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Theorem hoidmvlelem5 41134
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 1883 . . . . 5 𝑠𝜑
2 nfre1 3034 . . . . 5 𝑠𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)
31, 2nfan 1868 . . . 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 8221 . . . . . . . 8 ((𝑋 ∈ Fin ∧ 𝑌𝑋) → 𝑌 ∈ Fin)
96, 7, 8syl2anc 694 . . . . . . 7 (𝜑𝑌 ∈ Fin)
10 snfi 8079 . . . . . . . 8 {𝑍} ∈ Fin
1110a1i 11 . . . . . . 7 (𝜑 → {𝑍} ∈ Fin)
12 unfi 8268 . . . . . . 7 ((𝑌 ∈ Fin ∧ {𝑍} ∈ Fin) → (𝑌 ∪ {𝑍}) ∈ Fin)
139, 11, 12syl2anc 694 . . . . . 6 (𝜑 → (𝑌 ∪ {𝑍}) ∈ Fin)
145, 13syl5eqel 2734 . . . . 5 (𝜑𝑊 ∈ Fin)
1514adantr 480 . . . 4 ((𝜑 ∧ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → 𝑊 ∈ Fin)
16 hoidmvlelem5.a . . . . 5 (𝜑𝐴:𝑊⟶ℝ)
1716adantr 480 . . . 4 ((𝜑 ∧ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → 𝐴:𝑊⟶ℝ)
18 hoidmvlelem5.b . . . . 5 (𝜑𝐵:𝑊⟶ℝ)
1918adantr 480 . . . 4 ((𝜑 ∧ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → 𝐵:𝑊⟶ℝ)
20 simpr 476 . . . 4 ((𝜑 ∧ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠))
213, 4, 15, 17, 19, 20hoidmvval0 41122 . . 3 ((𝜑 ∧ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → (𝐴(𝐿𝑊)𝐵) = 0)
22 nnex 11064 . . . . . 6 ℕ ∈ V
2322a1i 11 . . . . 5 (𝜑 → ℕ ∈ V)
24 icossicc 12298 . . . . . . 7 (0[,)+∞) ⊆ (0[,]+∞)
2514adantr 480 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → 𝑊 ∈ Fin)
26 hoidmvlelem5.c . . . . . . . . . 10 (𝜑𝐶:ℕ⟶(ℝ ↑𝑚 𝑊))
2726ffvelrnda 6399 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ∈ (ℝ ↑𝑚 𝑊))
28 elmapi 7921 . . . . . . . . 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 6399 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗) ∈ (ℝ ↑𝑚 𝑊))
32 elmapi 7921 . . . . . . . . 9 ((𝐷𝑗) ∈ (ℝ ↑𝑚 𝑊) → (𝐷𝑗):𝑊⟶ℝ)
3331, 32syl 17 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗):𝑊⟶ℝ)
344, 25, 29, 33hoidmvcl 41117 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)) ∈ (0[,)+∞))
3524, 34sseldi 3634 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)) ∈ (0[,]+∞))
36 eqid 2651 . . . . . 6 (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗))) = (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))
3735, 36fmptd 6425 . . . . 5 (𝜑 → (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗))):ℕ⟶(0[,]+∞))
3823, 37sge0ge0 40919 . . . 4 (𝜑 → 0 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
3938adantr 480 . . 3 ((𝜑 ∧ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → 0 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
4021, 39eqbrtrd 4707 . 2 ((𝜑 ∧ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → (𝐴(𝐿𝑊)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
41 icossxr 12296 . . . . . . 7 (0[,)+∞) ⊆ ℝ*
424, 14, 16, 18hoidmvcl 41117 . . . . . . 7 (𝜑 → (𝐴(𝐿𝑊)𝐵) ∈ (0[,)+∞))
4341, 42sseldi 3634 . . . . . 6 (𝜑 → (𝐴(𝐿𝑊)𝐵) ∈ ℝ*)
4443adantr 480 . . . . 5 ((𝜑 ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (𝐴(𝐿𝑊)𝐵) ∈ ℝ*)
4523, 37sge0xrcl 40920 . . . . . 6 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ*)
4645adantr 480 . . . . 5 ((𝜑 ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ*)
47 rge0ssre 12318 . . . . . . . . 9 (0[,)+∞) ⊆ ℝ
4847, 42sseldi 3634 . . . . . . . 8 (𝜑 → (𝐴(𝐿𝑊)𝐵) ∈ ℝ)
49 ltpnf 11992 . . . . . . . 8 ((𝐴(𝐿𝑊)𝐵) ∈ ℝ → (𝐴(𝐿𝑊)𝐵) < +∞)
5048, 49syl 17 . . . . . . 7 (𝜑 → (𝐴(𝐿𝑊)𝐵) < +∞)
5150adantr 480 . . . . . 6 ((𝜑 ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (𝐴(𝐿𝑊)𝐵) < +∞)
52 id 22 . . . . . . . 8 ((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞ → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞)
