Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hoidmvlelem5 Structured version   Visualization version   GIF version

Theorem hoidmvlelem5 39489
 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 1830 . . . . 5 𝑠𝜑
2 nfre1 2988 . . . . 5 𝑠𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)
31, 2nfan 1816 . . . 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 8065 . . . . . . . 8 ((𝑋 ∈ Fin ∧ 𝑌𝑋) → 𝑌 ∈ Fin)
96, 7, 8syl2anc 691 . . . . . . 7 (𝜑𝑌 ∈ Fin)
10 snfi 7923 . . . . . . . 8 {𝑍} ∈ Fin
1110a1i 11 . . . . . . 7 (𝜑 → {𝑍} ∈ Fin)
12 unfi 8112 . . . . . . 7 ((𝑌 ∈ Fin ∧ {𝑍} ∈ Fin) → (𝑌 ∪ {𝑍}) ∈ Fin)
139, 11, 12syl2anc 691 . . . . . 6 (𝜑 → (𝑌 ∪ {𝑍}) ∈ Fin)
145, 13syl5eqel 2692 . . . . 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 39477 . . 3 ((𝜑 ∧ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → (𝐴(𝐿𝑊)𝐵) = 0)
22 nnex 10903 . . . . . 6 ℕ ∈ V
2322a1i 11 . . . . 5 (𝜑 → ℕ ∈ V)
24 icossicc 12131 . . . . . . 7 (0[,)+∞) ⊆ (0[,]+∞)
2514adantr 480 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → 𝑊 ∈ Fin)
26 hoidmvlelem5.c . . . . . . . . . 10 (𝜑𝐶:ℕ⟶(ℝ ↑𝑚 𝑊))
2726ffvelrnda 6267 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ∈ (ℝ ↑𝑚 𝑊))
28 elmapi 7765 . . . . . . . . 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 6267 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗) ∈ (ℝ ↑𝑚 𝑊))
32 elmapi 7765 . . . . . . . . 9 ((𝐷𝑗) ∈ (ℝ ↑𝑚 𝑊) → (𝐷𝑗):𝑊⟶ℝ)
3331, 32syl 17 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗):𝑊⟶ℝ)
344, 25, 29, 33hoidmvcl 39472 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)) ∈ (0[,)+∞))
3524, 34sseldi 3566 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)) ∈ (0[,]+∞))
36 eqid 2610 . . . . . 6 (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗))) = (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))
3735, 36fmptd 6292 . . . . 5 (𝜑 → (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗))):ℕ⟶(0[,]+∞))
3823, 37sge0ge0 39277 . . . 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 4605 . 2 ((𝜑 ∧ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → (𝐴(𝐿𝑊)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
41 icossxr 12129 . . . . . . 7 (0[,)+∞) ⊆ ℝ*
424, 14, 16, 18hoidmvcl 39472 . . . . . . 7 (𝜑 → (𝐴(𝐿𝑊)𝐵) ∈ (0[,)+∞))
4341, 42sseldi 3566 . . . . . 6 (𝜑 → (𝐴(𝐿𝑊)𝐵) ∈ ℝ*)
4443adantr 480 . . . . 5 ((𝜑 ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (𝐴(𝐿𝑊)𝐵) ∈ ℝ*)
4523, 37sge0xrcl 39278 . . . . . 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 12151 . . . . . . . . 9 (0[,)+∞) ⊆ ℝ
4847, 42sseldi 3566 . . . . . . . 8 (𝜑 → (𝐴(𝐿𝑊)𝐵) ∈ ℝ)
49 ltpnf 11830 . . . . . . . 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 2616 . . . . . . 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 4609 . . . . 5 ((𝜑 ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (𝐴(𝐿𝑊)𝐵) < (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
5644, 46, 55xrltled 38427 . . . 4 ((𝜑 ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (𝐴(𝐿𝑊)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
5756adantlr 747 . . 3 (((𝜑 ∧ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (𝐴(𝐿𝑊)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
58 simpll 786 . . . 4 (((𝜑 ∧ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → 𝜑)
59 simpr 476 . . . . . 6 ((𝜑 ∧ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠))
6016ffvelrnda 6267 . . . . . . . . . 10 ((𝜑𝑠𝑊) → (𝐴𝑠) ∈ ℝ)
6118ffvelrnda 6267 . . . . . . . . . 10 ((𝜑𝑠𝑊) → (𝐵𝑠) ∈ ℝ)
6260, 61ltnled 10063 . . . . . . . . 9 ((𝜑𝑠𝑊) → ((𝐴𝑠) < (𝐵𝑠) ↔ ¬ (𝐵𝑠) ≤ (𝐴𝑠)))
