hoidmvlelem5 - Mathbox for Glauco Siliprandi

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Theorem hoidmvlelem5 46614

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, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))
hoidmvlelem5.f	⊢ (𝜑 → 𝑋 ∈ Fin)
hoidmvlelem5.y	⊢ (𝜑 → 𝑌 ⊆ 𝑋)
hoidmvlelem5.z	⊢ (𝜑 → 𝑍 ∈ (𝑋 ∖ 𝑌))
hoidmvlelem5.w	⊢ 𝑊 = (𝑌 ∪ {𝑍})
hoidmvlelem5.a	⊢ (𝜑 → 𝐴:𝑊⟶ℝ)
hoidmvlelem5.b	⊢ (𝜑 → 𝐵:𝑊⟶ℝ)
hoidmvlelem5.c	⊢ (𝜑 → 𝐶:ℕ⟶(ℝ ↑_m 𝑊))
hoidmvlelem5.d	⊢ (𝜑 → 𝐷:ℕ⟶(ℝ ↑_m 𝑊))
hoidmvlelem5.i	⊢ (𝜑 → ∀𝑒 ∈ (ℝ ↑_m 𝑌)∀𝑓 ∈ (ℝ ↑_m 𝑌)∀𝑔 ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ)∀ℎ ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ)(X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑌)𝑓) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
hoidmvlelem5.s	⊢ (𝜑 → X𝑘 ∈ 𝑊 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
hoidmvlelem5.n	⊢ (𝜑 → 𝑌 ≠ ∅)

**Assertion**
Ref	Expression
hoidmvlelem5	⊢ (𝜑 → (𝐴(𝐿‘𝑊)𝐵) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))))

Distinct variable groups: 𝐴,𝑎,𝑏,ℎ,𝑗,𝑘 𝐶,ℎ,𝑗,𝑥 ℎ,𝑊,𝑗,𝑥,𝑘 ℎ,𝐿,𝑗 𝑍,𝑎,𝑏,ℎ,𝑗,𝑥 ℎ,𝑌,𝑗 𝐶,𝑔 𝑊,𝑎,𝑏 𝐷,ℎ,𝑗 𝑘,𝑍,𝑥 𝜑,ℎ,𝑗 𝐵,𝑎,𝑏,ℎ,𝑗 𝑌,𝑎,𝑏,𝑥 𝜑,𝑎,𝑏,𝑘,𝑥 𝐵,𝑘 𝑔,𝑍 𝑔,𝑊,𝑘 𝑒,𝑌,𝑓,𝑔,𝑘,ℎ,𝑗 𝐵,𝑓,𝑔 𝐷,𝑘 𝑒,𝐿,𝑓,𝑔 𝐶,𝑎,𝑏,𝑘 𝐷,𝑎,𝑏,𝑥 𝐷,𝑔 𝐴,𝑒,𝑓,𝑔 𝑥,𝐵 𝑥,𝐴 𝐿,𝑎,𝑏,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,ℎ,𝑗,𝑘,𝑎,𝑏) 𝑍(𝑒,𝑓)

Proof of Theorem hoidmvlelem5 Dummy variables 𝑤 𝑧 𝑟 𝑐 𝑠 𝑑 𝑙 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1914	. . . . 5 ⊢ Ⅎ𝑠𝜑
2		nfre1 3285	. . . . 5 ⊢ Ⅎ𝑠∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)
3	1, 2	nfan 1899	. . . 4 ⊢ Ⅎ𝑠(𝜑 ∧ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠))
4		hoidmvlelem5.l	. . . 4 ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑_m 𝑥), 𝑏 ∈ (ℝ ↑_m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))
5		hoidmvlelem5.w	. . . . . 6 ⊢ 𝑊 = (𝑌 ∪ {𝑍})
6		hoidmvlelem5.f	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ Fin)
7		hoidmvlelem5.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
8		ssfi 9213	. . . . . . . 8 ⊢ ((𝑋 ∈ Fin ∧ 𝑌 ⊆ 𝑋) → 𝑌 ∈ Fin)
9	6, 7, 8	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ Fin)
10		snfi 9083	. . . . . . . 8 ⊢ {𝑍} ∈ Fin
11	10	a1i 11	. . . . . . 7 ⊢ (𝜑 → {𝑍} ∈ Fin)
12		unfi 9211	. . . . . . 7 ⊢ ((𝑌 ∈ Fin ∧ {𝑍} ∈ Fin) → (𝑌 ∪ {𝑍}) ∈ Fin)
13	9, 11, 12	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝑌 ∪ {𝑍}) ∈ Fin)
14	5, 13	eqeltrid 2845	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ Fin)
15	14	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) → 𝑊 ∈ Fin)
16		hoidmvlelem5.a	. . . . 5 ⊢ (𝜑 → 𝐴:𝑊⟶ℝ)
17	16	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) → 𝐴:𝑊⟶ℝ)
18		hoidmvlelem5.b	. . . . 5 ⊢ (𝜑 → 𝐵:𝑊⟶ℝ)
19	18	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) → 𝐵:𝑊⟶ℝ)
20		simpr 484	. . . 4 ⊢ ((𝜑 ∧ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) → ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠))
21	3, 4, 15, 17, 19, 20	hoidmvval0 46602	. . 3 ⊢ ((𝜑 ∧ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) → (𝐴(𝐿‘𝑊)𝐵) = 0)
22		nnex 12272	. . . . . 6 ⊢ ℕ ∈ V
23	22	a1i 11	. . . . 5 ⊢ (𝜑 → ℕ ∈ V)
24		icossicc 13476	. . . . . . 7 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
25	14	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑊 ∈ Fin)
26		hoidmvlelem5.c	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶:ℕ⟶(ℝ ↑_m 𝑊))
27	26	ffvelcdmda 7104	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ (ℝ ↑_m 𝑊))
28		elmapi 8889	. . . . . . . . 9 ⊢ ((𝐶‘𝑗) ∈ (ℝ ↑_m 𝑊) → (𝐶‘𝑗):𝑊⟶ℝ)
29	27, 28	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗):𝑊⟶ℝ)
30		hoidmvlelem5.d	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷:ℕ⟶(ℝ ↑_m 𝑊))
31	30	ffvelcdmda 7104	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ (ℝ ↑_m 𝑊))