5352eqcomd 2657 . . . . . . 7 ((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞ → +∞ = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
5453adantl 481 . . . . . 6 ((𝜑 ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → +∞ = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
5551, 54breqtrd 4711 . . . . 5 ((𝜑 ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (𝐴(𝐿𝑊)𝐵) < (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
5644, 46, 55xrltled 39800 . . . 4 ((𝜑 ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (𝐴(𝐿𝑊)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
5756adantlr 751 . . 3 (((𝜑 ∧ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (𝐴(𝐿𝑊)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
58 simpll 805 . . . 4 (((𝜑 ∧ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → 𝜑)
59 simpr 476 . . . . . 6 ((𝜑 ∧ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠))
6016ffvelrnda 6399 . . . . . . . . . 10 ((𝜑𝑠𝑊) → (𝐴𝑠) ∈ ℝ)
6118ffvelrnda 6399 . . . . . . . . . 10 ((𝜑𝑠𝑊) → (𝐵𝑠) ∈ ℝ)
6260, 61ltnled 10222 . . . . . . . . 9 ((𝜑𝑠𝑊) → ((𝐴𝑠) < (𝐵𝑠) ↔ ¬ (𝐵𝑠) ≤ (𝐴𝑠)))
6362ralbidva 3014 . . . . . . . 8 (𝜑 → (∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠) ↔ ∀𝑠𝑊 ¬ (𝐵𝑠) ≤ (𝐴𝑠)))
64 ralnex 3021 . . . . . . . . 9 (∀𝑠𝑊 ¬ (𝐵𝑠) ≤ (𝐴𝑠) ↔ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠))
6564a1i 11 . . . . . . . 8 (𝜑 → (∀𝑠𝑊 ¬ (𝐵𝑠) ≤ (𝐴𝑠) ↔ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)))
6663, 65bitrd 268 . . . . . . 7 (𝜑 → (∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠) ↔ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)))
6766adantr 480 . . . . . 6 ((𝜑 ∧ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → (∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠) ↔ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)))
6859, 67mpbird 247 . . . . 5 ((𝜑 ∧ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠))
6968adantr 480 . . . 4 (((𝜑 ∧ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠))
70 simpr 476 . . . . . 6 ((𝜑 ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞)
7122a1i 11 . . . . . . 7 ((𝜑 ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → ℕ ∈ V)
7237adantr 480 . . . . . . 7 ((𝜑 ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗))):ℕ⟶(0[,]+∞))
7371, 72sge0repnf 40921 . . . . . 6 ((𝜑 ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → ((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ ↔ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞))
7470, 73mpbird 247 . . . . 5 ((𝜑 ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)
7574adantlr 751 . . . 4 (((𝜑 ∧ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)
76 simpll 805 . . . . . . 7 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → (𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)))
77 fveq2 6229 . . . . . . . . . . . . 13 (𝑗 = 𝑖 → (𝐶𝑗) = (𝐶𝑖))
78 fveq2 6229 . . . . . . . . . . . . 13 (𝑗 = 𝑖 → (𝐷𝑗) = (𝐷𝑖))
7977, 78oveq12d 6708 . . . . . . . . . . . 12 (𝑗 = 𝑖 → ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)) = ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))
8079cbvmptv 4783 . . . . . . . . . . 11 (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗))) = (𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))
8180fveq2i 6232 . . . . . . . . . 10 ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖))))
8281eleq1i 2721 . . . . . . . . 9 ((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ ↔ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ)