6362ralbidva 2968 . . . . . . . 8 (𝜑 → (∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠) ↔ ∀𝑠𝑊 ¬ (𝐵𝑠) ≤ (𝐴𝑠)))
64 ralnex 2975 . . . . . . . . 9 (∀𝑠𝑊 ¬ (𝐵𝑠) ≤ (𝐴𝑠) ↔ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠))
6564a1i 11 . . . . . . . 8 (𝜑 → (∀𝑠𝑊 ¬ (𝐵𝑠) ≤ (𝐴𝑠) ↔ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)))
6663, 65bitrd 267 . . . . . . 7 (𝜑 → (∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠) ↔ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)))
6766adantr 480 . . . . . 6 ((𝜑 ∧ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → (∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠) ↔ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)))
6859, 67mpbird 246 . . . . 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 39279 . . . . . 6 ((𝜑 ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → ((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ ↔ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞))
7470, 73mpbird 246 . . . . 5 ((𝜑 ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)
7574adantlr 747 . . . 4 (((𝜑 ∧ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)
76 simpll 786 . . . . . . 7 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → (𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)))
77 fveq2 6103 . . . . . . . . . . . . 13 (𝑗 = 𝑖 → (𝐶𝑗) = (𝐶𝑖))
78 fveq2 6103 . . . . . . . . . . . . 13 (𝑗 = 𝑖 → (𝐷𝑗) = (𝐷𝑖))
7977, 78oveq12d 6567 . . . . . . . . . . . 12 (𝑗 = 𝑖 → ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)) = ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))
8079cbvmptv 4678 . . . . . . . . . . 11 (𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗))) = (𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))
8180fveq2i 6106 . . . . . . . . . 10 ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖))))
8281eleq1i 2679 . . . . . . . . 9 ((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ ↔ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ)
8382biimpi 205 . . . . . . . 8 ((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ → (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ)
8483ad2antlr 759 . . . . . . 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 762 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝑋 ∈ Fin)
877ad3antrrr 762 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝑌𝑋)
88 hoidmvlelem5.n . . . . . . . . 9 (𝜑𝑌 ≠ ∅)
8988ad3antrrr 762 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝑌 ≠ ∅)
90 hoidmvlelem5.z . . . . . . . . 9 (𝜑𝑍 ∈ (𝑋𝑌))
9190ad3antrrr 762 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝑍 ∈ (𝑋𝑌))
9216ad3antrrr 762 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝐴:𝑊⟶ℝ)
9318ad3antrrr 762 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝐵:𝑊⟶ℝ)
94 fveq2 6103 . . . . . . . . . . . . . 14 (𝑠 = 𝑘 → (𝐴𝑠) = (𝐴𝑘))
95 fveq2 6103 . . . . . . . . . . . . . 14 (𝑠 = 𝑘 → (𝐵𝑠) = (𝐵𝑘))
9694, 95breq12d 4596 . . . . . . . . . . . . 13 (𝑠 = 𝑘 → ((𝐴𝑠) < (𝐵𝑠) ↔ (𝐴𝑘) < (𝐵𝑘)))
9796cbvralv 3147 . . . . . . . . . . . 12 (∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠) ↔ ∀𝑘𝑊 (𝐴𝑘) < (𝐵𝑘))
9897biimpi 205 . . . . . . . . . . 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 2914 . . . . . . . . . 10 ((∀𝑘𝑊 (𝐴𝑘) < (𝐵𝑘) ∧ 𝑘𝑊) → (𝐴𝑘) < (𝐵𝑘))
10299, 100, 101syl2anc 691 . . . . . . . . 9 ((∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠) ∧ 𝑘𝑊) → (𝐴𝑘) < (𝐵𝑘))
103102ad5ant25 1298 . . . . . . . 8 (((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) ∧ 𝑘𝑊) → (𝐴𝑘) < (𝐵𝑘))