32		elmapi 8889	. . . . . . . . 9 ⊢ ((𝐷‘𝑗) ∈ (ℝ ↑_m 𝑊) → (𝐷‘𝑗):𝑊⟶ℝ)
33	31, 32	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗):𝑊⟶ℝ)
34	4, 25, 29, 33	hoidmvcl 46597	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)) ∈ (0[,)+∞))
35	24, 34	sselid 3981	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)) ∈ (0[,]+∞))
36	35	fmpttd 7135	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗))):ℕ⟶(0[,]+∞))
37	23, 36	sge0ge0 46399	. . . 4 ⊢ (𝜑 → 0 ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))))
38	37	adantr 480	. . 3 ⊢ ((𝜑 ∧ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) → 0 ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))))
39	21, 38	eqbrtrd 5165	. 2 ⊢ ((𝜑 ∧ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) → (𝐴(𝐿‘𝑊)𝐵) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))))
40		icossxr 13472	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℝ^*
41	4, 14, 16, 18	hoidmvcl 46597	. . . . . . 7 ⊢ (𝜑 → (𝐴(𝐿‘𝑊)𝐵) ∈ (0[,)+∞))
42	40, 41	sselid 3981	. . . . . 6 ⊢ (𝜑 → (𝐴(𝐿‘𝑊)𝐵) ∈ ℝ^*)
43	42	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞) → (𝐴(𝐿‘𝑊)𝐵) ∈ ℝ^*)
44	23, 36	sge0xrcl 46400	. . . . . 6 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ^*)
45	44	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ^*)
46		rge0ssre 13496	. . . . . . . . 9 ⊢ (0[,)+∞) ⊆ ℝ
47	46, 41	sselid 3981	. . . . . . . 8 ⊢ (𝜑 → (𝐴(𝐿‘𝑊)𝐵) ∈ ℝ)
48		ltpnf 13162	. . . . . . . 8 ⊢ ((𝐴(𝐿‘𝑊)𝐵) ∈ ℝ → (𝐴(𝐿‘𝑊)𝐵) < +∞)
49	47, 48	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐴(𝐿‘𝑊)𝐵) < +∞)
50	49	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞) → (𝐴(𝐿‘𝑊)𝐵) < +∞)
51		id 22	. . . . . . . 8 ⊢ ((Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞ → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞)
52	51	eqcomd 2743	. . . . . . 7 ⊢ ((Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞ → +∞ = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))))
53	52	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞) → +∞ = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))))
54	50, 53	breqtrd 5169	. . . . 5 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞) → (𝐴(𝐿‘𝑊)𝐵) < (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))))
55	43, 45, 54	xrltled 13192	. . . 4 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞) → (𝐴(𝐿‘𝑊)𝐵) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))))
56	55	adantlr 715	. . 3 ⊢ (((𝜑 ∧ ¬ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞) → (𝐴(𝐿‘𝑊)𝐵) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))))
57		simpll 767	. . . 4 ⊢ (((𝜑 ∧ ¬ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) ∧ ¬ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞) → 𝜑)
58		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) → ¬ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠))
59	16	ffvelcdmda 7104	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑊) → (𝐴‘𝑠) ∈ ℝ)
60	18	ffvelcdmda 7104	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑊) → (𝐵‘𝑠) ∈ ℝ)
61	59, 60	ltnled 11408	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑊) → ((𝐴‘𝑠) < (𝐵‘𝑠) ↔ ¬ (𝐵‘𝑠) ≤ (𝐴‘𝑠)))
62	61	ralbidva 3176	. . . . . . . 8 ⊢ (𝜑 → (∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠) ↔ ∀𝑠 ∈ 𝑊 ¬ (𝐵‘𝑠) ≤ (𝐴‘𝑠)))
63		ralnex 3072	. . . . . . . . 9 ⊢ (∀𝑠 ∈ 𝑊 ¬ (𝐵‘𝑠) ≤ (𝐴‘𝑠) ↔ ¬ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠))
64	63	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (∀𝑠 ∈ 𝑊 ¬ (𝐵‘𝑠) ≤ (𝐴‘𝑠) ↔ ¬ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)))
65	62, 64	bitrd 279	. . . . . . 7 ⊢ (𝜑 → (∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠) ↔ ¬ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)))
