hoidmvlelem2 - Mathbox for Glauco Siliprandi

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Theorem hoidmvlelem2 45611

Description: This is the contradiction proven in step (d) in the proof of Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)

**Hypotheses**
Ref	Expression
hoidmvlelem2.l	⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑_m 𝑥), 𝑏 ∈ (ℝ ↑_m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))
hoidmvlelem2.x	⊢ (𝜑 → 𝑋 ∈ Fin)
hoidmvlelem2.y	⊢ (𝜑 → 𝑌 ⊆ 𝑋)
hoidmvlelem2.z	⊢ (𝜑 → 𝑍 ∈ (𝑋 ∖ 𝑌))
hoidmvlelem2.w	⊢ 𝑊 = (𝑌 ∪ {𝑍})
hoidmvlelem2.a	⊢ (𝜑 → 𝐴:𝑊⟶ℝ)
hoidmvlelem2.b	⊢ (𝜑 → 𝐵:𝑊⟶ℝ)
hoidmvlelem2.c	⊢ (𝜑 → 𝐶:ℕ⟶(ℝ ↑_m 𝑊))
hoidmvlelem2.f	⊢ 𝐹 = (𝑦 ∈ 𝑌 ↦ 0)
hoidmvlelem2.j	⊢ 𝐽 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹))
hoidmvlelem2.d	⊢ (𝜑 → 𝐷:ℕ⟶(ℝ ↑_m 𝑊))
hoidmvlelem2.k	⊢ 𝐾 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹))
hoidmvlelem2.r	⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ)
hoidmvlelem2.h	⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑_m 𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))))
hoidmvlelem2.g	⊢ 𝐺 = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌))
hoidmvlelem2.e	⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
hoidmvlelem2.u	⊢ 𝑈 = {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))}
hoidmvlelem2.su	⊢ (𝜑 → 𝑆 ∈ 𝑈)
hoidmvlelem2.sb	⊢ (𝜑 → 𝑆 < (𝐵‘𝑍))
hoidmvlelem2.p	⊢ 𝑃 = (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
hoidmvlelem2.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
hoidmvlelem2.le	⊢ (𝜑 → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗)))
hoidmvlelem2.O	⊢ 𝑂 = ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ↦ ((𝐷‘𝑖)‘𝑍))
hoidmvlelem2.v	⊢ 𝑉 = ({(𝐵‘𝑍)} ∪ 𝑂)
hoidmvlelem2.q	⊢ 𝑄 = inf(𝑉, ℝ, < )

**Assertion**
Ref	Expression
hoidmvlelem2	⊢ (𝜑 → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)

Distinct variable groups: 𝐴,𝑎,𝑏,𝑘 𝑧,𝐴 𝐵,𝑎,𝑏,𝑘 𝑦,𝐵 𝑧,𝐵 𝐶,𝑎,𝑏,𝑗,𝑘 𝐶,𝑖,𝑗 𝑧,𝐶,𝑗 𝐷,𝑎,𝑏,𝑗,𝑘 𝐷,𝑐,𝑗,𝑘 𝐷,𝑖 𝑦,𝐷,𝑗 𝑧,𝐷 𝑧,𝐸 𝐹,𝑎,𝑏,𝑘 𝑧,𝐺 𝐻,𝑎,𝑏,𝑘 𝑧,𝐻 𝐽,𝑎,𝑏,𝑘 𝐾,𝑎,𝑏,𝑘 𝑧,𝐿 𝑖,𝑀,𝑗 𝑖,𝑂 𝑃,𝑎,𝑏,𝑘,𝑥 𝑄,𝑎,𝑏,𝑗,𝑘,𝑥 𝑄,𝑐,𝑥 𝑢,𝑄 𝑧,𝑄 𝑆,𝑎,𝑏,𝑗,𝑘,𝑥 𝑆,𝑐 𝑆,𝑖,𝑥 𝑢,𝑆 𝑧,𝑆 𝑢,𝑈 𝑥,𝑉,𝑦 𝑊,𝑎,𝑏,𝑗,𝑘,𝑥 𝑊,𝑐 𝑧,𝑊 𝑌,𝑎,𝑏,𝑗,𝑘,𝑥 𝑌,𝑐 𝑦,𝑌 𝑍,𝑐,𝑗,𝑘,𝑥 𝑖,𝑍 𝑦,𝑍 𝑧,𝑍 𝜑,𝑎,𝑏,𝑗,𝑘,𝑥 𝜑,𝑐 𝜑,𝑖 𝜑,𝑦 Allowed substitution hints: 𝜑(𝑧,𝑢) 𝐴(𝑥,𝑦,𝑢,𝑖,𝑗,𝑐) 𝐵(𝑥,𝑢,𝑖,𝑗,𝑐) 𝐶(𝑥,𝑦,𝑢,𝑐) 𝐷(𝑥,𝑢) 𝑃(𝑦,𝑧,𝑢,𝑖,𝑗,𝑐) 𝑄(𝑦,𝑖) 𝑆(𝑦) 𝑈(𝑥,𝑦,𝑧,𝑖,𝑗,𝑘,𝑎,𝑏,𝑐) 𝐸(𝑥,𝑦,𝑢,𝑖,𝑗,𝑘,𝑎,𝑏,𝑐) 𝐹(𝑥,𝑦,𝑧,𝑢,𝑖,𝑗,𝑐) 𝐺(𝑥,𝑦,𝑢,𝑖,𝑗,𝑘,𝑎,𝑏,𝑐) 𝐻(𝑥,𝑦,𝑢,𝑖,𝑗,𝑐) 𝐽(𝑥,𝑦,𝑧,𝑢,𝑖,𝑗,𝑐) 𝐾(𝑥,𝑦,𝑧,𝑢,𝑖,𝑗,𝑐) 𝐿(𝑥,𝑦,𝑢,𝑖,𝑗,𝑘,𝑎,𝑏,𝑐) 𝑀(𝑥,𝑦,𝑧,𝑢,𝑘,𝑎,𝑏,𝑐) 𝑂(𝑥,𝑦,𝑧,𝑢,𝑗,𝑘,𝑎,𝑏,𝑐) 𝑉(𝑧,𝑢,𝑖,𝑗,𝑘,𝑎,𝑏,𝑐) 𝑊(𝑦,𝑢,𝑖) 𝑋(𝑥,𝑦,𝑧,𝑢,𝑖,𝑗,𝑘,𝑎,𝑏,𝑐) 𝑌(𝑧,𝑢,𝑖) 𝑍(𝑢,𝑎,𝑏)

Proof of Theorem hoidmvlelem2 Dummy variable 𝑙 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hoidmvlelem2.a	. . . . . . 7 ⊢ (𝜑 → 𝐴:𝑊⟶ℝ)
2		hoidmvlelem2.z	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ (𝑋 ∖ 𝑌))
3		snidg 4662	. . . . . . . . . 10 ⊢ (𝑍 ∈ (𝑋 ∖ 𝑌) → 𝑍 ∈ {𝑍})
4	2, 3	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ {𝑍})
5		elun2 4177	. . . . . . . . 9 ⊢ (𝑍 ∈ {𝑍} → 𝑍 ∈ (𝑌 ∪ {𝑍}))
6	4, 5	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ (𝑌 ∪ {𝑍}))
7		hoidmvlelem2.w	. . . . . . . 8 ⊢ 𝑊 = (𝑌 ∪ {𝑍})
8	6, 7	eleqtrrdi 2843	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
9	1, 8	ffvelcdmd 7087	. . . . . 6 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ)
10		hoidmvlelem2.b	. . . . . . 7 ⊢ (𝜑 → 𝐵:𝑊⟶ℝ)
11	10, 8	ffvelcdmd 7087	. . . . . 6 ⊢ (𝜑 → (𝐵‘𝑍) ∈ ℝ)
12		hoidmvlelem2.v	. . . . . . . 8 ⊢ 𝑉 = ({(𝐵‘𝑍)} ∪ 𝑂)
13	11	snssd 4812	. . . . . . . . 9 ⊢ (𝜑 → {(𝐵‘𝑍)} ⊆ ℝ)
14		hoidmvlelem2.O	. . . . . . . . . 10 ⊢ 𝑂 = ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ↦ ((𝐷‘𝑖)‘𝑍))
15		nfv 1916	. . . . . . . . . . 11 ⊢ Ⅎ𝑖𝜑
16		eqid 2731	. . . . . . . . . . 11 ⊢ (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ↦ ((𝐷‘𝑖)‘𝑍)) = (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ↦ ((𝐷‘𝑖)‘𝑍))
17		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))}) → 𝜑)
18		fz1ssnn 13537	. . . . . . . . . . . . . 14 ⊢ (1...𝑀) ⊆ ℕ
19		elrabi 3677	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} → 𝑖 ∈ (1...𝑀))
20	18, 19	sselid 3980	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} → 𝑖 ∈ ℕ)
21	20	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))}) → 𝑖 ∈ ℕ)
22		eleq1w 2815	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑖 → (𝑗 ∈ ℕ ↔ 𝑖 ∈ ℕ))
23	22	anbi2d 628	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑖 → ((𝜑 ∧ 𝑗 ∈ ℕ) ↔ (𝜑 ∧ 𝑖 ∈ ℕ)))
24		fveq2 6891	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑖 → (𝐷‘𝑗) = (𝐷‘𝑖))
25	24	fveq1d 6893	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑖 → ((𝐷‘𝑗)‘𝑍) = ((𝐷‘𝑖)‘𝑍))
26	25	eleq1d 2817	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑖 → (((𝐷‘𝑗)‘𝑍) ∈ ℝ ↔ ((𝐷‘𝑖)‘𝑍) ∈ ℝ))
27	23, 26	imbi12d 344	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑖 → (((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐷‘𝑗)‘𝑍) ∈ ℝ) ↔ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝐷‘𝑖)‘𝑍) ∈ ℝ)))
28		hoidmvlelem2.d	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐷:ℕ⟶(ℝ ↑_m 𝑊))
29	28	ffvelcdmda 7086	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ (ℝ ↑_m 𝑊))
30		elmapi 8847	. . . . . . . . . . . . . . 15 ⊢ ((𝐷‘𝑗) ∈ (ℝ ↑_m 𝑊) → (𝐷‘𝑗):𝑊⟶ℝ)
31	29, 30	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗):𝑊⟶ℝ)
32	8	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑍 ∈ 𝑊)
33	31, 32	ffvelcdmd 7087	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐷‘𝑗)‘𝑍) ∈ ℝ)
34	27, 33	chvarvv 2001	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝐷‘𝑖)‘𝑍) ∈ ℝ)
35	17, 21, 34	syl2anc 583	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))}) → ((𝐷‘𝑖)‘𝑍) ∈ ℝ)
36	15, 16, 35	rnmptssd 44194	. . . . . . . . . 10 ⊢ (𝜑 → ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ↦ ((𝐷‘𝑖)‘𝑍)) ⊆ ℝ)
37	14, 36	eqsstrid 4030	. . . . . . . . 9 ⊢ (𝜑 → 𝑂 ⊆ ℝ)
38	13, 37	unssd 4186	. . . . . . . 8 ⊢ (𝜑 → ({(𝐵‘𝑍)} ∪ 𝑂) ⊆ ℝ)
39	12, 38	eqsstrid 4030	. . . . . . 7 ⊢ (𝜑 → 𝑉 ⊆ ℝ)
40		hoidmvlelem2.q	. . . . . . . 8 ⊢ 𝑄 = inf(𝑉, ℝ, < )
41		ltso 11299	. . . . . . . . . 10 ⊢ < Or ℝ
42	41	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → < Or ℝ)
43		snfi 9048	. . . . . . . . . . . 12 ⊢ {(𝐵‘𝑍)} ∈ Fin
44	43	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → {(𝐵‘𝑍)} ∈ Fin)
45		fzfi 13942	. . . . . . . . . . . . . . 15 ⊢ (1...𝑀) ∈ Fin
46		rabfi 9273	. . . . . . . . . . . . . . 15 ⊢ ((1...𝑀) ∈ Fin → {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ∈ Fin)
47	45, 46	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ∈ Fin
48	47	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ∈ Fin)
49	16	rnmptfi 44169	. . . . . . . . . . . . 13 ⊢ ({𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ∈ Fin → ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ↦ ((𝐷‘𝑖)‘𝑍)) ∈ Fin)
50	48, 49	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ↦ ((𝐷‘𝑖)‘𝑍)) ∈ Fin)
51	14, 50	eqeltrid 2836	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑂 ∈ Fin)
52		unfi 9176	. . . . . . . . . . 11 ⊢ (({(𝐵‘𝑍)} ∈ Fin ∧ 𝑂 ∈ Fin) → ({(𝐵‘𝑍)} ∪ 𝑂) ∈ Fin)
53	44, 51, 52	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → ({(𝐵‘𝑍)} ∪ 𝑂) ∈ Fin)
54	12, 53	eqeltrid 2836	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ∈ Fin)
55		fvex 6904	. . . . . . . . . . . . . 14 ⊢ (𝐵‘𝑍) ∈ V
56	55	snid 4664	. . . . . . . . . . . . 13 ⊢ (𝐵‘𝑍) ∈ {(𝐵‘𝑍)}
57		elun1 4176	. . . . . . . . . . . . 13 ⊢ ((𝐵‘𝑍) ∈ {(𝐵‘𝑍)} → (𝐵‘𝑍) ∈ ({(𝐵‘𝑍)} ∪ 𝑂))
58	56, 57	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝐵‘𝑍) ∈ ({(𝐵‘𝑍)} ∪ 𝑂)
59	12	eqcomi 2740	. . . . . . . . . . . 12 ⊢ ({(𝐵‘𝑍)} ∪ 𝑂) = 𝑉
60	58, 59	eleqtri 2830	. . . . . . . . . . 11 ⊢ (𝐵‘𝑍) ∈ 𝑉
61	60	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵‘𝑍) ∈ 𝑉)
62		ne0i 4334	. . . . . . . . . 10 ⊢ ((𝐵‘𝑍) ∈ 𝑉 → 𝑉 ≠ ∅)
63	61, 62	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ≠ ∅)
64		fiinfcl 9500	. . . . . . . . 9 ⊢ (( < Or ℝ ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅ ∧ 𝑉 ⊆ ℝ)) → inf(𝑉, ℝ, < ) ∈ 𝑉)
65	42, 54, 63, 39, 64	syl13anc 1371	. . . . . . . 8 ⊢ (𝜑 → inf(𝑉, ℝ, < ) ∈ 𝑉)
66	40, 65	eqeltrid 2836	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ 𝑉)
67	39, 66	sseldd 3983	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℝ)
68		hoidmvlelem2.u	. . . . . . . . . . . 12 ⊢ 𝑈 = {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))}
69		ssrab2 4077	. . . . . . . . . . . 12 ⊢ {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))} ⊆ ((𝐴‘𝑍)[,](𝐵‘𝑍))
70	68, 69	eqsstri 4016	. . . . . . . . . . 11 ⊢ 𝑈 ⊆ ((𝐴‘𝑍)[,](𝐵‘𝑍))
71	70	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ⊆ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
72	9, 11	iccssred 13416	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴‘𝑍)[,](𝐵‘𝑍)) ⊆ ℝ)
73	71, 72	sstrd 3992	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ⊆ ℝ)
74		hoidmvlelem2.su	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ 𝑈)
75	73, 74	sseldd 3983	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℝ)
76	9	rexrd 11269	. . . . . . . . 9 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ^*)
77	11	rexrd 11269	. . . . . . . . 9 ⊢ (𝜑 → (𝐵‘𝑍) ∈ ℝ^*)
78	70, 74	sselid 3980	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
79		iccgelb 13385	. . . . . . . . 9 ⊢ (((𝐴‘𝑍) ∈ ℝ^* ∧ (𝐵‘𝑍) ∈ ℝ^* ∧ 𝑆 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍))) → (𝐴‘𝑍) ≤ 𝑆)
80	76, 77, 78, 79	syl3anc 1370	. . . . . . . 8 ⊢ (𝜑 → (𝐴‘𝑍) ≤ 𝑆)
81		hoidmvlelem2.sb	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 < (𝐵‘𝑍))
82	81	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 = (𝐵‘𝑍)) → 𝑆 < (𝐵‘𝑍))
83		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝐵‘𝑍) → 𝑥 = (𝐵‘𝑍))
84	83	eqcomd 2737	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝐵‘𝑍) → (𝐵‘𝑍) = 𝑥)
85	84	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 = (𝐵‘𝑍)) → (𝐵‘𝑍) = 𝑥)
86	82, 85	breqtrd 5174	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 = (𝐵‘𝑍)) → 𝑆 < 𝑥)
87	86	adantlr 712	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = (𝐵‘𝑍)) → 𝑆 < 𝑥)
88		simpll 764	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ ¬ 𝑥 = (𝐵‘𝑍)) → 𝜑)
89		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝑉 → 𝑥 ∈ 𝑉)
90	89, 12	eleqtrdi 2842	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑉 → 𝑥 ∈ ({(𝐵‘𝑍)} ∪ 𝑂))
91	90	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑉 ∧ ¬ 𝑥 = (𝐵‘𝑍)) → 𝑥 ∈ ({(𝐵‘𝑍)} ∪ 𝑂))
92		elsni 4645	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ {(𝐵‘𝑍)} → 𝑥 = (𝐵‘𝑍))
93	92	con3i 154	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑥 = (𝐵‘𝑍) → ¬ 𝑥 ∈ {(𝐵‘𝑍)})
94	93	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑉 ∧ ¬ 𝑥 = (𝐵‘𝑍)) → ¬ 𝑥 ∈ {(𝐵‘𝑍)})
95		elunnel1 4149	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ({(𝐵‘𝑍)} ∪ 𝑂) ∧ ¬ 𝑥 ∈ {(𝐵‘𝑍)}) → 𝑥 ∈ 𝑂)
96	91, 94, 95	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑉 ∧ ¬ 𝑥 = (𝐵‘𝑍)) → 𝑥 ∈ 𝑂)
97	96	adantll 711	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ ¬ 𝑥 = (𝐵‘𝑍)) → 𝑥 ∈ 𝑂)
98		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝑂 → 𝑥 ∈ 𝑂)