8382biimpi 206 . . . . . . . 8 ((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ → (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ)
8483ad2antlr 763 . . . . . . 7 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ)
85 simpr 476 . . . . . . 7 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝑟 ∈ ℝ+)
866ad3antrrr 766 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝑋 ∈ Fin)
877ad3antrrr 766 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝑌𝑋)
88 hoidmvlelem5.n . . . . . . . . 9 (𝜑𝑌 ≠ ∅)
8988ad3antrrr 766 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝑌 ≠ ∅)
90 hoidmvlelem5.z . . . . . . . . 9 (𝜑𝑍 ∈ (𝑋𝑌))
9190ad3antrrr 766 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝑍 ∈ (𝑋𝑌))
9216ad3antrrr 766 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝐴:𝑊⟶ℝ)
9318ad3antrrr 766 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝐵:𝑊⟶ℝ)
94 fveq2 6229 . . . . . . . . . . . . . 14 (𝑠 = 𝑘 → (𝐴𝑠) = (𝐴𝑘))
95 fveq2 6229 . . . . . . . . . . . . . 14 (𝑠 = 𝑘 → (𝐵𝑠) = (𝐵𝑘))
9694, 95breq12d 4698 . . . . . . . . . . . . 13 (𝑠 = 𝑘 → ((𝐴𝑠) < (𝐵𝑠) ↔ (𝐴𝑘) < (𝐵𝑘)))
9796cbvralv 3201 . . . . . . . . . . . 12 (∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠) ↔ ∀𝑘𝑊 (𝐴𝑘) < (𝐵𝑘))
9897biimpi 206 . . . . . . . . . . 11 (∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠) → ∀𝑘𝑊 (𝐴𝑘) < (𝐵𝑘))
9998adantr 480 . . . . . . . . . 10 ((∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠) ∧ 𝑘𝑊) → ∀𝑘𝑊 (𝐴𝑘) < (𝐵𝑘))
100 simpr 476 . . . . . . . . . 10 ((∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠) ∧ 𝑘𝑊) → 𝑘𝑊)
101 rspa 2959 . . . . . . . . . 10 ((∀𝑘𝑊 (𝐴𝑘) < (𝐵𝑘) ∧ 𝑘𝑊) → (𝐴𝑘) < (𝐵𝑘))
10299, 100, 101syl2anc 694 . . . . . . . . 9 ((∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠) ∧ 𝑘𝑊) → (𝐴𝑘) < (𝐵𝑘))
103102ad5ant25 1342 . . . . . . . 8 (((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) ∧ 𝑘𝑊) → (𝐴𝑘) < (𝐵𝑘))
10426ad3antrrr 766 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝐶:ℕ⟶(ℝ ↑𝑚 𝑊))
10530ad3antrrr 766 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝐷:ℕ⟶(ℝ ↑𝑚 𝑊))
10682biimpri 218 . . . . . . . . 9 ((Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)
107106ad2antlr 763 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)
108 fveq1 6228 . . . . . . . . . . . . 13 (𝑑 = 𝑐 → (𝑑𝑖) = (𝑐𝑖))
109108breq1d 4695 . . . . . . . . . . . . . 14 (𝑑 = 𝑐 → ((𝑑𝑖) ≤ 𝑥 ↔ (𝑐𝑖) ≤ 𝑥))
110109, 108ifbieq1d 4142 . . . . . . . . . . . . 13 (𝑑 = 𝑐 → if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥) = if((𝑐𝑖) ≤ 𝑥, (𝑐𝑖), 𝑥))
111108, 110ifeq12d 4139 . . . . . . . . . . . 12 (𝑑 = 𝑐 → if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥)) = if(𝑖𝑌, (𝑐𝑖), if((𝑐𝑖) ≤ 𝑥, (𝑐𝑖), 𝑥)))
112111mpteq2dv 4778 . . . . . . . . . . 11 (𝑑 = 𝑐 → (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥))) = (𝑖𝑊 ↦ if(𝑖𝑌, (𝑐𝑖), if((𝑐𝑖) ≤ 𝑥, (𝑐𝑖), 𝑥))))
113 eleq1 2718 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → (𝑖𝑌𝑗𝑌))
114 fveq2 6229 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → (𝑐𝑖) = (𝑐𝑗))
115114breq1d 4695 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → ((𝑐𝑖) ≤ 𝑥 ↔ (𝑐𝑗) ≤ 𝑥))
116115, 114ifbieq1d 4142 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → if((𝑐𝑖) ≤ 𝑥, (𝑐𝑖), 𝑥) = if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥))
117113, 114, 116ifbieq12d 4146 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → if(𝑖𝑌, (𝑐𝑖), if((𝑐𝑖) ≤ 𝑥, (𝑐𝑖), 𝑥)) = if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)))
118117cbvmptv 4783 . . . . . . . . . . . 12 (𝑖𝑊 ↦ if(𝑖𝑌, (𝑐𝑖), if((𝑐𝑖) ≤ 𝑥, (𝑐𝑖), 𝑥))) = (𝑗𝑊 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)))