66	65	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) → (∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠) ↔ ¬ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)))
67	58, 66	mpbird 257	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) → ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠))
68	67	adantr 480	. . . 4 ⊢ (((𝜑 ∧ ¬ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) ∧ ¬ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞) → ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠))
69		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞) → ¬ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞)
70	22	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞) → ℕ ∈ V)
71	36	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞) → (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗))):ℕ⟶(0[,]+∞))
72	70, 71	sge0repnf 46401	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞) → ((Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ ↔ ¬ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞))
73	69, 72	mpbird 257	. . . . 5 ⊢ ((𝜑 ∧ ¬ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ)
74	73	adantlr 715	. . . 4 ⊢ (((𝜑 ∧ ¬ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) ∧ ¬ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ)
75		simpll 767	. . . . . . 7 ⊢ ((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → (𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)))
76		fveq2 6906	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑖 → (𝐶‘𝑗) = (𝐶‘𝑖))
77		fveq2 6906	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑖 → (𝐷‘𝑗) = (𝐷‘𝑖))
78	76, 77	oveq12d 7449	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑖 → ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)) = ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))
79	78	cbvmptv 5255	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗))) = (𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))
80	79	fveq2i 6909	. . . . . . . . . 10 ⊢ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = (Σ^{^}‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖))))
81	80	eleq1i 2832	. . . . . . . . 9 ⊢ ((Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ ↔ (Σ^{^}‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ)
82	81	biimpi 216	. . . . . . . 8 ⊢ ((Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ → (Σ^{^}‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ)
83	82	ad2antlr 727	. . . . . . 7 ⊢ ((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → (Σ^{^}‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ)
84		simpr 484	. . . . . . 7 ⊢ ((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → 𝑟 ∈ ℝ⁺)
85	6	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧ (Σ^{^}‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → 𝑋 ∈ Fin)
86	7	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧ (Σ^{^}‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → 𝑌 ⊆ 𝑋)
87		hoidmvlelem5.n	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ≠ ∅)
88	87	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧ (Σ^{^}‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → 𝑌 ≠ ∅)
89		hoidmvlelem5.z	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ (𝑋 ∖ 𝑌))
90	89	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧ (Σ^{^}‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → 𝑍 ∈ (𝑋 ∖ 𝑌))
91	16	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧ (Σ^{^}‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → 𝐴:𝑊⟶ℝ)
92	18	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧ (Σ^{^}‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → 𝐵:𝑊⟶ℝ)