99	98, 14	eleqtrdi 2842	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑂 → 𝑥 ∈ ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ↦ ((𝐷‘𝑖)‘𝑍)))
100		vex 3477	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ V
101	16	elrnmpt 5955	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ V → (𝑥 ∈ ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ↦ ((𝐷‘𝑖)‘𝑍)) ↔ ∃𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))}𝑥 = ((𝐷‘𝑖)‘𝑍)))
102	100, 101	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ↦ ((𝐷‘𝑖)‘𝑍)) ↔ ∃𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))}𝑥 = ((𝐷‘𝑖)‘𝑍))
103	99, 102	sylib 217	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑂 → ∃𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))}𝑥 = ((𝐷‘𝑖)‘𝑍))
104	103	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑂) → ∃𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))}𝑥 = ((𝐷‘𝑖)‘𝑍))
105		fveq2 6891	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 = 𝑖 → (𝐶‘𝑗) = (𝐶‘𝑖))
106	105	fveq1d 6893	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 = 𝑖 → ((𝐶‘𝑗)‘𝑍) = ((𝐶‘𝑖)‘𝑍))
107	106	eleq1d 2817	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 = 𝑖 → (((𝐶‘𝑗)‘𝑍) ∈ ℝ ↔ ((𝐶‘𝑖)‘𝑍) ∈ ℝ))
108	23, 107	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = 𝑖 → (((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)‘𝑍) ∈ ℝ) ↔ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝐶‘𝑖)‘𝑍) ∈ ℝ)))
109		hoidmvlelem2.c	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝐶:ℕ⟶(ℝ ↑_m 𝑊))
110	109	ffvelcdmda 7086	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ (ℝ ↑_m 𝑊))
111		elmapi 8847	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐶‘𝑗) ∈ (ℝ ↑_m 𝑊) → (𝐶‘𝑗):𝑊⟶ℝ)
112	110, 111	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗):𝑊⟶ℝ)
113	112, 32	ffvelcdmd 7087	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)‘𝑍) ∈ ℝ)
114	108, 113	chvarvv 2001	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝐶‘𝑖)‘𝑍) ∈ ℝ)
115	114	rexrd 11269	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝐶‘𝑖)‘𝑍) ∈ ℝ^*)
116	17, 21, 115	syl2anc 583	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))}) → ((𝐶‘𝑖)‘𝑍) ∈ ℝ^*)
117	34	rexrd 11269	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝐷‘𝑖)‘𝑍) ∈ ℝ^*)
118	17, 21, 117	syl2anc 583	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))}) → ((𝐷‘𝑖)‘𝑍) ∈ ℝ^*)
119	106, 25	oveq12d 7430	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 = 𝑖 → (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) = (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)))
120	119	eleq2d 2818	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 = 𝑖 → (𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) ↔ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))))
121	120	elrab 3683	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ↔ (𝑖 ∈ (1...𝑀) ∧ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))))
122	121	biimpi 215	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} → (𝑖 ∈ (1...𝑀) ∧ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))))
123	122	simprd 495	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} → 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)))
124	123	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))}) → 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)))
125		icoltub 44520	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐶‘𝑖)‘𝑍) ∈ ℝ^* ∧ ((𝐷‘𝑖)‘𝑍) ∈ ℝ^* ∧ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))) → 𝑆 < ((𝐷‘𝑖)‘𝑍))
126	116, 118, 124, 125	syl3anc 1370	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))}) → 𝑆 < ((𝐷‘𝑖)‘𝑍))
127	126	3adant3 1131	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ∧ 𝑥 = ((𝐷‘𝑖)‘𝑍)) → 𝑆 < ((𝐷‘𝑖)‘𝑍))
128		id 22	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = ((𝐷‘𝑖)‘𝑍) → 𝑥 = ((𝐷‘𝑖)‘𝑍))
129	128	eqcomd 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ((𝐷‘𝑖)‘𝑍) → ((𝐷‘𝑖)‘𝑍) = 𝑥)
130	129	3ad2ant3 1134	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ∧ 𝑥 = ((𝐷‘𝑖)‘𝑍)) → ((𝐷‘𝑖)‘𝑍) = 𝑥)
131	127, 130	breqtrd 5174	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ∧ 𝑥 = ((𝐷‘𝑖)‘𝑍)) → 𝑆 < 𝑥)
132	131	3exp 1118	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} → (𝑥 = ((𝐷‘𝑖)‘𝑍) → 𝑆 < 𝑥)))
133	132	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑂) → (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} → (𝑥 = ((𝐷‘𝑖)‘𝑍) → 𝑆 < 𝑥)))
134	133	rexlimdv 3152	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑂) → (∃𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))}𝑥 = ((𝐷‘𝑖)‘𝑍) → 𝑆 < 𝑥))
135	104, 134	mpd 15	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑂) → 𝑆 < 𝑥)
136	88, 97, 135	syl2anc 583	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ ¬ 𝑥 = (𝐵‘𝑍)) → 𝑆 < 𝑥)
137	87, 136	pm2.61dan 810	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑆 < 𝑥)
138	137	ralrimiva 3145	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 𝑆 < 𝑥)
139		breq2 5152	. . . . . . . . . . 11 ⊢ (𝑥 = inf(𝑉, ℝ, < ) → (𝑆 < 𝑥 ↔ 𝑆 < inf(𝑉, ℝ, < )))
140	139	rspcva 3610	. . . . . . . . . 10 ⊢ ((inf(𝑉, ℝ, < ) ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑉 𝑆 < 𝑥) → 𝑆 < inf(𝑉, ℝ, < ))
141	65, 138, 140	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 < inf(𝑉, ℝ, < ))
142	40	eqcomi 2740	. . . . . . . . . 10 ⊢ inf(𝑉, ℝ, < ) = 𝑄
143	142	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → inf(𝑉, ℝ, < ) = 𝑄)
144	141, 143	breqtrd 5174	. . . . . . . 8 ⊢ (𝜑 → 𝑆 < 𝑄)
145	9, 75, 67, 80, 144	lelttrd 11377	. . . . . . 7 ⊢ (𝜑 → (𝐴‘𝑍) < 𝑄)
146	9, 67, 145	ltled 11367	. . . . . 6 ⊢ (𝜑 → (𝐴‘𝑍) ≤ 𝑄)
147		fiminre 12166	. . . . . . . . 9 ⊢ ((𝑉 ⊆ ℝ ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ∃𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 𝑥 ≤ 𝑦)
148	39, 54, 63, 147	syl3anc 1370	. . . . . . . 8 ⊢ (𝜑 → ∃𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 𝑥 ≤ 𝑦)
149		lbinfle 12174	. . . . . . . 8 ⊢ ((𝑉 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 𝑥 ≤ 𝑦 ∧ (𝐵‘𝑍) ∈ 𝑉) → inf(𝑉, ℝ, < ) ≤ (𝐵‘𝑍))
150	39, 148, 61, 149	syl3anc 1370	. . . . . . 7 ⊢ (𝜑 → inf(𝑉, ℝ, < ) ≤ (𝐵‘𝑍))
151	40, 150	eqbrtrid 5183	. . . . . 6 ⊢ (𝜑 → 𝑄 ≤ (𝐵‘𝑍))
152	9, 11, 67, 146, 151	eliccd 44516	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
153	67	recnd 11247	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ ℂ)
154	75	recnd 11247	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ ℂ)
155	9	recnd 11247	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℂ)
156	153, 154, 155	npncand 11600	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄 − 𝑆) + (𝑆 − (𝐴‘𝑍))) = (𝑄 − (𝐴‘𝑍)))
157	156	eqcomd 2737	. . . . . . . 8 ⊢ (𝜑 → (𝑄 − (𝐴‘𝑍)) = ((𝑄 − 𝑆) + (𝑆 − (𝐴‘𝑍))))
158	157	oveq2d 7428	. . . . . . 7 ⊢ (𝜑 → (𝐺 · (𝑄 − (𝐴‘𝑍))) = (𝐺 · ((𝑄 − 𝑆) + (𝑆 − (𝐴‘𝑍)))))
159		rge0ssre 13438	. . . . . . . . . 10 ⊢ (0[,)+∞) ⊆ ℝ
160		hoidmvlelem2.g	. . . . . . . . . . 11 ⊢ 𝐺 = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌))
161		hoidmvlelem2.l	. . . . . . . . . . . 12 ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑_m 𝑥), 𝑏 ∈ (ℝ ↑_m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))
162		hoidmvlelem2.x	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ Fin)
163		hoidmvlelem2.y	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
164	162, 163	ssfid 9271	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ Fin)
165		ssun1 4172	. . . . . . . . . . . . . . 15 ⊢ 𝑌 ⊆ (𝑌 ∪ {𝑍})
166	165, 7	sseqtrri 4019	. . . . . . . . . . . . . 14 ⊢ 𝑌 ⊆ 𝑊
167	166	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ⊆ 𝑊)
168	1, 167	fssresd 6758	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ↾ 𝑌):𝑌⟶ℝ)
169	10, 167	fssresd 6758	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 ↾ 𝑌):𝑌⟶ℝ)
170	161, 164, 168, 169	hoidmvcl 45597	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ∈ (0[,)+∞))
171	160, 170	eqeltrid 2836	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ (0[,)+∞))
172	159, 171	sselid 3980	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ ℝ)
173	172	recnd 11247	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ℂ)
174	153, 154	subcld 11576	. . . . . . . 8 ⊢ (𝜑 → (𝑄 − 𝑆) ∈ ℂ)
175	154, 155	subcld 11576	. . . . . . . 8 ⊢ (𝜑 → (𝑆 − (𝐴‘𝑍)) ∈ ℂ)
176	173, 174, 175	adddid 11243	. . . . . . 7 ⊢ (𝜑 → (𝐺 · ((𝑄 − 𝑆) + (𝑆 − (𝐴‘𝑍)))) = ((𝐺 · (𝑄 − 𝑆)) + (𝐺 · (𝑆 − (𝐴‘𝑍)))))
177	173, 174	mulcld 11239	. . . . . . . 8 ⊢ (𝜑 → (𝐺 · (𝑄 − 𝑆)) ∈ ℂ)
178	173, 175	mulcld 11239	. . . . . . . 8 ⊢ (𝜑 → (𝐺 · (𝑆 − (𝐴‘𝑍))) ∈ ℂ)
179	177, 178	addcomd 11421	. . . . . . 7 ⊢ (𝜑 → ((𝐺 · (𝑄 − 𝑆)) + (𝐺 · (𝑆 − (𝐴‘𝑍)))) = ((𝐺 · (𝑆 − (𝐴‘𝑍))) + (𝐺 · (𝑄 − 𝑆))))
180	158, 176, 179	3eqtrd 2775	. . . . . 6 ⊢ (𝜑 → (𝐺 · (𝑄 − (𝐴‘𝑍))) = ((𝐺 · (𝑆 − (𝐴‘𝑍))) + (𝐺 · (𝑄 − 𝑆))))
181	67, 75	jca 511	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄 ∈ ℝ ∧ 𝑆 ∈ ℝ))
182		resubcl 11529	. . . . . . . . . . . . 13 ⊢ ((𝑄 ∈ ℝ ∧ 𝑆 ∈ ℝ) → (𝑄 − 𝑆) ∈ ℝ)
183	181, 182	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄 − 𝑆) ∈ ℝ)
184	172, 183	jca 511	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 ∈ ℝ ∧ (𝑄 − 𝑆) ∈ ℝ))
185		remulcl 11199	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ ℝ ∧ (𝑄 − 𝑆) ∈ ℝ) → (𝐺 · (𝑄 − 𝑆)) ∈ ℝ)
186	184, 185	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 · (𝑄 − 𝑆)) ∈ ℝ)
187	75, 9	jca 511	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆 ∈ ℝ ∧ (𝐴‘𝑍) ∈ ℝ))
188		resubcl 11529	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ ℝ ∧ (𝐴‘𝑍) ∈ ℝ) → (𝑆 − (𝐴‘𝑍)) ∈ ℝ)
189	187, 188	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆 − (𝐴‘𝑍)) ∈ ℝ)
190	172, 189	jca 511	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 ∈ ℝ ∧ (𝑆 − (𝐴‘𝑍)) ∈ ℝ))
191		remulcl 11199	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ ℝ ∧ (𝑆 − (𝐴‘𝑍)) ∈ ℝ) → (𝐺 · (𝑆 − (𝐴‘𝑍))) ∈ ℝ)
192	190, 191	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 · (𝑆 − (𝐴‘𝑍))) ∈ ℝ)
193	186, 192	jca 511	. . . . . . . . 9 ⊢ (𝜑 → ((𝐺 · (𝑄 − 𝑆)) ∈ ℝ ∧ (𝐺 · (𝑆 − (𝐴‘𝑍))) ∈ ℝ))
194		readdcl 11197	. . . . . . . . 9 ⊢ (((𝐺 · (𝑄 − 𝑆)) ∈ ℝ ∧ (𝐺 · (𝑆 − (𝐴‘𝑍))) ∈ ℝ) → ((𝐺 · (𝑄 − 𝑆)) + (𝐺 · (𝑆 − (𝐴‘𝑍)))) ∈ ℝ)
195	193, 194	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((𝐺 · (𝑄 − 𝑆)) + (𝐺 · (𝑆 − (𝐴‘𝑍)))) ∈ ℝ)
196	179, 195	eqeltrrd 2833	. . . . . . 7 ⊢ (𝜑 → ((𝐺 · (𝑆 − (𝐴‘𝑍))) + (𝐺 · (𝑄 − 𝑆))) ∈ ℝ)
197		1red 11220	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℝ)
198		hoidmvlelem2.e	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
199	198	rpred 13021	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ ℝ)
200	197, 199	readdcld 11248	. . . . . . . . 9 ⊢ (𝜑 → (1 + 𝐸) ∈ ℝ)
201	2	eldifbd 3961	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝑍 ∈ 𝑌)
202	8, 201	eldifd 3959	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ (𝑊 ∖ 𝑌))
203		hoidmvlelem2.r	. . . . . . . . . 10 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ)
204		hoidmvlelem2.h	. . . . . . . . . 10 ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑_m 𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))))
205	161, 164, 202, 7, 109, 28, 203, 204, 75	sge0hsphoire 45604	. . . . . . . . 9 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) ∈ ℝ)
