syl2anc 584 . . . . . 6 (𝜑 → (𝑌 ∪ {𝑍}) ∈ Fin)
145, 13eqeltrid 2843 . . . . 5 (𝜑𝑊 ∈ Fin)
1514adantr 481 . . . 4 ((𝜑 ∧ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → 𝑊 ∈ Fin)
16 hoidmvlelem5.a . . . . 5 (𝜑𝐴:𝑊⟶ℝ)
1716adantr 481 . . . 4 ((𝜑 ∧ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → 𝐴:𝑊⟶ℝ)
18 hoidmvlelem5.b . . . . 5 (𝜑𝐵:𝑊⟶ℝ)
1918adantr 481 . . . 4 ((𝜑 ∧ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → 𝐵:𝑊⟶ℝ)
20 simpr 485 . . . 4 ((𝜑 ∧ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠))
213, 4, 15, 17, 19, 20hoidmvval0 44125 . . 3 ((𝜑 ∧ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → (𝐴(𝐿𝑊)𝐵) = 0)
22 nnex 11979 . . . . . 6 ℕ ∈ V
2322a1i 11 . . . . 5 (𝜑 → ℕ ∈ V)
24 icossicc 13168 . . . . . . 7 (0[,)+∞) ⊆ (0[,]+∞)
2514adantr 481 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → 𝑊 ∈ Fin)
26 hoidmvlelem5.c . . . . . . . . . 10 (𝜑𝐶:ℕ⟶(ℝ ↑m 𝑊))
2726ffvelrnda 6961 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ∈ (ℝ ↑m 𝑊))
28 elmapi 8637 . . . . . . . . 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 6961 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗) ∈ (ℝ ↑m 𝑊))
32 elmapi 8637 . . . . . . . . 9 ((𝐷𝑗) ∈ (ℝ ↑m 𝑊) → (𝐷𝑗):𝑊⟶ℝ)
3331, 32syl 17 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗):𝑊⟶ℝ)
344, 25, 29, 33hoidmvcl 44120 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)) ∈ (0[,)+∞))
3524, 34sselid 3919 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)) ∈ (0[,]+∞))
3635fmpttd 6989 . . . . 5 (𝜑 → (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗))):ℕ⟶(0[,]+∞))
3723, 36sge0ge0 43922 . . . 4 (𝜑 → 0 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
3837adantr 481 . . 3 ((𝜑 ∧ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → 0 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
3921, 38eqbrtrd 5096 . 2 ((𝜑 ∧ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → (𝐴(𝐿𝑊)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
40 icossxr 13164 . . . . . . 7 (0[,)+∞) ⊆ ℝ*
414, 14, 16, 18hoidmvcl 44120 . . . . . . 7 (𝜑 → (𝐴(𝐿𝑊)𝐵) ∈ (0[,)+∞))
4240, 41sselid 3919 . . . . . 6 (𝜑 → (𝐴(𝐿𝑊)𝐵) ∈ ℝ*)
4342adantr 481 . . . . 5 ((𝜑 ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (𝐴(𝐿𝑊)𝐵) ∈ ℝ*)
4423, 36sge0xrcl 43923 . . . . . 6 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ*)
4544adantr 481 . . . . 5 ((𝜑 ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ*)
46 rge0ssre 13188 . . . . . . . . 9 (0[,)+∞) ⊆ ℝ
4746, 41sselid 3919 . . . . . . . 8 (𝜑 → (𝐴(𝐿𝑊)𝐵) ∈ ℝ)
48 ltpnf 12856 . . . . . . . 8 ((𝐴(𝐿𝑊)𝐵) ∈ ℝ → (𝐴(𝐿𝑊)𝐵) < +∞)
4947, 48syl 17 . . . . . . 7 (𝜑 → (𝐴(𝐿𝑊)𝐵) < +∞)
5049adantr 481 . . . . . 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 2744 . . . . . . 7 ((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞ → +∞ = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
5352adantl 482 . . . . . 6 ((𝜑 ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → +∞ = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
5450, 53breqtrd 5100 . . . . 5 ((𝜑 ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (𝐴(𝐿𝑊)𝐵) < (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
5543, 45, 54xrltled 12884 . . . 4 ((𝜑 ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (𝐴(𝐿𝑊)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
5655adantlr 712 . . 3 (((𝜑 ∧ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (𝐴(𝐿𝑊)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
57 simpll 764 . . . 4 (((𝜑 ∧ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → 𝜑)
58 simpr 485 . . . . . 6 ((𝜑 ∧ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠))
5916ffvelrnda 6961 . . . . . . . . . 10 ((𝜑𝑠𝑊) → (𝐴𝑠) ∈ ℝ)
6018ffvelrnda 6961 . . . . . . . . . 10 ((𝜑𝑠𝑊) → (𝐵𝑠) ∈ ℝ)
6159, 60ltnled 11122 . . . . . . . . 9 ((𝜑𝑠𝑊) → ((𝐴𝑠) < (𝐵𝑠) ↔ ¬ (𝐵𝑠) ≤ (𝐴𝑠)))
6261ralbidva 3111 . . . . . . . 8 (𝜑 → (∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠) ↔ ∀𝑠𝑊 ¬ (𝐵𝑠) ≤ (𝐴𝑠)))