93		fveq2 6906	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑘 → (𝐴‘𝑠) = (𝐴‘𝑘))
94		fveq2 6906	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑘 → (𝐵‘𝑠) = (𝐵‘𝑘))
95	93, 94	breq12d 5156	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑘 → ((𝐴‘𝑠) < (𝐵‘𝑠) ↔ (𝐴‘𝑘) < (𝐵‘𝑘)))
96	95	cbvralvw 3237	. . . . . . . . . . . 12 ⊢ (∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠) ↔ ∀𝑘 ∈ 𝑊 (𝐴‘𝑘) < (𝐵‘𝑘))
97	96	biimpi 216	. . . . . . . . . . 11 ⊢ (∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠) → ∀𝑘 ∈ 𝑊 (𝐴‘𝑘) < (𝐵‘𝑘))
98	97	adantr 480	. . . . . . . . . 10 ⊢ ((∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠) ∧ 𝑘 ∈ 𝑊) → ∀𝑘 ∈ 𝑊 (𝐴‘𝑘) < (𝐵‘𝑘))
99		simpr 484	. . . . . . . . . 10 ⊢ ((∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠) ∧ 𝑘 ∈ 𝑊) → 𝑘 ∈ 𝑊)
100		rspa 3248	. . . . . . . . . 10 ⊢ ((∀𝑘 ∈ 𝑊 (𝐴‘𝑘) < (𝐵‘𝑘) ∧ 𝑘 ∈ 𝑊) → (𝐴‘𝑘) < (𝐵‘𝑘))
101	98, 99, 100	syl2anc 584	. . . . . . . . 9 ⊢ ((∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠) ∧ 𝑘 ∈ 𝑊) → (𝐴‘𝑘) < (𝐵‘𝑘))
102	101	ad5ant25 762	. . . . . . . 8 ⊢ (((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧ (Σ^{^}‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ 𝑊) → (𝐴‘𝑘) < (𝐵‘𝑘))
103	26	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧ (Σ^{^}‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → 𝐶:ℕ⟶(ℝ ↑_m 𝑊))
104	30	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧ (Σ^{^}‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → 𝐷:ℕ⟶(ℝ ↑_m 𝑊))
105	81	biimpri 228	. . . . . . . . 9 ⊢ ((Σ^{^}‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ)
106	105	ad2antlr 727	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧ (Σ^{^}‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ)
107		fveq1 6905	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑐 → (𝑑‘𝑖) = (𝑐‘𝑖))
108	107	breq1d 5153	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑐 → ((𝑑‘𝑖) ≤ 𝑥 ↔ (𝑐‘𝑖) ≤ 𝑥))
109	108, 107	ifbieq1d 4550	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑐 → if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥) = if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥))
110	107, 109	ifeq12d 4547	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑐 → if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)) = if(𝑖 ∈ 𝑌, (𝑐‘𝑖), if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥)))
111	110	mpteq2dv 5244	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑐 → (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥))) = (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑐‘𝑖), if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥))))
112		eleq1w 2824	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑗 → (𝑖 ∈ 𝑌 ↔ 𝑗 ∈ 𝑌))
113		fveq2 6906	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑗 → (𝑐‘𝑖) = (𝑐‘𝑗))
114	113	breq1d 5153	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑗 → ((𝑐‘𝑖) ≤ 𝑥 ↔ (𝑐‘𝑗) ≤ 𝑥))
115	114, 113	ifbieq1d 4550	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑗 → if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥) = if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥))
116	112, 113, 115	ifbieq12d 4554	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → if(𝑖 ∈ 𝑌, (𝑐‘𝑖), if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥)) = if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))
117	116	cbvmptv 5255	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑐‘𝑖), if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥))) = (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))
118	117	a1i 11	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑐 → (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑐‘𝑖), if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥))) = (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥))))