206	200, 205	remulcld 11249	. . . . . . . 8 ⊢ (𝜑 → ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) ∈ ℝ)
207		fzfid 13943	. . . . . . . . . 10 ⊢ (𝜑 → (1...𝑀) ∈ Fin)
208	183	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → (𝑄 − 𝑆) ∈ ℝ)
209		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → 𝜑)
210		elfznn 13535	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (1...𝑀) → 𝑗 ∈ ℕ)
211	210	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → 𝑗 ∈ ℕ)
212		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℕ)
213		ovexd 7447	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ ℕ → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) ∈ V)
214		hoidmvlelem2.p	. . . . . . . . . . . . . . . . 17 ⊢ 𝑃 = (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
215	214	fvmpt2 7009	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ ℕ ∧ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) ∈ V) → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
216	212, 213, 215	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℕ → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
217	216	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
218	164	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑌 ∈ Fin)
219	166	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑌 ⊆ 𝑊)
220	112, 219	fssresd 6758	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗) ↾ 𝑌):𝑌⟶ℝ)
221	220	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐶‘𝑗) ↾ 𝑌):𝑌⟶ℝ)
222		iftrue 4534	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) = ((𝐶‘𝑗) ↾ 𝑌))
223	222	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) = ((𝐶‘𝑗) ↾ 𝑌))
224	223	feq1d 6702	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ ↔ ((𝐶‘𝑗) ↾ 𝑌):𝑌⟶ℝ))
225	221, 224	mpbird 257	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
226		0red 11222	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 0 ∈ ℝ)
227		hoidmvlelem2.f	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐹 = (𝑦 ∈ 𝑌 ↦ 0)
228	226, 227	fmptd 7115	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹:𝑌⟶ℝ)
229	228	ad2antrr 723	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → 𝐹:𝑌⟶ℝ)
230		iffalse 4537	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) = 𝐹)
231	230	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) = 𝐹)
232	231	feq1d 6702	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ ↔ 𝐹:𝑌⟶ℝ))
233	229, 232	mpbird 257	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
234	225, 233	pm2.61dan 810	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
235		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
236		fvex 6904	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐶‘𝑗) ∈ V
237	236	resex 6029	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐶‘𝑗) ↾ 𝑌) ∈ V
238	237	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝐶‘𝑗) ↾ 𝑌) ∈ V)
239	162, 163	ssexd 5324	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑌 ∈ V)
240		mptexg 7225	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑌 ∈ V → (𝑦 ∈ 𝑌 ↦ 0) ∈ V)
241	239, 240	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ 0) ∈ V)
242	227, 241	eqeltrid 2836	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐹 ∈ V)
243	238, 242	ifcld 4574	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ∈ V)
244	243	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ∈ V)
245		hoidmvlelem2.j	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐽 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹))
246	245	fvmpt2 7009	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ ℕ ∧ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ∈ V) → (𝐽‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹))
247	235, 244, 246	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹))
248	247	feq1d 6702	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐽‘𝑗):𝑌⟶ℝ ↔ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ))
249	234, 248	mpbird 257	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗):𝑌⟶ℝ)
250	31, 219	fssresd 6758	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐷‘𝑗) ↾ 𝑌):𝑌⟶ℝ)
251	250	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐷‘𝑗) ↾ 𝑌):𝑌⟶ℝ)
252		iftrue 4534	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) = ((𝐷‘𝑗) ↾ 𝑌))
253	252	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) = ((𝐷‘𝑗) ↾ 𝑌))
254	253	feq1d 6702	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ ↔ ((𝐷‘𝑗) ↾ 𝑌):𝑌⟶ℝ))
255	251, 254	mpbird 257	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
256		iffalse 4537	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) = 𝐹)
257	256	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) = 𝐹)
258	257	feq1d 6702	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ ↔ 𝐹:𝑌⟶ℝ))
259	229, 258	mpbird 257	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
260	255, 259	pm2.61dan 810	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
261		fvex 6904	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐷‘𝑗) ∈ V
262	261	resex 6029	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐷‘𝑗) ↾ 𝑌) ∈ V
263	262	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝐷‘𝑗) ↾ 𝑌) ∈ V)
264	263, 242	ifcld 4574	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) ∈ V)
265	264	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) ∈ V)
266		hoidmvlelem2.k	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐾 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹))
267	266	fvmpt2 7009	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ ℕ ∧ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) ∈ V) → (𝐾‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹))
268	235, 265, 267	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐾‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹))
269	268	feq1d 6702	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐾‘𝑗):𝑌⟶ℝ ↔ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ))
270	260, 269	mpbird 257	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐾‘𝑗):𝑌⟶ℝ)
271	161, 218, 249, 270	hoidmvcl 45597	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) ∈ (0[,)+∞))
272	217, 271	eqeltrd 2832	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ (0[,)+∞))
273	159, 272	sselid 3980	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ ℝ)
274	209, 211, 273	syl2anc 583	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → (𝑃‘𝑗) ∈ ℝ)
275	208, 274	remulcld 11249	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝑄 − 𝑆) · (𝑃‘𝑗)) ∈ ℝ)
276	207, 275	fsumrecl 15685	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)) ∈ ℝ)
277	200, 276	remulcld 11249	. . . . . . . 8 ⊢ (𝜑 → ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗))) ∈ ℝ)
278	206, 277	readdcld 11248	. . . . . . 7 ⊢ (𝜑 → (((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) + ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)))) ∈ ℝ)
279	161, 164, 202, 7, 109, 28, 203, 204, 67	sge0hsphoire 45604	. . . . . . . 8 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) ∈ ℝ)
280	200, 279	remulcld 11249	. . . . . . 7 ⊢ (𝜑 → ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))) ∈ ℝ)
281	74, 68	eleqtrdi 2842	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))})
282		oveq1 7419	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑆 → (𝑧 − (𝐴‘𝑍)) = (𝑆 − (𝐴‘𝑍)))
283	282	oveq2d 7428	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑆 → (𝐺 · (𝑧 − (𝐴‘𝑍))) = (𝐺 · (𝑆 − (𝐴‘𝑍))))
284		fveq2 6891	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑆 → (𝐻‘𝑧) = (𝐻‘𝑆))
285	284	fveq1d 6893	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑆 → ((𝐻‘𝑧)‘(𝐷‘𝑗)) = ((𝐻‘𝑆)‘(𝐷‘𝑗)))
286	285	oveq2d 7428	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑆 → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))) = ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))
287	286	mpteq2dv 5250	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑆 → (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))) = (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))
288	287	fveq2d 6895	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑆 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))))
289	288	oveq2d 7428	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑆 → ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))))) = ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))))
290	283, 289	breq12d 5161	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑆 → ((𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))))) ↔ (𝐺 · (𝑆 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))))))
291	290	elrab 3683	. . . . . . . . . 10 ⊢ (𝑆 ∈ {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))} ↔ (𝑆 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∧ (𝐺 · (𝑆 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))))))
292	281, 291	sylib 217	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∧ (𝐺 · (𝑆 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))))))
293	292	simprd 495	. . . . . . . 8 ⊢ (𝜑 → (𝐺 · (𝑆 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))))
294	207, 274	fsumrecl 15685	. . . . . . . . . . 11 ⊢ (𝜑 → Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗) ∈ ℝ)
295	200, 294	remulcld 11249	. . . . . . . . . 10 ⊢ (𝜑 → ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗)) ∈ ℝ)
296		0red 11222	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ∈ ℝ)
297	75, 67	posdifd 11806	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆 < 𝑄 ↔ 0 < (𝑄 − 𝑆)))
298	144, 297	mpbid 231	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < (𝑄 − 𝑆))
299	296, 183, 298	ltled 11367	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (𝑄 − 𝑆))
300		hoidmvlelem2.le	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗)))
301	172, 295, 183, 299, 300	lemul1ad 12158	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 · (𝑄 − 𝑆)) ≤ (((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗)) · (𝑄 − 𝑆)))
302	200	recnd 11247	. . . . . . . . . . 11 ⊢ (𝜑 → (1 + 𝐸) ∈ ℂ)
303	294	recnd 11247	. . . . . . . . . . 11 ⊢ (𝜑 → Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗) ∈ ℂ)
304	302, 303, 174	mulassd 11242	. . . . . . . . . 10 ⊢ (𝜑 → (((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗)) · (𝑄 − 𝑆)) = ((1 + 𝐸) · (Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗) · (𝑄 − 𝑆))))
305	274	recnd 11247	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → (𝑃‘𝑗) ∈ ℂ)
306	207, 174, 305	fsummulc1 15736	. . . . . . . . . . . 12 ⊢ (𝜑 → (Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗) · (𝑄 − 𝑆)) = Σ𝑗 ∈ (1...𝑀)((𝑃‘𝑗) · (𝑄 − 𝑆)))
307	174	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → (𝑄 − 𝑆) ∈ ℂ)
308	305, 307	mulcomd 11240	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝑃‘𝑗) · (𝑄 − 𝑆)) = ((𝑄 − 𝑆) · (𝑃‘𝑗)))
309	308	sumeq2dv 15654	. . . . . . . . . . . 12 ⊢ (𝜑 → Σ𝑗 ∈ (1...𝑀)((𝑃‘𝑗) · (𝑄 − 𝑆)) = Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)))