63 ralnex 3167 . . . . . . . . 9 (∀𝑠𝑊 ¬ (𝐵𝑠) ≤ (𝐴𝑠) ↔ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠))
6463a1i 11 . . . . . . . 8 (𝜑 → (∀𝑠𝑊 ¬ (𝐵𝑠) ≤ (𝐴𝑠) ↔ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)))
6562, 64bitrd 278 . . . . . . 7 (𝜑 → (∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠) ↔ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)))
6665adantr 481 . . . . . 6 ((𝜑 ∧ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → (∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠) ↔ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)))
6758, 66mpbird 256 . . . . 5 ((𝜑 ∧ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠))
6867adantr 481 . . . 4 (((𝜑 ∧ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠))
69 simpr 485 . . . . . 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 481 . . . . . . 7 ((𝜑 ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗))):ℕ⟶(0[,]+∞))
7270, 71sge0repnf 43924 . . . . . 6 ((𝜑 ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → ((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ ↔ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞))
7369, 72mpbird 256 . . . . 5 ((𝜑 ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)
7473adantlr 712 . . . 4 (((𝜑 ∧ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)
75 simpll 764 . . . . . . 7 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → (𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)))
76 fveq2 6774 . . . . . . . . . . . . 13 (𝑗 = 𝑖 → (𝐶𝑗) = (𝐶𝑖))
77 fveq2 6774 . . . . . . . . . . . . 13 (𝑗 = 𝑖 → (𝐷𝑗) = (𝐷𝑖))
7876, 77oveq12d 7293 . . . . . . . . . . . 12 (𝑗 = 𝑖 → ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)) = ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))
7978cbvmptv 5187 . . . . . . . . . . 11 (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗))) = (𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))
8079fveq2i 6777 . . . . . . . . . 10 ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖))))
8180eleq1i 2829 . . . . . . . . 9 ((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ ↔ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ)
8281biimpi 215 . . . . . . . 8 ((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ → (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ)
8382ad2antlr 724 . . . . . . 7 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ)
84 simpr 485 . . . . . . 7 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝑟 ∈ ℝ+)
856ad3antrrr 727 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝑋 ∈ Fin)
867ad3antrrr 727 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝑌𝑋)
87 hoidmvlelem5.n . . . . . . . . 9 (𝜑𝑌 ≠ ∅)
8887ad3antrrr 727 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝑌 ≠ ∅)
89 hoidmvlelem5.z . . . . . . . . 9 (𝜑𝑍 ∈ (𝑋𝑌))
9089ad3antrrr 727 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝑍 ∈ (𝑋𝑌))
9116ad3antrrr 727 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝐴:𝑊⟶ℝ)
9218ad3antrrr 727 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝐵:𝑊⟶ℝ)
93 fveq2 6774 . . . . . . . . . . . . . 14 (𝑠 = 𝑘 → (𝐴𝑠) = (𝐴𝑘))
94 fveq2 6774 . . . . . . . . . . . . . 14 (𝑠 = 𝑘 → (𝐵𝑠) = (𝐵𝑘))
9593, 94breq12d 5087 . . . . . . . . . . . . 13 (𝑠 = 𝑘 → ((𝐴𝑠) < (𝐵𝑠) ↔ (𝐴𝑘) < (𝐵𝑘)))
9695cbvralvw 3383 . . . . . . . . . . . 12 (∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠) ↔ ∀𝑘𝑊 (𝐴𝑘) < (𝐵𝑘))
9796biimpi 215 . . . . . . . . . . 11 (∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠) → ∀𝑘𝑊 (𝐴𝑘) < (𝐵𝑘))
9897adantr 481 . . . . . . . . . 10 ((∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠) ∧ 𝑘𝑊) → ∀𝑘𝑊 (𝐴𝑘) < (𝐵𝑘))
99 simpr 485 . . . . . . . . . 10 ((∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠) ∧ 𝑘𝑊) → 𝑘𝑊)
100 rspa 3132 . . . . . . . . . 10 ((∀𝑘𝑊 (𝐴𝑘) < (𝐵𝑘) ∧ 𝑘𝑊) → (𝐴𝑘) < (𝐵𝑘))
10198, 99, 100syl2anc 584 . . . . . . . . 9 ((∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠) ∧ 𝑘𝑊) → (𝐴𝑘) < (𝐵𝑘))