10426ad3antrrr 762 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝐶:ℕ⟶(ℝ ↑𝑚 𝑊))
10530ad3antrrr 762 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝐷:ℕ⟶(ℝ ↑𝑚 𝑊))
10682biimpri 217 . . . . . . . . 9 ((Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)
107106ad2antlr 759 . . . . . . . 8 ((((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶𝑖)(𝐿𝑊)(𝐷𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)
108 fveq1 6102 . . . . . . . . . . . . 13 (𝑑 = 𝑐 → (𝑑𝑖) = (𝑐𝑖))
109108breq1d 4593 . . . . . . . . . . . . . 14 (𝑑 = 𝑐 → ((𝑑𝑖) ≤ 𝑥 ↔ (𝑐𝑖) ≤ 𝑥))
110109, 108ifbieq1d 4059 . . . . . . . . . . . . 13 (𝑑 = 𝑐 → if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥) = if((𝑐𝑖) ≤ 𝑥, (𝑐𝑖), 𝑥))
111108, 110ifeq12d 4056 . . . . . . . . . . . 12 (𝑑 = 𝑐 → if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥)) = if(𝑖𝑌, (𝑐𝑖), if((𝑐𝑖) ≤ 𝑥, (𝑐𝑖), 𝑥)))
112111mpteq2dv 4673 . . . . . . . . . . 11 (𝑑 = 𝑐 → (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥))) = (𝑖𝑊 ↦ if(𝑖𝑌, (𝑐𝑖), if((𝑐𝑖) ≤ 𝑥, (𝑐𝑖), 𝑥))))
113 eleq1 2676 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → (𝑖𝑌𝑗𝑌))
114 fveq2 6103 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → (𝑐𝑖) = (𝑐𝑗))
115114breq1d 4593 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → ((𝑐𝑖) ≤ 𝑥 ↔ (𝑐𝑗) ≤ 𝑥))
116115, 114ifbieq1d 4059 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → if((𝑐𝑖) ≤ 𝑥, (𝑐𝑖), 𝑥) = if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥))
117113, 114, 116ifbieq12d 4063 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → if(𝑖𝑌, (𝑐𝑖), if((𝑐𝑖) ≤ 𝑥, (𝑐𝑖), 𝑥)) = if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)))
118117cbvmptv 4678 . . . . . . . . . . . 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 2644 . . . . . . . . . 10 (𝑑 = 𝑐 → (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥))) = (𝑗𝑊 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥))))
121120cbvmptv 4678 . . . . . . . . 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 4669 . . . . . . . 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 2610 . . . . . . . 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 6556 . . . . . . . . . . 11 (𝑤 = 𝑧 → (𝑤 − (𝐴𝑍)) = (𝑧 − (𝐴𝑍)))
126125oveq2d 6565 . . . . . . . . . 10 (𝑤 = 𝑧 → (((𝐴𝑌)(𝐿𝑌)(𝐵𝑌)) · (𝑤 − (𝐴𝑍))) = (((𝐴𝑌)(𝐿𝑌)(𝐵𝑌)) · (𝑧 − (𝐴𝑍))))
127 breq2 4587 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = 𝑥 → ((𝑑𝑖) ≤ 𝑤 ↔ (𝑑𝑖) ≤ 𝑥))
128 eqidd 2611 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = 𝑥 → (𝑑𝑖) = (𝑑𝑖))
129 id 22 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = 𝑥𝑤 = 𝑥)
130127, 128, 129ifbieq12d 4063 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = 𝑥 → if((𝑑𝑖) ≤ 𝑤, (𝑑𝑖), 𝑤) = if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥))
131130ifeq2d 4055 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑥 → if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑤, (𝑑𝑖), 𝑤)) = if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥)))
132131mpteq2dv 4673 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑥 → (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑤, (𝑑𝑖), 𝑤))) = (𝑖𝑊 ↦ if(𝑖𝑌, (𝑑𝑖), if((𝑑𝑖) ≤ 𝑥, (𝑑𝑖), 𝑥))))
133132mpteq2dv 4673 . . . . . . . . . . . . . . . . . . 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 4678 . . . . . . . . . . . . . . . . . 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 6109 . . . . . . . . . . . . . . . 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 6105 . . . . . . . . . . . . . . 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 6565 . . . . . . . . . . . . . 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 4673 . . . . . . . . . . . . 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 6103 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑗 → (𝐶𝑙) = (𝐶𝑗))