119	111, 118	eqtrd 2777	. . . . . . . . . 10 ⊢ (𝑑 = 𝑐 → (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥))) = (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥))))
120	119	cbvmptv 5255	. . . . . . . . 9 ⊢ (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))) = (𝑐 ∈ (ℝ ↑_m 𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥))))
121	120	mpteq2i 5247	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥))))) = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑_m 𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))))
122		eqid 2737	. . . . . . . 8 ⊢ ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌))
123		simpr 484	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧ (Σ^{^}‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → 𝑟 ∈ ℝ⁺)
124		oveq1 7438	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → (𝑤 − (𝐴‘𝑍)) = (𝑧 − (𝐴‘𝑍)))
125	124	oveq2d 7447	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → (((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) · (𝑤 − (𝐴‘𝑍))) = (((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) · (𝑧 − (𝐴‘𝑍))))
126		breq2 5147	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = 𝑥 → ((𝑑‘𝑖) ≤ 𝑤 ↔ (𝑑‘𝑖) ≤ 𝑥))
127		eqidd 2738	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = 𝑥 → (𝑑‘𝑖) = (𝑑‘𝑖))
128		id 22	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = 𝑥 → 𝑤 = 𝑥)
129	126, 127, 128	ifbieq12d 4554	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 = 𝑥 → if((𝑑‘𝑖) ≤ 𝑤, (𝑑‘𝑖), 𝑤) = if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥))
130	129	ifeq2d 4546	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 = 𝑥 → if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑤, (𝑑‘𝑖), 𝑤)) = if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))
131	130	mpteq2dv 5244	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = 𝑥 → (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑤, (𝑑‘𝑖), 𝑤))) = (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥))))
132	131	mpteq2dv 5244	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = 𝑥 → (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑤, (𝑑‘𝑖), 𝑤)))) = (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))
133	132	cbvmptv 5255	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑤, (𝑑‘𝑖), 𝑤))))) = (𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))
134	133	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑤, (𝑑‘𝑖), 𝑤))))) = (𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥))))))
135		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑧 → 𝑤 = 𝑧)
136	134, 135	fveq12d 6913	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑧 → ((𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑤, (𝑑‘𝑖), 𝑤)))))‘𝑤) = ((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))‘𝑧))
137	136	fveq1d 6908	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑧 → (((𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑤, (𝑑‘𝑖), 𝑤)))))‘𝑤)‘(𝐷‘𝑙)) = (((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))‘𝑧)‘(𝐷‘𝑙)))
138	137	oveq2d 7447	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑧 → ((𝐶‘𝑙)(𝐿‘𝑊)(((𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑤, (𝑑‘𝑖), 𝑤)))))‘𝑤)‘(𝐷‘𝑙))) = ((𝐶‘𝑙)(𝐿‘𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))‘𝑧)‘(𝐷‘𝑙))))
139	138	mpteq2dv 5244	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑧 → (𝑙 ∈ ℕ ↦ ((𝐶‘𝑙)(𝐿‘𝑊)(((𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑤, (𝑑‘𝑖), 𝑤)))))‘𝑤)‘(𝐷‘𝑙)))) = (𝑙 ∈ ℕ ↦ ((𝐶‘𝑙)(𝐿‘𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))‘𝑧)‘(𝐷‘𝑙)))))