102101ad5ant25 759 . . . . . . . 8 (((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) ∧ 𝑘𝑊) → (𝐴𝑘) < (𝐵𝑘))
10326ad3antrrr 727 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝐶:ℕ⟶(ℝ ↑m 𝑊))
10430ad3antrrr 727 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝐷:ℕ⟶(ℝ ↑m 𝑊))
10581biimpri 227 . . . . . . . . 9 ((Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)
106105ad2antlr 724 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)
107 fveq1 6773 . . . . . . . . . . . . 13 (𝑑 = 𝑐 → (𝑑𝑖) = (𝑐𝑖))
108107breq1d 5084 . . . . . . . . . . . . . 14 (𝑑 = 𝑐 → ((𝑑𝑖) ≤ 𝑥 ↔ (𝑐𝑖) ≤ 𝑥))
109108, 107ifbieq1d 4483 . . . . . . . . . . . . 13 (𝑑 = 𝑐 → if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥) = if((𝑐𝑖) ≤ 𝑥, (𝑐𝑖), 𝑥))
110107, 109ifeq12d 4480 . . . . . . . . . . . 12 (𝑑 = 𝑐 → if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥)) = if(𝑖𝑌, (𝑐𝑖), if((𝑐𝑖) ≤ 𝑥, (𝑐𝑖), 𝑥)))
111110mpteq2dv 5176 . . . . . . . . . . 11 (𝑑 = 𝑐 → (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥))) = (𝑖𝑊 ↦ if(𝑖𝑌, (𝑐𝑖), if((𝑐𝑖) ≤ 𝑥, (𝑐𝑖), 𝑥))))
112 eleq1w 2821 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → (𝑖𝑌𝑗𝑌))
113 fveq2 6774 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → (𝑐𝑖) = (𝑐𝑗))
114113breq1d 5084 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → ((𝑐𝑖) ≤ 𝑥 ↔ (𝑐𝑗) ≤ 𝑥))
115114, 113ifbieq1d 4483 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → if((𝑐𝑖) ≤ 𝑥, (𝑐𝑖), 𝑥) = if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥))
116112, 113, 115ifbieq12d 4487 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → if(𝑖𝑌, (𝑐𝑖), if((𝑐𝑖) ≤ 𝑥, (𝑐𝑖), 𝑥)) = if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)))
117116cbvmptv 5187 . . . . . . . . . . . 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 2778 . . . . . . . . . 10 (𝑑 = 𝑐 → (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥))) = (𝑗𝑊 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥))))
120119cbvmptv 5187 . . . . . . . . 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 5179 . . . . . . . 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 2738 . . . . . . . 8 ((𝐴𝑌)(𝐿𝑌)(𝐵𝑌)) = ((𝐴𝑌)(𝐿𝑌)(𝐵𝑌))
123 simpr 485 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝑟 ∈ ℝ+)
124 oveq1 7282 . . . . . . . . . . 11 (𝑤 = 𝑧 → (𝑤 − (𝐴𝑍)) = (𝑧 − (𝐴𝑍)))
125124oveq2d 7291 . . . . . . . . . 10 (𝑤 = 𝑧 → (((𝐴𝑌)(𝐿𝑌)(𝐵𝑌)) · (𝑤 − (𝐴𝑍))) = (((𝐴𝑌)(𝐿𝑌)(𝐵𝑌)) · (𝑧 − (𝐴𝑍))))
126 breq2 5078 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = 𝑥 → ((𝑑𝑖) ≤ 𝑤 ↔ (𝑑𝑖) ≤ 𝑥))
127 eqidd 2739 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = 𝑥 → (𝑑𝑖) = (𝑑𝑖))
128 id 22 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = 𝑥𝑤 = 𝑥)
129126, 127, 128ifbieq12d 4487 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = 𝑥 → if((𝑑𝑖) ≤ 𝑤, (𝑑𝑖), 𝑤) = if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥))
130129ifeq2d 4479 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑥 → if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑤, (𝑑𝑖), 𝑤)) = if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥)))
131130mpteq2dv 5176 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑥 → (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑤, (𝑑𝑖), 𝑤))) = (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥))))
132131mpteq2dv 5176 . . . . . . . . . . . . . . . . . . 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 5187 . . . . . . . . . . . . . . . . . 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 6781 . . . . . . . . . . . . . . . 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 6776 . . . . . . . . . . . . . . 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 7291 . . . . . . . . . . . . . 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 5176 . . . . . . . . . . . . 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 6774 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑗 → (𝐶𝑙) = (𝐶𝑗))