142 fveq2 6103 . . . . . . . . . . . . . . . . 17 (𝑙 = 𝑗 → (𝐷𝑙) = (𝐷𝑗))
143142fveq2d 6107 . . . . . . . . . . . . . . . 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 6567 . . . . . . . . . . . . . . 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 4678 . . . . . . . . . . . . . 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 2644 . . . . . . . . . . . 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 6107 . . . . . . . . . . 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 6565 . . . . . . . . . 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 4596 . . . . . . . . 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 3172 . . . . . . . 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 2610 . . . . . . . 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 762 . . . . . . . 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 762 . . . . . . . 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 39488 . . . . . . 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 1317 . . . . . 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 2949 . . . . 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 1830 . . . . . 6 𝑟((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)
16143ad2antrr 758 . . . . . 6 (((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) → (𝐴(𝐿𝑊)𝐵) ∈ ℝ*)
162 0xr 9965 . . . . . . . 8 0 ∈ ℝ*
163162a1i 11 . . . . . . 7 (((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) → 0 ∈ ℝ*)
164 pnfxr 9971 . . . . . . . 8 +∞ ∈ ℝ*
165164a1i 11 . . . . . . 7 (((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) → +∞ ∈ ℝ*)
16645ad2antrr 758 . . . . . . 7 (((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ*)
16738ad2antrr 758 . . . . . . 7 (((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) → 0 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
168 ltpnf 11830 . . . . . . . 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 12095 . . . . . 6 (((𝜑 ∧ ∀𝑠𝑊 (𝐴𝑠) < (𝐵𝑠)) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ) → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ (0[,)+∞))
171160, 161, 170xralrple2 38511 . . . . 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 246 . . . 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 1317 . . 3 (((𝜑 ∧ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) = +∞) → (𝐴(𝐿𝑊)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
17457, 173pm2.61dan 828 . 2 ((𝜑 ∧ ¬ ∃𝑠𝑊 (𝐵𝑠) ≤ (𝐴𝑠)) → (𝐴(𝐿𝑊)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
17540, 174pm2.61dan 828 1 (𝜑 → (𝐴(𝐿𝑊)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  ifcif 4036  {csn 4125  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643   ↾ cres 5040  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   ↑𝑚 cmap 7744  Xcixp 7794  Fincfn 7841  supcsup 8229  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820  +∞cpnf 9950  ℝ*cxr 9952   < clt 9953   ≤ cle 9954   − cmin 10145  ℕcn 10897  ℝ+crp 11708  [,)cico 12048  [,]cicc 12049  ∏cprod 14474  volcvol 23039  Σ^csumge0 39255 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-prod 14475  df-rest 15906  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-top 20521  df-bases 20522  df-topon 20523  df-cmp 21000  df-ovol 23040  df-vol 23041  df-sumge0 39256 This theorem is referenced by:  hoidmvle  39490
 Copyright terms: Public domain W3C validator