310	306, 309	eqtrd 2771	. . . . . . . . . . 11 ⊢ (𝜑 → (Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗) · (𝑄 − 𝑆)) = Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)))
311	310	oveq2d 7428	. . . . . . . . . 10 ⊢ (𝜑 → ((1 + 𝐸) · (Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗) · (𝑄 − 𝑆))) = ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗))))
312	304, 311	eqtrd 2771	. . . . . . . . 9 ⊢ (𝜑 → (((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗)) · (𝑄 − 𝑆)) = ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗))))
313	301, 312	breqtrd 5174	. . . . . . . 8 ⊢ (𝜑 → (𝐺 · (𝑄 − 𝑆)) ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗))))
314	192, 186, 206, 277, 293, 313	leadd12dd 44325	. . . . . . 7 ⊢ (𝜑 → ((𝐺 · (𝑆 − (𝐴‘𝑍))) + (𝐺 · (𝑄 − 𝑆))) ≤ (((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) + ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)))))
315		hoidmvlelem2.m	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑀 ∈ ℕ)
316		nnsplit 44367	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℕ → ℕ = ((1...𝑀) ∪ (ℤ_≥‘(𝑀 + 1))))
317	315, 316	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ℕ = ((1...𝑀) ∪ (ℤ_≥‘(𝑀 + 1))))
318		uncom 4153	. . . . . . . . . . . . . . . . 17 ⊢ ((1...𝑀) ∪ (ℤ_≥‘(𝑀 + 1))) = ((ℤ_≥‘(𝑀 + 1)) ∪ (1...𝑀))
319	318	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((1...𝑀) ∪ (ℤ_≥‘(𝑀 + 1))) = ((ℤ_≥‘(𝑀 + 1)) ∪ (1...𝑀)))
320	317, 319	eqtr2d 2772	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((ℤ_≥‘(𝑀 + 1)) ∪ (1...𝑀)) = ℕ)
321	320	eqcomd 2737	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ℕ = ((ℤ_≥‘(𝑀 + 1)) ∪ (1...𝑀)))
322	321	mpteq1d 5243	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))) = (𝑗 ∈ ((ℤ_≥‘(𝑀 + 1)) ∪ (1...𝑀)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))
323	322	fveq2d 6895	. . . . . . . . . . . 12 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) = (Σ^{^}‘(𝑗 ∈ ((ℤ_≥‘(𝑀 + 1)) ∪ (1...𝑀)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))))
324		nfv 1916	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝜑
325		fvexd 6906	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℤ_≥‘(𝑀 + 1)) ∈ V)
326		ovexd 7447	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1...𝑀) ∈ V)
327		incom 4201	. . . . . . . . . . . . . . 15 ⊢ ((ℤ_≥‘(𝑀 + 1)) ∩ (1...𝑀)) = ((1...𝑀) ∩ (ℤ_≥‘(𝑀 + 1)))
328		nnuzdisj 44364	. . . . . . . . . . . . . . 15 ⊢ ((1...𝑀) ∩ (ℤ_≥‘(𝑀 + 1))) = ∅
329	327, 328	eqtri 2759	. . . . . . . . . . . . . 14 ⊢ ((ℤ_≥‘(𝑀 + 1)) ∩ (1...𝑀)) = ∅
330	329	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((ℤ_≥‘(𝑀 + 1)) ∩ (1...𝑀)) = ∅)
331		icossicc 13418	. . . . . . . . . . . . . 14 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
332		ssid 4004	. . . . . . . . . . . . . . 15 ⊢ (0[,)+∞) ⊆ (0[,)+∞)
333		simpl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘(𝑀 + 1))) → 𝜑)
334	315	peano2nnd 12234	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑀 + 1) ∈ ℕ)
335		uznnssnn 12884	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 + 1) ∈ ℕ → (ℤ_≥‘(𝑀 + 1)) ⊆ ℕ)
336	334, 335	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (ℤ_≥‘(𝑀 + 1)) ⊆ ℕ)
337	336	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘(𝑀 + 1))) → (ℤ_≥‘(𝑀 + 1)) ⊆ ℕ)
338		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘(𝑀 + 1))) → 𝑗 ∈ (ℤ_≥‘(𝑀 + 1)))
339	337, 338	sseldd 3983	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘(𝑀 + 1))) → 𝑗 ∈ ℕ)
340		snfi 9048	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {𝑍} ∈ Fin
341	340	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → {𝑍} ∈ Fin)
342		unfi 9176	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑌 ∈ Fin ∧ {𝑍} ∈ Fin) → (𝑌 ∪ {𝑍}) ∈ Fin)
343	164, 341, 342	syl2anc 583	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑌 ∪ {𝑍}) ∈ Fin)
344	7, 343	eqeltrid 2836	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑊 ∈ Fin)
345	344	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑊 ∈ Fin)
346		eleq1w 2815	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = 𝑙 → (𝑗 ∈ 𝑌 ↔ 𝑙 ∈ 𝑌))
347		fveq2 6891	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = 𝑙 → (𝑐‘𝑗) = (𝑐‘𝑙))
348	347	breq1d 5158	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 = 𝑙 → ((𝑐‘𝑗) ≤ 𝑥 ↔ (𝑐‘𝑙) ≤ 𝑥))
349	348, 347	ifbieq1d 4552	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = 𝑙 → if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥) = if((𝑐‘𝑙) ≤ 𝑥, (𝑐‘𝑙), 𝑥))
350	346, 347, 349	ifbieq12d 4556	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = 𝑙 → if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)) = if(𝑙 ∈ 𝑌, (𝑐‘𝑙), if((𝑐‘𝑙) ≤ 𝑥, (𝑐‘𝑙), 𝑥)))
351	350	cbvmptv 5261	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥))) = (𝑙 ∈ 𝑊 ↦ if(𝑙 ∈ 𝑌, (𝑐‘𝑙), if((𝑐‘𝑙) ≤ 𝑥, (𝑐‘𝑙), 𝑥)))
352	351	mpteq2i 5253	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 ∈ (ℝ ↑_m 𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))) = (𝑐 ∈ (ℝ ↑_m 𝑊) ↦ (𝑙 ∈ 𝑊 ↦ if(𝑙 ∈ 𝑌, (𝑐‘𝑙), if((𝑐‘𝑙) ≤ 𝑥, (𝑐‘𝑙), 𝑥))))
353	352	mpteq2i 5253	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑_m 𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥))))) = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑_m 𝑊) ↦ (𝑙 ∈ 𝑊 ↦ if(𝑙 ∈ 𝑌, (𝑐‘𝑙), if((𝑐‘𝑙) ≤ 𝑥, (𝑐‘𝑙), 𝑥)))))
354	204, 353	eqtri 2759	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑_m 𝑊) ↦ (𝑙 ∈ 𝑊 ↦ if(𝑙 ∈ 𝑌, (𝑐‘𝑙), if((𝑐‘𝑙) ≤ 𝑥, (𝑐‘𝑙), 𝑥)))))
355	75	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑆 ∈ ℝ)
356	354, 355, 345, 31	hsphoif 45591	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐻‘𝑆)‘(𝐷‘𝑗)):𝑊⟶ℝ)
357	161, 345, 112, 356	hoidmvcl 45597	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ (0[,)+∞))
358	333, 339, 357	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘(𝑀 + 1))) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ (0[,)+∞))
359	332, 358	sselid 3980	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘(𝑀 + 1))) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ (0[,)+∞))
360	331, 359	sselid 3980	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘(𝑀 + 1))) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ (0[,]+∞))
361	209, 211, 357	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ (0[,)+∞))
362	331, 361	sselid 3980	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ (0[,]+∞))
363	324, 325, 326, 330, 360, 362	sge0splitmpt 45426	. . . . . . . . . . . 12 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ((ℤ_≥‘(𝑀 + 1)) ∪ (1...𝑀)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) = ((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) +_𝑒 (Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))))
364		nnex 12223	. . . . . . . . . . . . . . 15 ⊢ ℕ ∈ V
365	364	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ℕ ∈ V)
366	331, 357	sselid 3980	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ (0[,]+∞))
367	324, 365, 366, 205, 336	sge0ssrempt 45420	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) ∈ ℝ)
368	18	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1...𝑀) ⊆ ℕ)
369	324, 365, 366, 205, 368	sge0ssrempt 45420	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) ∈ ℝ)
370		rexadd 13216	. . . . . . . . . . . . 13 ⊢ (((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) ∈ ℝ ∧ (Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) ∈ ℝ) → ((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) +_𝑒 (Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) = ((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + (Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))))
371	367, 369, 370	syl2anc 583	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) +_𝑒 (Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) = ((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + (Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))))
372	323, 363, 371	3eqtrd 2775	. . . . . . . . . . 11 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) = ((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + (Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))))
373	372	oveq2d 7428	. . . . . . . . . 10 ⊢ (𝜑 → ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) = ((1 + 𝐸) · ((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + (Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))))))
374	373	oveq1d 7427	. . . . . . . . 9 ⊢ (𝜑 → (((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) + ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)))) = (((1 + 𝐸) · ((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + (Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))))) + ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)))))
375	372, 205	eqeltrrd 2833	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + (Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) ∈ ℝ)
376	375	recnd 11247	. . . . . . . . . . 11 ⊢ (𝜑 → ((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + (Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) ∈ ℂ)
377	276	recnd 11247	. . . . . . . . . . 11 ⊢ (𝜑 → Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)) ∈ ℂ)
378	302, 376, 377	adddid 11243	. . . . . . . . . 10 ⊢ (𝜑 → ((1 + 𝐸) · (((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + (Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) + Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)))) = (((1 + 𝐸) · ((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + (Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))))) + ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)))))
379	378	eqcomd 2737	. . . . . . . . 9 ⊢ (𝜑 → (((1 + 𝐸) · ((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + (Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))))) + ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)))) = ((1 + 𝐸) · (((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + (Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) + Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)))))
380	367	recnd 11247	. . . . . . . . . . . 12 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) ∈ ℂ)
381	369	recnd 11247	. . . . . . . . . . . 12 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) ∈ ℂ)
382	380, 381, 377	addassd 11241	. . . . . . . . . . 11 ⊢ (𝜑 → (((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + (Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) + Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗))) = ((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + ((Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)))))
383	207, 361	sge0fsummpt 45405	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) = Σ𝑗 ∈ (1...𝑀)((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))
384	383	oveq1d 7427	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗))) = (Σ𝑗 ∈ (1...𝑀)((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗))))
385		ax-resscn 11171	. . . . . . . . . . . . . . . . . 18 ⊢ ℝ ⊆ ℂ
386	159, 385	sstri 3991	. . . . . . . . . . . . . . . . 17 ⊢ (0[,)+∞) ⊆ ℂ
387	386, 357	sselid 3980	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ ℂ)