119118a1i 11 . . . . . . . . . . 11 (𝑑 = 𝑐 → (𝑖𝑊 ↦ if(𝑖𝑌, (𝑐𝑖), if((𝑐𝑖) ≤ 𝑥, (𝑐𝑖), 𝑥))) = (𝑗𝑊 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥))))
120112, 119eqtrd 2685 . . . . . . . . . 10 (𝑑 = 𝑐 → (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥))) = (𝑗𝑊 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥))))
121120cbvmptv 4783 . . . . . . . . 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))))
122121mpteq2i 4774 . . . . . . . 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)))))
123 eqid 2651 . . . . . . . 8 ((𝐴𝑌)(𝐿𝑌)(𝐵𝑌)) = ((𝐴𝑌)(𝐿𝑌)(𝐵𝑌))
124 simpr 476 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝑟 ∈ ℝ+)
125 oveq1 6697 . . . . . . . . . . 11 (𝑤 = 𝑧 → (𝑤 − (𝐴𝑍)) = (𝑧 − (𝐴𝑍)))
126125oveq2d 6706 . . . . . . . . . 10 (𝑤 = 𝑧 → (((𝐴𝑌)(𝐿𝑌)(𝐵𝑌)) · (𝑤 − (𝐴𝑍))) = (((𝐴𝑌)(𝐿𝑌)(𝐵𝑌)) · (𝑧 − (𝐴𝑍))))
127 breq2 4689 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = 𝑥 → ((𝑑𝑖) ≤ 𝑤 ↔ (𝑑𝑖) ≤ 𝑥))
128 eqidd 2652 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = 𝑥 → (𝑑𝑖) = (𝑑𝑖))
129 id 22 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = 𝑥𝑤 = 𝑥)
130127, 128, 129ifbieq12d 4146 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = 𝑥 → if((𝑑𝑖) ≤ 𝑤, (𝑑𝑖), 𝑤) = if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥))
131130ifeq2d 4138 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑥 → if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑤, (𝑑𝑖), 𝑤)) = if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥)))
132131mpteq2dv 4778 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑥 → (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑤, (𝑑𝑖), 𝑤))) = (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥))))
133132mpteq2dv 4778 . . . . . . . . . . . . . . . . . . 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)))))
134133cbvmptv 4783 . . . . . . . . . . . . . . . . . 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)))))
135134a1i 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))))))
136 id 22 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑧𝑤 = 𝑧)
137135, 136fveq12d 6235 . . . . . . . . . . . . . . . 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))
138137fveq1d 6231 . . . . . . . . . . . . . . 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)))
139138oveq2d 6706 . . . . . . . . . . . . . 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))))
140139mpteq2dv 4778 . . . . . . . . . . . . 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)))))
141 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑗 → (𝐶𝑙) = (𝐶𝑗))
142 fveq2 6229 . . . . . . . . . . . . . . . . 17 (𝑙 = 𝑗 → (𝐷𝑙) = (𝐷𝑗))
143142fveq2d 6233 . . . . . . . . . . . . . . . 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)))
144141, 143oveq12d 6708 . . . . . . . . . . . . . . 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))))
145144cbvmptv 4783 . . . . . . . . . . . . . 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))))
146145a1i 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)))))
147140, 146eqtrd 2685 . . . . . . . . . . . 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)))))
148147fveq2d 6233 . . . . . . . . . . 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))))))
149148oveq2d 6706 . . . . . . . . . 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)))))))
150126, 149breq12d 4698 . . . . . . . . 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))))))))