141 2fveq3 6779 . . . . . . . . . . . . . . . 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 7293 . . . . . . . . . . . . . . 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 5187 . . . . . . . . . . . . . 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 2778 . . . . . . . . . . . 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 6778 . . . . . . . . . . 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 7291 . . . . . . . . . 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 5087 . . . . . . . . 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 3426 . . . . . . . 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 2738 . . . . . . . 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 727 . . . . . . . 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 727 . . . . . . . 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 44136 . . . . . . 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 835 . . . . . 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 3103 . . . . 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 1917 . . . . . 6 𝑟((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)
15942ad2antrr 723 . . . . . 6 (((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) → (𝐴(𝐿𝑊)𝐵) ∈ ℝ*)
160 0xr 11022 . . . . . . . 8 0 ∈ ℝ*
161160a1i 11 . . . . . . 7 (((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) → 0 ∈ ℝ*)
162 pnfxr 11029 . . . . . . . 8 +∞ ∈ ℝ*
163162a1i 11 . . . . . . 7 (((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) → +∞ ∈ ℝ*)
16444ad2antrr 723 . . . . . . 7 (((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ*)
16537ad2antrr 723 . . . . . . 7 (((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) → 0 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
166 ltpnf 12856 . . . . . . . 8 ((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) < +∞)
167166adantl 482 . . . . . . 7 (((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) < +∞)
168161, 163, 164, 165, 167elicod 13129 . . . . . 6 (((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ (0[,)+∞))
169158, 159, 168xralrple2 42893 . . . . 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 256 . . . 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 835 . . 3 (((𝜑 ∧ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (𝐴(𝐿𝑊)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
17256, 171pm2.61dan 810 . 2 ((𝜑 ∧ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → (𝐴(𝐿𝑊)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
17339, 172pm2.61dan 810 1 (𝜑 → (𝐴(𝐿𝑊)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  wne 2943  wral 3064  wrex 3065  {crab 3068  Vcvv 3432  cdif 3884  cun 3885  wss 3887  c0 4256  ifcif 4459  {csn 4561   ciun 4924   class class class wbr 5074  cmpt 5157  cres 5591  wf 6429  cfv 6433  (class class class)co 7275  cmpo 7277  m cmap 8615  Xcixp 8685  Fincfn 8733  supcsup 9199  cr 10870  0cc0 10871  1c1 10872   + caddc 10874   · cmul 10876  +∞cpnf 11006  *cxr 11008   < clt 11009  cle 11010  cmin 11205  cn 11973  +crp 12730  [,)cico 13081  [,]cicc 13082  cprod 15615  volcvol 24627  Σ^csumge0 43900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-er 8498  df-map 8617  df-pm 8618  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fi 9170  df-sup 9201  df-inf 9202  df-oi 9269  df-dju 9659  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-z 12320  df-uz 12583  df-q 12689  df-rp 12731  df-xneg 12848  df-xadd 12849  df-xmul 12850  df-ioo 13083  df-ico 13085  df-icc 13086  df-fz 13240  df-fzo 13383  df-fl 13512  df-seq 13722  df-exp 13783  df-hash 14045  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-clim 15197  df-rlim 15198  df-sum 15398  df-prod 15616  df-rest 17133  df-topgen 17154  df-psmet 20589  df-xmet 20590  df-met 20591  df-bl 20592  df-mopn 20593  df-top 22043  df-topon 22060  df-bases 22096  df-cmp 22538  df-ovol 24628  df-vol 24629  df-sumge0 43901
This theorem is referenced by:  hoidmvle  44138
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