388	209, 211, 387	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ ℂ)
389	183	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑄 − 𝑆) ∈ ℝ)
390	389, 273	remulcld 11249	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑄 − 𝑆) · (𝑃‘𝑗)) ∈ ℝ)
391	390	recnd 11247	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑄 − 𝑆) · (𝑃‘𝑗)) ∈ ℂ)
392	211, 391	syldan 590	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝑄 − 𝑆) · (𝑃‘𝑗)) ∈ ℂ)
393	207, 388, 392	fsumadd 15691	. . . . . . . . . . . . . 14 ⊢ (𝜑 → Σ𝑗 ∈ (1...𝑀)(((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) = (Σ𝑗 ∈ (1...𝑀)((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗))))
394	393	eqcomd 2737	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Σ𝑗 ∈ (1...𝑀)((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗))) = Σ𝑗 ∈ (1...𝑀)(((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))))
395	384, 394	eqtrd 2771	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗))) = Σ𝑗 ∈ (1...𝑀)(((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))))
396	395	oveq2d 7428	. . . . . . . . . . 11 ⊢ (𝜑 → ((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + ((Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)))) = ((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + Σ𝑗 ∈ (1...𝑀)(((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗)))))
397	382, 396	eqtrd 2771	. . . . . . . . . 10 ⊢ (𝜑 → (((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + (Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) + Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗))) = ((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + Σ𝑗 ∈ (1...𝑀)(((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗)))))
398	397	oveq2d 7428	. . . . . . . . 9 ⊢ (𝜑 → ((1 + 𝐸) · (((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + (Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) + Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)))) = ((1 + 𝐸) · ((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + Σ𝑗 ∈ (1...𝑀)(((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))))))
399	374, 379, 398	3eqtrd 2775	. . . . . . . 8 ⊢ (𝜑 → (((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) + ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)))) = ((1 + 𝐸) · ((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + Σ𝑗 ∈ (1...𝑀)(((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))))))
400	159, 357	sselid 3980	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ ℝ)
401	400, 390	readdcld 11248	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) ∈ ℝ)
402	209, 211, 401	syl2anc 583	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) ∈ ℝ)
403	207, 402	fsumrecl 15685	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑗 ∈ (1...𝑀)(((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) ∈ ℝ)
404	367, 403	readdcld 11248	. . . . . . . . 9 ⊢ (𝜑 → ((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + Σ𝑗 ∈ (1...𝑀)(((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗)))) ∈ ℝ)
405		0le1 11742	. . . . . . . . . . 11 ⊢ 0 ≤ 1
406	405	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ 1)
407	198	rpge0d 13025	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ 𝐸)
408	197, 199, 406, 407	addge0d 11795	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (1 + 𝐸))
409	67	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑄 ∈ ℝ)
410	354, 409, 345, 31	hsphoif 45591	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐻‘𝑄)‘(𝐷‘𝑗)):𝑊⟶ℝ)
411	161, 345, 112, 410	hoidmvcl 45597	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))) ∈ (0[,)+∞))
412	331, 411	sselid 3980	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))) ∈ (0[,]+∞))
413	324, 365, 412, 279, 336	sge0ssrempt 45420	. . . . . . . . . . 11 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) ∈ ℝ)
414	159, 411	sselid 3980	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))) ∈ ℝ)
415	209, 211, 414	syl2anc 583	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))) ∈ ℝ)
416	207, 415	fsumrecl 15685	. . . . . . . . . . 11 ⊢ (𝜑 → Σ𝑗 ∈ (1...𝑀)((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))) ∈ ℝ)
417	333, 339, 412	syl2anc 583	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘(𝑀 + 1))) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))) ∈ (0[,]+∞))
418	202	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑍 ∈ (𝑊 ∖ 𝑌))
419	75, 67, 144	ltled 11367	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ≤ 𝑄)
420	419	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑆 ≤ 𝑄)
421	161, 345, 418, 7, 355, 409, 420, 354, 112, 31	hsphoidmvle2 45600	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ≤ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))
422	333, 339, 421	syl2anc 583	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘(𝑀 + 1))) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ≤ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))
423	324, 325, 360, 417, 422	sge0lempt 45425	. . . . . . . . . . 11 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) ≤ (Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))))
424	209	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) = 0) → 𝜑)
425	211	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) = 0) → 𝑗 ∈ ℕ)
426		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) = 0) → (𝑃‘𝑗) = 0)
427		oveq2 7420	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃‘𝑗) = 0 → ((𝑄 − 𝑆) · (𝑃‘𝑗)) = ((𝑄 − 𝑆) · 0))
428	427	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) = 0) → ((𝑄 − 𝑆) · (𝑃‘𝑗)) = ((𝑄 − 𝑆) · 0))
429	174	mul01d 11418	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑄 − 𝑆) · 0) = 0)
430	429	ad2antrr 723	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) = 0) → ((𝑄 − 𝑆) · 0) = 0)
431	428, 430	eqtrd 2771	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) = 0) → ((𝑄 − 𝑆) · (𝑃‘𝑗)) = 0)
432	431	oveq2d 7428	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) = 0) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) = (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + 0))
433	387	addridd 11419	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + 0) = ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))
434	433	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) = 0) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + 0) = ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))
435	432, 434	eqtrd 2771	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) = 0) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) = ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))
436	421	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) = 0) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ≤ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))
437	435, 436	eqbrtrd 5170	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) = 0) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) ≤ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))
438	424, 425, 426, 437	syl21anc 835	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) = 0) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) ≤ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))
439		simpl 482	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ ¬ (𝑃‘𝑗) = 0) → (𝜑 ∧ 𝑗 ∈ (1...𝑀)))
440		neqne 2947	. . . . . . . . . . . . . . 15 ⊢ (¬ (𝑃‘𝑗) = 0 → (𝑃‘𝑗) ≠ 0)
441	440	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ ¬ (𝑃‘𝑗) = 0) → (𝑃‘𝑗) ≠ 0)
442	402	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) ∈ ℝ)
443	209	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → 𝜑)
444	211	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → 𝑗 ∈ ℕ)
445		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → (𝑃‘𝑗) ≠ 0)
446	2	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑍 ∈ (𝑋 ∖ 𝑌))
447	201	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ¬ 𝑍 ∈ 𝑌)
448		eqid 2731	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑘))) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑘)))
449	161, 218, 446, 447, 7, 112, 356, 448	hoiprodp1 45603	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) = (∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑘))) · (vol‘(((𝐶‘𝑗)‘𝑍)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑍)))))
450	449	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) = (∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑘))) · (vol‘(((𝐶‘𝑗)‘𝑍)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑍)))))
451	217	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
452	218	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → 𝑌 ∈ Fin)
453	217	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
454		fveq2 6891	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑌 = ∅ → (𝐿‘𝑌) = (𝐿‘∅))
455	454	oveqd 7429	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑌 = ∅ → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) = ((𝐽‘𝑗)(𝐿‘∅)(𝐾‘𝑗)))
456	455	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) = ((𝐽‘𝑗)(𝐿‘∅)(𝐾‘𝑗)))
457	249	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → (𝐽‘𝑗):𝑌⟶ℝ)
458		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑌 = ∅ → 𝑌 = ∅)
459	458	eqcomd 2737	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑌 = ∅ → ∅ = 𝑌)
460	459	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → ∅ = 𝑌)
461	460	feq2d 6703	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → ((𝐽‘𝑗):∅⟶ℝ ↔ (𝐽‘𝑗):𝑌⟶ℝ))
462	457, 461	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → (𝐽‘𝑗):∅⟶ℝ)
463	270	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → (𝐾‘𝑗):𝑌⟶ℝ)
464	460	feq2d 6703	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → ((𝐾‘𝑗):∅⟶ℝ ↔ (𝐾‘𝑗):𝑌⟶ℝ))
465	463, 464	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → (𝐾‘𝑗):∅⟶ℝ)
466	161, 462, 465	hoidmv0val 45598	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → ((𝐽‘𝑗)(𝐿‘∅)(𝐾‘𝑗)) = 0)
467	453, 456, 466	3eqtrd 2775	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → (𝑃‘𝑗) = 0)
468	467	adantlr 712	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ 𝑌 = ∅) → (𝑃‘𝑗) = 0)
469		neneq 2945	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑃‘𝑗) ≠ 0 → ¬ (𝑃‘𝑗) = 0)
470	469	ad2antlr 724	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ 𝑌 = ∅) → ¬ (𝑃‘𝑗) = 0)
471	468, 470	pm2.65da 814	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ¬ 𝑌 = ∅)
472	471	neqned 2946	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → 𝑌 ≠ ∅)
473	249	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (𝐽‘𝑗):𝑌⟶ℝ)
474	270	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (𝐾‘𝑗):𝑌⟶ℝ)
475	161, 452, 472, 473, 474	hoidmvn0val 45599	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))))
476	247	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (𝐽‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹))
477	217	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
478	247	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝐽‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹))