151150cbvrabv 3230 . . . . . . . 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))))))}
152 eqid 2651 . . . . . . . 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, < )
153 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))))))
154153ad3antrrr 766 . . . . . . . 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))))))
155 hoidmvlelem5.s . . . . . . . . 9 (𝜑X𝑘𝑊 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑊 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))
156155ad3antrrr 766 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → X𝑘𝑊 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑊 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))
1574, 86, 87, 89, 91, 5, 92, 93, 103, 104, 105, 107, 122, 123, 124, 151, 152, 154, 156hoidmvlelem4 41133 . . . . . . 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))))))
15876, 84, 85, 157syl21anc 1365 . . . . . 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))))))
159158ralrimiva 2995 . . . . 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))))))
160 nfv 1883 . . . . . 6 𝑟((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)
16143ad2antrr 762 . . . . . 6 (((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) → (𝐴(𝐿𝑊)𝐵) ∈ ℝ*)
162 0xr 10124 . . . . . . . 8 0 ∈ ℝ*
163162a1i 11 . . . . . . 7 (((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) → 0 ∈ ℝ*)
164 pnfxr 10130 . . . . . . . 8 +∞ ∈ ℝ*
165164a1i 11 . . . . . . 7 (((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) → +∞ ∈ ℝ*)
16645ad2antrr 762 . . . . . . 7 (((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ*)
16738ad2antrr 762 . . . . . . 7 (((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) → 0 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
168 ltpnf 11992 . . . . . . . 8 ((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) < +∞)
169168adantl 481 . . . . . . 7 (((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) < +∞)
170163, 165, 166, 167, 169elicod 12262 . . . . . 6 (((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ (0[,)+∞))
171160, 161, 170xralrple2 39883 . . . . 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)))))))
172159, 171mpbird 247 . . . 4 (((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) → (𝐴(𝐿𝑊)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
17358, 69, 75, 172syl21anc 1365 . . 3 (((𝜑 ∧ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (𝐴(𝐿𝑊)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
17457, 173pm2.61dan 849 . 2 ((𝜑 ∧ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → (𝐴(𝐿𝑊)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
17540, 174pm2.61dan 849 1 (𝜑 → (𝐴(𝐿𝑊)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  cdif 3604  cun 3605  wss 3607  c0 3948  ifcif 4119  {csn 4210   ciun 4552   class class class wbr 4685  cmpt 4762  cres 5145  wf 5922  cfv 5926  (class class class)co 6690  cmpt2 6692  𝑚 cmap 7899  Xcixp 7950  Fincfn 7997  supcsup 8387  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979  +∞cpnf 10109  *cxr 10111   < clt 10112  cle 10113  cmin 10304  cn 11058  +crp 11870  [,)cico 12215  [,]cicc 12216  cprod 14679  volcvol 23278  Σ^csumge0 40897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-rlim 14264  df-sum 14461  df-prod 14680  df-rest 16130  df-topgen 16151  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-top 20747  df-topon 20764  df-bases 20798  df-cmp 21238  df-ovol 23279  df-vol 23280  df-sumge0 40898
This theorem is referenced by:  hoidmvle  41135
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