140		fveq2 6906	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 = 𝑗 → (𝐶‘𝑙) = (𝐶‘𝑗))
141		2fveq3 6911	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 = 𝑗 → (((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))‘𝑧)‘(𝐷‘𝑙)) = (((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))‘𝑧)‘(𝐷‘𝑗)))
142	140, 141	oveq12d 7449	. . . . . . . . . . . . . . 15 ⊢ (𝑙 = 𝑗 → ((𝐶‘𝑙)(𝐿‘𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))‘𝑧)‘(𝐷‘𝑙))) = ((𝐶‘𝑗)(𝐿‘𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))‘𝑧)‘(𝐷‘𝑗))))
143	142	cbvmptv 5255	. . . . . . . . . . . . . 14 ⊢ (𝑙 ∈ ℕ ↦ ((𝐶‘𝑙)(𝐿‘𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))‘𝑧)‘(𝐷‘𝑙)))) = (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))‘𝑧)‘(𝐷‘𝑗))))
144	143	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑧 → (𝑙 ∈ ℕ ↦ ((𝐶‘𝑙)(𝐿‘𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))‘𝑧)‘(𝐷‘𝑙)))) = (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))‘𝑧)‘(𝐷‘𝑗)))))
145	139, 144	eqtrd 2777	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → (𝑙 ∈ ℕ ↦ ((𝐶‘𝑙)(𝐿‘𝑊)(((𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑤, (𝑑‘𝑖), 𝑤)))))‘𝑤)‘(𝐷‘𝑙)))) = (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))‘𝑧)‘(𝐷‘𝑗)))))
146	145	fveq2d 6910	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → (Σ^{^}‘(𝑙 ∈ ℕ ↦ ((𝐶‘𝑙)(𝐿‘𝑊)(((𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑤, (𝑑‘𝑖), 𝑤)))))‘𝑤)‘(𝐷‘𝑙))))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))‘𝑧)‘(𝐷‘𝑗))))))
147	146	oveq2d 7447	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → ((1 + 𝑟) · (Σ^{^}‘(𝑙 ∈ ℕ ↦ ((𝐶‘𝑙)(𝐿‘𝑊)(((𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑤, (𝑑‘𝑖), 𝑤)))))‘𝑤)‘(𝐷‘𝑙)))))) = ((1 + 𝑟) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))‘𝑧)‘(𝐷‘𝑗)))))))
148	125, 147	breq12d 5156	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → ((((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) · (𝑤 − (𝐴‘𝑍))) ≤ ((1 + 𝑟) · (Σ^{^}‘(𝑙 ∈ ℕ ↦ ((𝐶‘𝑙)(𝐿‘𝑊)(((𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑤, (𝑑‘𝑖), 𝑤)))))‘𝑤)‘(𝐷‘𝑙)))))) ↔ (((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝑟) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))‘𝑧)‘(𝐷‘𝑗))))))))
149	148	cbvrabv 3447	. . . . . . . 8 ⊢ {𝑤 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) · (𝑤 − (𝐴‘𝑍))) ≤ ((1 + 𝑟) · (Σ^{^}‘(𝑙 ∈ ℕ ↦ ((𝐶‘𝑙)(𝐿‘𝑊)(((𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑤, (𝑑‘𝑖), 𝑤)))))‘𝑤)‘(𝐷‘𝑙))))))} = {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝑟) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))‘𝑧)‘(𝐷‘𝑗))))))}
150		eqid 2737	. . . . . . . 8 ⊢ sup({𝑤 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) · (𝑤 − (𝐴‘𝑍))) ≤ ((1 + 𝑟) · (Σ^{^}‘(𝑙 ∈ ℕ ↦ ((𝐶‘𝑙)(𝐿‘𝑊)(((𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑤, (𝑑‘𝑖), 𝑤)))))‘𝑤)‘(𝐷‘𝑙))))))}, ℝ, < ) = sup({𝑤 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) · (𝑤 − (𝐴‘𝑍))) ≤ ((1 + 𝑟) · (Σ^{^}‘(𝑙 ∈ ℕ ↦ ((𝐶‘𝑙)(𝐿‘𝑊)(((𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑_m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑤, (𝑑‘𝑖), 𝑤)))))‘𝑤)‘(𝐷‘𝑙))))))}, ℝ, < )
151		hoidmvlelem5.i	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑒 ∈ (ℝ ↑_m 𝑌)∀𝑓 ∈ (ℝ ↑_m 𝑌)∀𝑔 ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ)∀ℎ ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ)(X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑌)𝑓) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
152	151	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧ (Σ^{^}‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → ∀𝑒 ∈ (ℝ ↑_m 𝑌)∀𝑓 ∈ (ℝ ↑_m 𝑌)∀𝑔 ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ)∀ℎ ∈ ((ℝ ↑_m 𝑌) ↑_m ℕ)(X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑌)𝑓) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