479	478, 231	eqtrd 2771	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝐽‘𝑗) = 𝐹)
480	268	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝐾‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹))
481	480, 257	eqtrd 2771	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝐾‘𝑗) = 𝐹)
482	479, 481	oveq12d 7430	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) = (𝐹(𝐿‘𝑌)𝐹))
483	161, 164, 228	hoidmvval0b 45605	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝜑 → (𝐹(𝐿‘𝑌)𝐹) = 0)
484	483	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝐹(𝐿‘𝑌)𝐹) = 0)
485	477, 482, 484	3eqtrd 2775	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝑃‘𝑗) = 0)
486	485	adantlr 712	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝑃‘𝑗) = 0)
487	469	ad2antlr 724	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ¬ (𝑃‘𝑗) = 0)
488	486, 487	condan 815	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)))
489	488	iftrued 4536	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) = ((𝐶‘𝑗) ↾ 𝑌))
490	476, 489	eqtrd 2771	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (𝐽‘𝑗) = ((𝐶‘𝑗) ↾ 𝑌))
491	490	fveq1d 6893	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐽‘𝑗)‘𝑘) = (((𝐶‘𝑗) ↾ 𝑌)‘𝑘))
492	491	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ 𝑘 ∈ 𝑌) → ((𝐽‘𝑗)‘𝑘) = (((𝐶‘𝑗) ↾ 𝑌)‘𝑘))
493		fvres 6910	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 ∈ 𝑌 → (((𝐶‘𝑗) ↾ 𝑌)‘𝑘) = ((𝐶‘𝑗)‘𝑘))
494	493	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ 𝑘 ∈ 𝑌) → (((𝐶‘𝑗) ↾ 𝑌)‘𝑘) = ((𝐶‘𝑗)‘𝑘))
495	492, 494	eqtrd 2771	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ 𝑘 ∈ 𝑌) → ((𝐽‘𝑗)‘𝑘) = ((𝐶‘𝑗)‘𝑘))
496	268	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (𝐾‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹))
497	488, 252	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) = ((𝐷‘𝑗) ↾ 𝑌))
498	496, 497	eqtrd 2771	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (𝐾‘𝑗) = ((𝐷‘𝑗) ↾ 𝑌))
499	498	fveq1d 6893	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐾‘𝑗)‘𝑘) = (((𝐷‘𝑗) ↾ 𝑌)‘𝑘))
500	499	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ 𝑘 ∈ 𝑌) → ((𝐾‘𝑗)‘𝑘) = (((𝐷‘𝑗) ↾ 𝑌)‘𝑘))
501		fvres 6910	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 ∈ 𝑌 → (((𝐷‘𝑗) ↾ 𝑌)‘𝑘) = ((𝐷‘𝑗)‘𝑘))
502	501	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ 𝑘 ∈ 𝑌) → (((𝐷‘𝑗) ↾ 𝑌)‘𝑘) = ((𝐷‘𝑗)‘𝑘))
503	500, 502	eqtrd 2771	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ 𝑘 ∈ 𝑌) → ((𝐾‘𝑗)‘𝑘) = ((𝐷‘𝑗)‘𝑘))
504	495, 503	oveq12d 7430	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ 𝑘 ∈ 𝑌) → (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)) = (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
505	504	fveq2d 6895	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ 𝑘 ∈ 𝑌) → (vol‘(((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))) = (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))
506	505	prodeq2dv 15872	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ∏𝑘 ∈ 𝑌 (vol‘(((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))
507	475, 506	eqtrd 2771	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))
508	355	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → 𝑆 ∈ ℝ)
509	345	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → 𝑊 ∈ Fin)
510	31	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → (𝐷‘𝑗):𝑊⟶ℝ)
511		elun1 4176	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑘 ∈ 𝑌 → 𝑘 ∈ (𝑌 ∪ {𝑍}))
512	511, 7	eleqtrrdi 2843	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑘 ∈ 𝑌 → 𝑘 ∈ 𝑊)
513	512	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → 𝑘 ∈ 𝑊)
514	354, 508, 509, 510, 513	hsphoival 45594	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → (((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑘) = if(𝑘 ∈ 𝑌, ((𝐷‘𝑗)‘𝑘), if(((𝐷‘𝑗)‘𝑘) ≤ 𝑆, ((𝐷‘𝑗)‘𝑘), 𝑆)))
515		iftrue 4534	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 ∈ 𝑌 → if(𝑘 ∈ 𝑌, ((𝐷‘𝑗)‘𝑘), if(((𝐷‘𝑗)‘𝑘) ≤ 𝑆, ((𝐷‘𝑗)‘𝑘), 𝑆)) = ((𝐷‘𝑗)‘𝑘))
516	515	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, ((𝐷‘𝑗)‘𝑘), if(((𝐷‘𝑗)‘𝑘) ≤ 𝑆, ((𝐷‘𝑗)‘𝑘), 𝑆)) = ((𝐷‘𝑗)‘𝑘))
517	514, 516	eqtrd 2771	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → (((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑘) = ((𝐷‘𝑗)‘𝑘))
518	517	oveq2d 7428	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → (((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑘)) = (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
519	518	fveq2d 6895	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑘))) = (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))
520	519	prodeq2dv 15872	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑘))) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))
521	520	eqcomd 2737	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑘))))
522	521	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑘))))
523	451, 507, 522	3eqtrrd 2776	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑘))) = (𝑃‘𝑗))
524	354, 355, 345, 31, 32	hsphoival 45594	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑍) = if(𝑍 ∈ 𝑌, ((𝐷‘𝑗)‘𝑍), if(((𝐷‘𝑗)‘𝑍) ≤ 𝑆, ((𝐷‘𝑗)‘𝑍), 𝑆)))
525	201	iffalsed 4539	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → if(𝑍 ∈ 𝑌, ((𝐷‘𝑗)‘𝑍), if(((𝐷‘𝑗)‘𝑍) ≤ 𝑆, ((𝐷‘𝑗)‘𝑍), 𝑆)) = if(((𝐷‘𝑗)‘𝑍) ≤ 𝑆, ((𝐷‘𝑗)‘𝑍), 𝑆))
526	525	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑍 ∈ 𝑌, ((𝐷‘𝑗)‘𝑍), if(((𝐷‘𝑗)‘𝑍) ≤ 𝑆, ((𝐷‘𝑗)‘𝑍), 𝑆)) = if(((𝐷‘𝑗)‘𝑍) ≤ 𝑆, ((𝐷‘𝑗)‘𝑍), 𝑆))
527	524, 526	eqtrd 2771	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑍) = if(((𝐷‘𝑗)‘𝑍) ≤ 𝑆, ((𝐷‘𝑗)‘𝑍), 𝑆))
528	527	oveq2d 7428	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝐶‘𝑗)‘𝑍)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑍)) = (((𝐶‘𝑗)‘𝑍)[,)if(((𝐷‘𝑗)‘𝑍) ≤ 𝑆, ((𝐷‘𝑗)‘𝑍), 𝑆)))
529	528	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (((𝐶‘𝑗)‘𝑍)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑍)) = (((𝐶‘𝑗)‘𝑍)[,)if(((𝐷‘𝑗)‘𝑍) ≤ 𝑆, ((𝐷‘𝑗)‘𝑍), 𝑆)))
530	113	rexrd 11269	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)‘𝑍) ∈ ℝ^*)
531	530	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐶‘𝑗)‘𝑍) ∈ ℝ^*)
532	33	rexrd 11269	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐷‘𝑗)‘𝑍) ∈ ℝ^*)
533	532	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐷‘𝑗)‘𝑍) ∈ ℝ^*)
534		icoltub 44520	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝐶‘𝑗)‘𝑍) ∈ ℝ^* ∧ ((𝐷‘𝑗)‘𝑍) ∈ ℝ^* ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → 𝑆 < ((𝐷‘𝑗)‘𝑍))
535	531, 533, 488, 534	syl3anc 1370	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → 𝑆 < ((𝐷‘𝑗)‘𝑍))
536	355	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → 𝑆 ∈ ℝ)
537	33	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐷‘𝑗)‘𝑍) ∈ ℝ)
538	536, 537	ltnled 11366	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (𝑆 < ((𝐷‘𝑗)‘𝑍) ↔ ¬ ((𝐷‘𝑗)‘𝑍) ≤ 𝑆))
539	535, 538	mpbid 231	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ¬ ((𝐷‘𝑗)‘𝑍) ≤ 𝑆)
540	539	iffalsed 4539	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → if(((𝐷‘𝑗)‘𝑍) ≤ 𝑆, ((𝐷‘𝑗)‘𝑍), 𝑆) = 𝑆)
541	540	oveq2d 7428	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (((𝐶‘𝑗)‘𝑍)[,)if(((𝐷‘𝑗)‘𝑍) ≤ 𝑆, ((𝐷‘𝑗)‘𝑍), 𝑆)) = (((𝐶‘𝑗)‘𝑍)[,)𝑆))
542	529, 541	eqtrd 2771	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (((𝐶‘𝑗)‘𝑍)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑍)) = (((𝐶‘𝑗)‘𝑍)[,)𝑆))
543	542	fveq2d 6895	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (vol‘(((𝐶‘𝑗)‘𝑍)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑍))) = (vol‘(((𝐶‘𝑗)‘𝑍)[,)𝑆)))
544		volico 44998	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐶‘𝑗)‘𝑍) ∈ ℝ ∧ 𝑆 ∈ ℝ) → (vol‘(((𝐶‘𝑗)‘𝑍)[,)𝑆)) = if(((𝐶‘𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶‘𝑗)‘𝑍)), 0))
545	113, 536, 544	syl2an 595	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0)) → (vol‘(((𝐶‘𝑗)‘𝑍)[,)𝑆)) = if(((𝐶‘𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶‘𝑗)‘𝑍)), 0))
546	545	anabss5 665	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (vol‘(((𝐶‘𝑗)‘𝑍)[,)𝑆)) = if(((𝐶‘𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶‘𝑗)‘𝑍)), 0))
547		iftrue 4534	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐶‘𝑗)‘𝑍) < 𝑆 → if(((𝐶‘𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶‘𝑗)‘𝑍)), 0) = (𝑆 − ((𝐶‘𝑗)‘𝑍)))
548	547	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ((𝐶‘𝑗)‘𝑍) < 𝑆) → if(((𝐶‘𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶‘𝑗)‘𝑍)), 0) = (𝑆 − ((𝐶‘𝑗)‘𝑍)))
549		iffalse 4537	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (¬ ((𝐶‘𝑗)‘𝑍) < 𝑆 → if(((𝐶‘𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶‘𝑗)‘𝑍)), 0) = 0)
550	549	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐶‘𝑗)‘𝑍) < 𝑆) → if(((𝐶‘𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶‘𝑗)‘𝑍)), 0) = 0)
551		simpll 764	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐶‘𝑗)‘𝑍) < 𝑆) → (𝜑 ∧ 𝑗 ∈ ℕ))
552		icogelb 13380	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝐶‘𝑗)‘𝑍) ∈ ℝ^* ∧ ((𝐷‘𝑗)‘𝑍) ∈ ℝ^* ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐶‘𝑗)‘𝑍) ≤ 𝑆)
553	531, 533, 488, 552	syl3anc 1370	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐶‘𝑗)‘𝑍) ≤ 𝑆)
554	553	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐶‘𝑗)‘𝑍) < 𝑆) → ((𝐶‘𝑗)‘𝑍) ≤ 𝑆)
555		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐶‘𝑗)‘𝑍) < 𝑆) → ¬ ((𝐶‘𝑗)‘𝑍) < 𝑆)
556	554, 555	jca 511	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐶‘𝑗)‘𝑍) < 𝑆) → (((𝐶‘𝑗)‘𝑍) ≤ 𝑆 ∧ ¬ ((𝐶‘𝑗)‘𝑍) < 𝑆))
557	551, 113	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐶‘𝑗)‘𝑍) < 𝑆) → ((𝐶‘𝑗)‘𝑍) ∈ ℝ)
558	551, 355	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐶‘𝑗)‘𝑍) < 𝑆) → 𝑆 ∈ ℝ)
559	557, 558	eqleltd 11363	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐶‘𝑗)‘𝑍) < 𝑆) → (((𝐶‘𝑗)‘𝑍) = 𝑆 ↔ (((𝐶‘𝑗)‘𝑍) ≤ 𝑆 ∧ ¬ ((𝐶‘𝑗)‘𝑍) < 𝑆)))
560	556, 559	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐶‘𝑗)‘𝑍) < 𝑆) → ((𝐶‘𝑗)‘𝑍) = 𝑆)
561		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝐶‘𝑗)‘𝑍) = 𝑆 → ((𝐶‘𝑗)‘𝑍) = 𝑆)
562	561	eqcomd 2737	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝐶‘𝑗)‘𝑍) = 𝑆 → 𝑆 = ((𝐶‘𝑗)‘𝑍))
563	562	oveq1d 7427	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐶‘𝑗)‘𝑍) = 𝑆 → (𝑆 − ((𝐶‘𝑗)‘𝑍)) = (((𝐶‘𝑗)‘𝑍) − ((𝐶‘𝑗)‘𝑍)))
564	563	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ((𝐶‘𝑗)‘𝑍) = 𝑆) → (𝑆 − ((𝐶‘𝑗)‘𝑍)) = (((𝐶‘𝑗)‘𝑍) − ((𝐶‘𝑗)‘𝑍)))
565	385, 113	sselid 3980	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)‘𝑍) ∈ ℂ)
566	565	subidd 11564	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝐶‘𝑗)‘𝑍) − ((𝐶‘𝑗)‘𝑍)) = 0)
567	566	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ((𝐶‘𝑗)‘𝑍) = 𝑆) → (((𝐶‘𝑗)‘𝑍) − ((𝐶‘𝑗)‘𝑍)) = 0)
568	564, 567	eqtr2d 2772	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ((𝐶‘𝑗)‘𝑍) = 𝑆) → 0 = (𝑆 − ((𝐶‘𝑗)‘𝑍)))
569	551, 560, 568	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐶‘𝑗)‘𝑍) < 𝑆) → 0 = (𝑆 − ((𝐶‘𝑗)‘𝑍)))
570	550, 569	eqtrd 2771	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐶‘𝑗)‘𝑍) < 𝑆) → if(((𝐶‘𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶‘𝑗)‘𝑍)), 0) = (𝑆 − ((𝐶‘𝑗)‘𝑍)))
571	548, 570	pm2.61dan 810	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → if(((𝐶‘𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶‘𝑗)‘𝑍)), 0) = (𝑆 − ((𝐶‘𝑗)‘𝑍)))
572	543, 546, 571	3eqtrd 2775	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (vol‘(((𝐶‘𝑗)‘𝑍)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑍))) = (𝑆 − ((𝐶‘𝑗)‘𝑍)))
573	523, 572	oveq12d 7430	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑘))) · (vol‘(((𝐶‘𝑗)‘𝑍)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑍)))) = ((𝑃‘𝑗) · (𝑆 − ((𝐶‘𝑗)‘𝑍))))
574	386, 272	sselid 3980	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ ℂ)
575	355, 113	resubcld 11647	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑆 − ((𝐶‘𝑗)‘𝑍)) ∈ ℝ)
576	575	recnd 11247	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑆 − ((𝐶‘𝑗)‘𝑍)) ∈ ℂ)
577	574, 576	mulcomd 11240	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑃‘𝑗) · (𝑆 − ((𝐶‘𝑗)‘𝑍))) = ((𝑆 − ((𝐶‘𝑗)‘𝑍)) · (𝑃‘𝑗)))
578	577	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ((𝑃‘𝑗) · (𝑆 − ((𝐶‘𝑗)‘𝑍))) = ((𝑆 − ((𝐶‘𝑗)‘𝑍)) · (𝑃‘𝑗)))
579	450, 573, 578	3eqtrd 2775	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) = ((𝑆 − ((𝐶‘𝑗)‘𝑍)) · (𝑃‘𝑗)))
580	579	oveq1d 7427	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) = (((𝑆 − ((𝐶‘𝑗)‘𝑍)) · (𝑃‘𝑗)) + ((𝑄 − 𝑆) · (𝑃‘𝑗))))
581	174	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑄 − 𝑆) ∈ ℂ)
582	576, 581, 574	adddird 11244	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝑆 − ((𝐶‘𝑗)‘𝑍)) + (𝑄 − 𝑆)) · (𝑃‘𝑗)) = (((𝑆 − ((𝐶‘𝑗)‘𝑍)) · (𝑃‘𝑗)) + ((𝑄 − 𝑆) · (𝑃‘𝑗))))
583	582	eqcomd 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝑆 − ((𝐶‘𝑗)‘𝑍)) · (𝑃‘𝑗)) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) = (((𝑆 − ((𝐶‘𝑗)‘𝑍)) + (𝑄 − 𝑆)) · (𝑃‘𝑗)))
584	583	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (((𝑆 − ((𝐶‘𝑗)‘𝑍)) · (𝑃‘𝑗)) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) = (((𝑆 − ((𝐶‘𝑗)‘𝑍)) + (𝑄 − 𝑆)) · (𝑃‘𝑗)))
585	576, 581	addcomd 11421	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑆 − ((𝐶‘𝑗)‘𝑍)) + (𝑄 − 𝑆)) = ((𝑄 − 𝑆) + (𝑆 − ((𝐶‘𝑗)‘𝑍))))
586	153	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑄 ∈ ℂ)
587	154	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑆 ∈ ℂ)
588	586, 587, 565	npncand 11600	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑄 − 𝑆) + (𝑆 − ((𝐶‘𝑗)‘𝑍))) = (𝑄 − ((𝐶‘𝑗)‘𝑍)))
589	585, 588	eqtrd 2771	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑆 − ((𝐶‘𝑗)‘𝑍)) + (𝑄 − 𝑆)) = (𝑄 − ((𝐶‘𝑗)‘𝑍)))
590	589	oveq1d 7427	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝑆 − ((𝐶‘𝑗)‘𝑍)) + (𝑄 − 𝑆)) · (𝑃‘𝑗)) = ((𝑄 − ((𝐶‘𝑗)‘𝑍)) · (𝑃‘𝑗)))
591	590	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (((𝑆 − ((𝐶‘𝑗)‘𝑍)) + (𝑄 − 𝑆)) · (𝑃‘𝑗)) = ((𝑄 − ((𝐶‘𝑗)‘𝑍)) · (𝑃‘𝑗)))
592	580, 584, 591	3eqtrd 2775	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) = ((𝑄 − ((𝐶‘𝑗)‘𝑍)) · (𝑃‘𝑗)))
593	443, 444, 445, 592	syl21anc 835	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) = ((𝑄 − ((𝐶‘𝑗)‘𝑍)) · (𝑃‘𝑗)))
594		eqid 2731	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑘))) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑘)))
595	161, 218, 32, 447, 7, 112, 410, 594	hoiprodp1 45603	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))) = (∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑘))) · (vol‘(((𝐶‘𝑗)‘𝑍)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑍)))))
596	209, 211, 595	syl2anc 583	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))) = (∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑘))) · (vol‘(((𝐶‘𝑗)‘𝑍)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑍)))))
597	596	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))) = (∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑘))) · (vol‘(((𝐶‘𝑗)‘𝑍)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑍)))))
598	507	eqcomd 2737	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
599	409	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → 𝑄 ∈ ℝ)
600	354, 599, 509, 510, 513	hsphoival 45594	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → (((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑘) = if(𝑘 ∈ 𝑌, ((𝐷‘𝑗)‘𝑘), if(((𝐷‘𝑗)‘𝑘) ≤ 𝑄, ((𝐷‘𝑗)‘𝑘), 𝑄)))
601		iftrue 4534	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 ∈ 𝑌 → if(𝑘 ∈ 𝑌, ((𝐷‘𝑗)‘𝑘), if(((𝐷‘𝑗)‘𝑘) ≤ 𝑄, ((𝐷‘𝑗)‘𝑘), 𝑄)) = ((𝐷‘𝑗)‘𝑘))
602	601	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, ((𝐷‘𝑗)‘𝑘), if(((𝐷‘𝑗)‘𝑘) ≤ 𝑄, ((𝐷‘𝑗)‘𝑘), 𝑄)) = ((𝐷‘𝑗)‘𝑘))
603	600, 602	eqtrd 2771	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → (((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑘) = ((𝐷‘𝑗)‘𝑘))
604	603	oveq2d 7428	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → (((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑘)) = (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
605	604	fveq2d 6895	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑘))) = (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))
606	605	prodeq2dv 15872	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑘))) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))
607	606	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑘))) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))
608	598, 607, 451	3eqtr4d 2781	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑘))) = (𝑃‘𝑗))
609	443, 444, 445, 608	syl21anc 835	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑘))) = (𝑃‘𝑗))
610	354, 409, 345, 31, 32	hsphoival 45594	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑍) = if(𝑍 ∈ 𝑌, ((𝐷‘𝑗)‘𝑍), if(((𝐷‘𝑗)‘𝑍) ≤ 𝑄, ((𝐷‘𝑗)‘𝑍), 𝑄)))
611	211, 610	syldan 590	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → (((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑍) = if(𝑍 ∈ 𝑌, ((𝐷‘𝑗)‘𝑍), if(((𝐷‘𝑗)‘𝑍) ≤ 𝑄, ((𝐷‘𝑗)‘𝑍), 𝑄)))
612	611	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → (((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑍) = if(𝑍 ∈ 𝑌, ((𝐷‘𝑗)‘𝑍), if(((𝐷‘𝑗)‘𝑍) ≤ 𝑄, ((𝐷‘𝑗)‘𝑍), 𝑄)))
613	201	iffalsed 4539	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → if(𝑍 ∈ 𝑌, ((𝐷‘𝑗)‘𝑍), if(((𝐷‘𝑗)‘𝑍) ≤ 𝑄, ((𝐷‘𝑗)‘𝑍), 𝑄)) = if(((𝐷‘𝑗)‘𝑍) ≤ 𝑄, ((𝐷‘𝑗)‘𝑍), 𝑄))
614	613	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → if(𝑍 ∈ 𝑌, ((𝐷‘𝑗)‘𝑍), if(((𝐷‘𝑗)‘𝑍) ≤ 𝑄, ((𝐷‘𝑗)‘𝑍), 𝑄)) = if(((𝐷‘𝑗)‘𝑍) ≤ 𝑄, ((𝐷‘𝑗)‘𝑍), 𝑄))
615	211, 33	syldan 590	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝐷‘𝑗)‘𝑍) ∈ ℝ)
616	615	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ ((𝐷‘𝑗)‘𝑍) = 𝑄) → ((𝐷‘𝑗)‘𝑍) ∈ ℝ)
617		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ ((𝐷‘𝑗)‘𝑍) = 𝑄) → ((𝐷‘𝑗)‘𝑍) = 𝑄)
618	616, 617	eqled 11322	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ ((𝐷‘𝑗)‘𝑍) = 𝑄) → ((𝐷‘𝑗)‘𝑍) ≤ 𝑄)
619	618	iftrued 4536	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ ((𝐷‘𝑗)‘𝑍) = 𝑄) → if(((𝐷‘𝑗)‘𝑍) ≤ 𝑄, ((𝐷‘𝑗)‘𝑍), 𝑄) = ((𝐷‘𝑗)‘𝑍))
620	619, 617	eqtrd 2771	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ ((𝐷‘𝑗)‘𝑍) = 𝑄) → if(((𝐷‘𝑗)‘𝑍) ≤ 𝑄, ((𝐷‘𝑗)‘𝑍), 𝑄) = 𝑄)
621	620	adantlr 712	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) ∧ ((𝐷‘𝑗)‘𝑍) = 𝑄) → if(((𝐷‘𝑗)‘𝑍) ≤ 𝑄, ((𝐷‘𝑗)‘𝑍), 𝑄) = 𝑄)
622	67	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → 𝑄 ∈ ℝ)
623	622	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ ¬ ((𝐷‘𝑗)‘𝑍) = 𝑄) → 𝑄 ∈ ℝ)
624	623	adantlr 712	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐷‘𝑗)‘𝑍) = 𝑄) → 𝑄 ∈ ℝ)
625	615	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ ¬ ((𝐷‘𝑗)‘𝑍) = 𝑄) → ((𝐷‘𝑗)‘𝑍) ∈ ℝ)
626	625	adantlr 712	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐷‘𝑗)‘𝑍) = 𝑄) → ((𝐷‘𝑗)‘𝑍) ∈ ℝ)
627	40	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → 𝑄 = inf(𝑉, ℝ, < ))
628	443, 39	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → 𝑉 ⊆ ℝ)
629	148	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ∃𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 𝑥 ≤ 𝑦)
630		simplr 766	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → 𝑗 ∈ (1...𝑀))
631	210, 488	sylanl2 678	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)))
632	630, 631	jca 511	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → (𝑗 ∈ (1...𝑀) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))))
633		rabid 3451	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑗 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ↔ (𝑗 ∈ (1...𝑀) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))))
634	632, 633	sylibr 233	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → 𝑗 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))})
635		eqidd 2732	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐷‘𝑗)‘𝑍) = ((𝐷‘𝑗)‘𝑍))
636		fveq2 6891	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑖 = 𝑗 → (𝐷‘𝑖) = (𝐷‘𝑗))
637	636	fveq1d 6893	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑖 = 𝑗 → ((𝐷‘𝑖)‘𝑍) = ((𝐷‘𝑗)‘𝑍))
638	637	eqeq2d 2742	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑖 = 𝑗 → (((𝐷‘𝑗)‘𝑍) = ((𝐷‘𝑖)‘𝑍) ↔ ((𝐷‘𝑗)‘𝑍) = ((𝐷‘𝑗)‘𝑍)))
639	638	rspcev 3612	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑗 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ∧ ((𝐷‘𝑗)‘𝑍) = ((𝐷‘𝑗)‘𝑍)) → ∃𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ((𝐷‘𝑗)‘𝑍) = ((𝐷‘𝑖)‘𝑍))
640	634, 635, 639	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ∃𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ((𝐷‘𝑗)‘𝑍) = ((𝐷‘𝑖)‘𝑍))
641		fvexd 6906	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐷‘𝑗)‘𝑍) ∈ V)
642	16, 640, 641	elrnmptd 5960	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐷‘𝑗)‘𝑍) ∈ ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ↦ ((𝐷‘𝑖)‘𝑍)))
643	642, 14	eleqtrrdi 2843	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐷‘𝑗)‘𝑍) ∈ 𝑂)
644		elun2 4177	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝐷‘𝑗)‘𝑍) ∈ 𝑂 → ((𝐷‘𝑗)‘𝑍) ∈ ({(𝐵‘𝑍)} ∪ 𝑂))
645	643, 644	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐷‘𝑗)‘𝑍) ∈ ({(𝐵‘𝑍)} ∪ 𝑂))
646	59	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ({(𝐵‘𝑍)} ∪ 𝑂) = 𝑉)
647	645, 646	eleqtrd 2834	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐷‘𝑗)‘𝑍) ∈ 𝑉)
648		lbinfle 12174	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑉 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 𝑥 ≤ 𝑦 ∧ ((𝐷‘𝑗)‘𝑍) ∈ 𝑉) → inf(𝑉, ℝ, < ) ≤ ((𝐷‘𝑗)‘𝑍))
649	628, 629, 647, 648	syl3anc 1370	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → inf(𝑉, ℝ, < ) ≤ ((𝐷‘𝑗)‘𝑍))
650	627, 649	eqbrtrd 5170	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → 𝑄 ≤ ((𝐷‘𝑗)‘𝑍))
651	650	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐷‘𝑗)‘𝑍) = 𝑄) → 𝑄 ≤ ((𝐷‘𝑗)‘𝑍))
652		neqne 2947	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (¬ ((𝐷‘𝑗)‘𝑍) = 𝑄 → ((𝐷‘𝑗)‘𝑍) ≠ 𝑄)
653	652	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐷‘𝑗)‘𝑍) = 𝑄) → ((𝐷‘𝑗)‘𝑍) ≠ 𝑄)
654	624, 626, 651, 653	leneltd 11373	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐷‘𝑗)‘𝑍) = 𝑄) → 𝑄 < ((𝐷‘𝑗)‘𝑍))
655	624, 626	ltnled 11366	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐷‘𝑗)‘𝑍) = 𝑄) → (𝑄 < ((𝐷‘𝑗)‘𝑍) ↔ ¬ ((𝐷‘𝑗)‘𝑍) ≤ 𝑄))