153		hoidmvlelem5.s	. . . . . . . . 9 ⊢ (𝜑 → X𝑘 ∈ 𝑊 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
154	153	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧ (Σ^{^}‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → X𝑘 ∈ 𝑊 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
155	4, 85, 86, 88, 90, 5, 91, 92, 102, 103, 104, 106, 121, 122, 123, 149, 150, 152, 154	hoidmvlelem4 46613	. . . . . . 7 ⊢ ((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧ (Σ^{^}‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → (𝐴(𝐿‘𝑊)𝐵) ≤ ((1 + 𝑟) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗))))))
156	75, 83, 84, 155	syl21anc 838	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → (𝐴(𝐿‘𝑊)𝐵) ≤ ((1 + 𝑟) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗))))))
157	156	ralrimiva 3146	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ) → ∀𝑟 ∈ ℝ⁺ (𝐴(𝐿‘𝑊)𝐵) ≤ ((1 + 𝑟) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗))))))
158		nfv 1914	. . . . . 6 ⊢ Ⅎ𝑟((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ)
159	42	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ) → (𝐴(𝐿‘𝑊)𝐵) ∈ ℝ^*)
160		0xr 11308	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
161	160	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ) → 0 ∈ ℝ^*)
162		pnfxr 11315	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
163	162	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ) → +∞ ∈ ℝ^*)
164	44	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ^*)
165	37	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ) → 0 ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))))
166		ltpnf 13162	. . . . . . . 8 ⊢ ((Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) < +∞)
167	166	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) < +∞)
168	161, 163, 164, 165, 167	elicod 13437	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ (0[,)+∞))
169	158, 159, 168	xralrple2 45365	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ) → ((𝐴(𝐿‘𝑊)𝐵) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ↔ ∀𝑟 ∈ ℝ⁺ (𝐴(𝐿‘𝑊)𝐵) ≤ ((1 + 𝑟) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))))))
170	157, 169	mpbird 257	. . . 4 ⊢ (((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ) → (𝐴(𝐿‘𝑊)𝐵) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))))
171	57, 68, 74, 170	syl21anc 838	. . 3 ⊢ (((𝜑 ∧ ¬ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) ∧ ¬ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞) → (𝐴(𝐿‘𝑊)𝐵) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))))
172	56, 171	pm2.61dan 813	. 2 ⊢ ((𝜑 ∧ ¬ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) → (𝐴(𝐿‘𝑊)𝐵) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))))
173	39, 172	pm2.61dan 813	1 ⊢ (𝜑 → (𝐴(𝐿‘𝑊)𝐵) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3436 Vcvv 3480 ∖ cdif 3948 ∪ cun 3949 ⊆ wss 3951 ∅c0 4333 ifcif 4525 {csn 4626 ∪ ciun 4991 class class class wbr 5143 ↦ cmpt 5225 ↾ cres 5687 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ↑_m cmap 8866 Xcixp 8937 Fincfn 8985 supcsup 9480 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 +∞cpnf 11292 ℝ^*cxr 11294 < clt 11295 ≤ cle 11296 − cmin 11492 ℕcn 12266 ℝ⁺crp 13034 [,)cico 13389 [,]cicc 13390 ∏cprod 15939 volcvol 25498 Σ^{^}csumge0 46377

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-rlim 15525 df-sum 15723 df-prod 15940 df-rest 17467 df-topgen 17488 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-top 22900 df-topon 22917 df-bases 22953 df-cmp 23395 df-ovol 25499 df-vol 25500 df-sumge0 46378

This theorem is referenced by: hoidmvle 46615

W3C validator