656	654, 655	mpbid 231	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐷‘𝑗)‘𝑍) = 𝑄) → ¬ ((𝐷‘𝑗)‘𝑍) ≤ 𝑄)
657	656	iffalsed 4539	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐷‘𝑗)‘𝑍) = 𝑄) → if(((𝐷‘𝑗)‘𝑍) ≤ 𝑄, ((𝐷‘𝑗)‘𝑍), 𝑄) = 𝑄)
658	621, 657	pm2.61dan 810	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → if(((𝐷‘𝑗)‘𝑍) ≤ 𝑄, ((𝐷‘𝑗)‘𝑍), 𝑄) = 𝑄)
659	612, 614, 658	3eqtrd 2775	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → (((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑍) = 𝑄)
660	659	oveq2d 7428	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → (((𝐶‘𝑗)‘𝑍)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑍)) = (((𝐶‘𝑗)‘𝑍)[,)𝑄))
661	660	fveq2d 6895	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → (vol‘(((𝐶‘𝑗)‘𝑍)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑍))) = (vol‘(((𝐶‘𝑗)‘𝑍)[,)𝑄)))
662	209, 211, 113	syl2anc 583	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝐶‘𝑗)‘𝑍) ∈ ℝ)
663	662	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐶‘𝑗)‘𝑍) ∈ ℝ)
664	443, 67	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → 𝑄 ∈ ℝ)
665		volico 44998	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐶‘𝑗)‘𝑍) ∈ ℝ ∧ 𝑄 ∈ ℝ) → (vol‘(((𝐶‘𝑗)‘𝑍)[,)𝑄)) = if(((𝐶‘𝑗)‘𝑍) < 𝑄, (𝑄 − ((𝐶‘𝑗)‘𝑍)), 0))
666	663, 664, 665	syl2anc 583	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → (vol‘(((𝐶‘𝑗)‘𝑍)[,)𝑄)) = if(((𝐶‘𝑗)‘𝑍) < 𝑄, (𝑄 − ((𝐶‘𝑗)‘𝑍)), 0))
667	443, 75	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → 𝑆 ∈ ℝ)
668	443, 444, 445, 553	syl21anc 835	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐶‘𝑗)‘𝑍) ≤ 𝑆)
669	443, 144	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → 𝑆 < 𝑄)
670	663, 667, 664, 668, 669	lelttrd 11377	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐶‘𝑗)‘𝑍) < 𝑄)
671	670	iftrued 4536	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → if(((𝐶‘𝑗)‘𝑍) < 𝑄, (𝑄 − ((𝐶‘𝑗)‘𝑍)), 0) = (𝑄 − ((𝐶‘𝑗)‘𝑍)))
672	661, 666, 671	3eqtrd 2775	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → (vol‘(((𝐶‘𝑗)‘𝑍)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑍))) = (𝑄 − ((𝐶‘𝑗)‘𝑍)))
673	609, 672	oveq12d 7430	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → (∏𝑘 ∈ 𝑌 (vol‘(((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑘))) · (vol‘(((𝐶‘𝑗)‘𝑍)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑍)))) = ((𝑃‘𝑗) · (𝑄 − ((𝐶‘𝑗)‘𝑍))))
674	209, 153	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → 𝑄 ∈ ℂ)
675	385, 662	sselid 3980	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝐶‘𝑗)‘𝑍) ∈ ℂ)
676	674, 675	subcld 11576	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → (𝑄 − ((𝐶‘𝑗)‘𝑍)) ∈ ℂ)
677	305, 676	mulcomd 11240	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝑃‘𝑗) · (𝑄 − ((𝐶‘𝑗)‘𝑍))) = ((𝑄 − ((𝐶‘𝑗)‘𝑍)) · (𝑃‘𝑗)))
678	677	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ((𝑃‘𝑗) · (𝑄 − ((𝐶‘𝑗)‘𝑍))) = ((𝑄 − ((𝐶‘𝑗)‘𝑍)) · (𝑃‘𝑗)))
679	597, 673, 678	3eqtrd 2775	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))) = ((𝑄 − ((𝐶‘𝑗)‘𝑍)) · (𝑃‘𝑗)))
680	593, 679	eqtr4d 2774	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) = ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))
681	442, 680	eqled 11322	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) ≤ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))
682	439, 441, 681	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ ¬ (𝑃‘𝑗) = 0) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) ≤ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))
683	438, 682	pm2.61dan 810	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) ≤ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))
684	207, 402, 415, 683	fsumle 15750	. . . . . . . . . . 11 ⊢ (𝜑 → Σ𝑗 ∈ (1...𝑀)(((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) ≤ Σ𝑗 ∈ (1...𝑀)((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))
685	367, 403, 413, 416, 423, 684	leadd12dd 44325	. . . . . . . . . 10 ⊢ (𝜑 → ((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + Σ𝑗 ∈ (1...𝑀)(((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗)))) ≤ ((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) + Σ𝑗 ∈ (1...𝑀)((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))
686	321	mpteq1d 5243	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))) = (𝑗 ∈ ((ℤ_≥‘(𝑀 + 1)) ∪ (1...𝑀)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))
687	686	fveq2d 6895	. . . . . . . . . . . 12 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) = (Σ^{^}‘(𝑗 ∈ ((ℤ_≥‘(𝑀 + 1)) ∪ (1...𝑀)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))))
688	211, 412	syldan 590	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))) ∈ (0[,]+∞))
689	324, 325, 326, 330, 417, 688	sge0splitmpt 45426	. . . . . . . . . . . 12 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ((ℤ_≥‘(𝑀 + 1)) ∪ (1...𝑀)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) = ((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) +_𝑒 (Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))))
690	687, 689	eqtrd 2771	. . . . . . . . . . 11 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) = ((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) +_𝑒 (Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))))
691	209, 211, 411	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))) ∈ (0[,)+∞))
692	207, 691	sge0fsummpt 45405	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) = Σ𝑗 ∈ (1...𝑀)((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))
693	692, 416	eqeltrd 2832	. . . . . . . . . . . 12 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) ∈ ℝ)
694		rexadd 13216	. . . . . . . . . . . 12 ⊢ (((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) ∈ ℝ ∧ (Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) ∈ ℝ) → ((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) +_𝑒 (Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))) = ((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) + (Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))))
695	413, 693, 694	syl2anc 583	. . . . . . . . . . 11 ⊢ (𝜑 → ((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) +_𝑒 (Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))) = ((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) + (Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))))
696	692	oveq2d 7428	. . . . . . . . . . 11 ⊢ (𝜑 → ((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) + (Σ^{^}‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))) = ((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) + Σ𝑗 ∈ (1...𝑀)((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))
697	690, 695, 696	3eqtrrd 2776	. . . . . . . . . 10 ⊢ (𝜑 → ((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) + Σ𝑗 ∈ (1...𝑀)((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))))
698	685, 697	breqtrd 5174	. . . . . . . . 9 ⊢ (𝜑 → ((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + Σ𝑗 ∈ (1...𝑀)(((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗)))) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))))
699	404, 279, 200, 408, 698	lemul2ad 12159	. . . . . . . 8 ⊢ (𝜑 → ((1 + 𝐸) · ((Σ^{^}‘(𝑗 ∈ (ℤ_≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) + Σ𝑗 ∈ (1...𝑀)(((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))))
700	399, 699	eqbrtrd 5170	. . . . . . 7 ⊢ (𝜑 → (((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) + ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))))
701	196, 278, 280, 314, 700	letrd 11376	. . . . . 6 ⊢ (𝜑 → ((𝐺 · (𝑆 − (𝐴‘𝑍))) + (𝐺 · (𝑄 − 𝑆))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))))
702	180, 701	eqbrtrd 5170	. . . . 5 ⊢ (𝜑 → (𝐺 · (𝑄 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))))
703	152, 702	jca 511	. . . 4 ⊢ (𝜑 → (𝑄 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∧ (𝐺 · (𝑄 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))))))
704		oveq1 7419	. . . . . . 7 ⊢ (𝑧 = 𝑄 → (𝑧 − (𝐴‘𝑍)) = (𝑄 − (𝐴‘𝑍)))
705	704	oveq2d 7428	. . . . . 6 ⊢ (𝑧 = 𝑄 → (𝐺 · (𝑧 − (𝐴‘𝑍))) = (𝐺 · (𝑄 − (𝐴‘𝑍))))
706		fveq2 6891	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑄 → (𝐻‘𝑧) = (𝐻‘𝑄))
707	706	fveq1d 6893	. . . . . . . . . 10 ⊢ (𝑧 = 𝑄 → ((𝐻‘𝑧)‘(𝐷‘𝑗)) = ((𝐻‘𝑄)‘(𝐷‘𝑗)))
708	707	oveq2d 7428	. . . . . . . . 9 ⊢ (𝑧 = 𝑄 → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))) = ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))
709	708	mpteq2dv 5250	. . . . . . . 8 ⊢ (𝑧 = 𝑄 → (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))) = (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))
710	709	fveq2d 6895	. . . . . . 7 ⊢ (𝑧 = 𝑄 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))))
711	710	oveq2d 7428	. . . . . 6 ⊢ (𝑧 = 𝑄 → ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))))) = ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))))
712	705, 711	breq12d 5161	. . . . 5 ⊢ (𝑧 = 𝑄 → ((𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))))) ↔ (𝐺 · (𝑄 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))))))
713	712	elrab 3683	. . . 4 ⊢ (𝑄 ∈ {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))} ↔ (𝑄 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∧ (𝐺 · (𝑄 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))))))
714	703, 713	sylibr 233	. . 3 ⊢ (𝜑 → 𝑄 ∈ {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))})
715	714, 68	eleqtrrdi 2843	. 2 ⊢ (𝜑 → 𝑄 ∈ 𝑈)
716		breq2 5152	. . 3 ⊢ (𝑢 = 𝑄 → (𝑆 < 𝑢 ↔ 𝑆 < 𝑄))
717	716	rspcev 3612	. 2 ⊢ ((𝑄 ∈ 𝑈 ∧ 𝑆 < 𝑄) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
718	715, 144, 717	syl2anc 583	1 ⊢ (𝜑 → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 {crab 3431 Vcvv 3473 ∖ cdif 3945 ∪ cun 3946 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 ifcif 4528 {csn 4628 class class class wbr 5148 ↦ cmpt 5231 Or wor 5587 ran crn 5677 ↾ cres 5678 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 ∈ cmpo 7414 ↑_m cmap 8824 Fincfn 8943 infcinf 9440 ℂcc 11112 ℝcr 11113 0cc0 11114 1c1 11115 + caddc 11117 · cmul 11119 +∞cpnf 11250 ℝ^*cxr 11252 < clt 11253 ≤ cle 11254 − cmin 11449 ℕcn 12217 ℤ_≥cuz 12827 ℝ⁺crp 12979 +_𝑒 cxad 13095 [,)cico 13331 [,]cicc 13332 ...cfz 13489 Σcsu 15637 ∏cprod 15854 volcvol 25213 Σ^{^}csumge0 45377

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9640 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-2o 8471 df-er 8707 df-map 8826 df-pm 8827 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fi 9410 df-sup 9441 df-inf 9442 df-oi 9509 df-dju 9900 df-card 9938 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-n0 12478 df-z 12564 df-uz 12828 df-q 12938 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-ioo 13333 df-ico 13335 df-icc 13336 df-fz 13490 df-fzo 13633 df-fl 13762 df-seq 13972 df-exp 14033 df-hash 14296 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-clim 15437 df-rlim 15438 df-sum 15638 df-prod 15855 df-rest 17373 df-topgen 17394 df-psmet 21137 df-xmet 21138 df-met 21139 df-bl 21140 df-mopn 21141 df-top 22617 df-topon 22634 df-bases 22670 df-cmp 23112 df-ovol 25214 df-vol 25215 df-sumge0 45378

This theorem is referenced by: hoidmvlelem3 45612

W3C validator