Theorem hoidmvlelem2 42885

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 4601	. . . . . . . . . 10 ⊢ (𝑍 ∈ (𝑋 ∖ 𝑌) → 𝑍 ∈ {𝑍})
4	2, 3	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ {𝑍})
5		elun2 4155	. . . . . . . . 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 2926	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
9	1, 8	ffvelrnd 6854	. . . . . 6 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ)
10		hoidmvlelem2.b	. . . . . . 7 ⊢ (𝜑 → 𝐵:𝑊⟶ℝ)
11	10, 8	ffvelrnd 6854	. . . . . 6 ⊢ (𝜑 → (𝐵‘𝑍) ∈ ℝ)
12		hoidmvlelem2.v	. . . . . . . 8 ⊢ 𝑉 = ({(𝐵‘𝑍)} ∪ 𝑂)
13	11	snssd 4744	. . . . . . . . 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 1915	. . . . . . . . . . 11 ⊢ Ⅎ𝑖𝜑
16		eqid 2823	. . . . . . . . . . 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 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))}) → 𝜑)
18		fz1ssnn 12941	. . . . . . . . . . . . . 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	sseldi 3967	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} → 𝑖 ∈ ℕ)
21	20	adantl 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))}) → 𝑖 ∈ ℕ)
22		eleq1w 2897	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑖 → (𝑗 ∈ ℕ ↔ 𝑖 ∈ ℕ))
23	22	anbi2d 630	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑖 → ((𝜑 ∧ 𝑗 ∈ ℕ) ↔ (𝜑 ∧ 𝑖 ∈ ℕ)))
24		fveq2 6672	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑖 → (𝐷‘𝑗) = (𝐷‘𝑖))
25	24	fveq1d 6674	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑖 → ((𝐷‘𝑗)‘𝑍) = ((𝐷‘𝑖)‘𝑍))
26	25	eleq1d 2899	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑖 → (((𝐷‘𝑗)‘𝑍) ∈ ℝ ↔ ((𝐷‘𝑖)‘𝑍) ∈ ℝ))
27	23, 26	imbi12d 347	. . . . . . . . . . . . 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	ffvelrnda 6853	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ (ℝ ↑m 𝑊))
30		elmapi 8430	. . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑍 ∈ 𝑊)
33	31, 32	ffvelrnd 6854	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐷‘𝑗)‘𝑍) ∈ ℝ)
34	27, 33	chvarvv 2005	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝐷‘𝑖)‘𝑍) ∈ ℝ)
35	17, 21, 34	syl2anc 586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))}) → ((𝐷‘𝑖)‘𝑍) ∈ ℝ)
36	15, 16, 35	rnmptssd 41465	. . . . . . . . . 10 ⊢ (𝜑 → ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ↦ ((𝐷‘𝑖)‘𝑍)) ⊆ ℝ)
37	14, 36	eqsstrid 4017	. . . . . . . . 9 ⊢ (𝜑 → 𝑂 ⊆ ℝ)
38	13, 37	unssd 4164	. . . . . . . 8 ⊢ (𝜑 → ({(𝐵‘𝑍)} ∪ 𝑂) ⊆ ℝ)
39	12, 38	eqsstrid 4017	. . . . . . 7 ⊢ (𝜑 → 𝑉 ⊆ ℝ)
40		hoidmvlelem2.q	. . . . . . . 8 ⊢ 𝑄 = inf(𝑉, ℝ, < )
41		ltso 10723	. . . . . . . . . 10 ⊢ < Or ℝ
42	41	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → < Or ℝ)
43		snfi 8596	. . . . . . . . . . . 12 ⊢ {(𝐵‘𝑍)} ∈ Fin
44	43	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → {(𝐵‘𝑍)} ∈ Fin)
45		fzfi 13343	. . . . . . . . . . . . . . 15 ⊢ (1...𝑀) ∈ Fin
46		rabfi 8745	. . . . . . . . . . . . . . 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 41434	. . . . . . . . . . . . 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 2919	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑂 ∈ Fin)
52		unfi 8787	. . . . . . . . . . 11 ⊢ (({(𝐵‘𝑍)} ∈ Fin ∧ 𝑂 ∈ Fin) → ({(𝐵‘𝑍)} ∪ 𝑂) ∈ Fin)
53	44, 51, 52	syl2anc 586	. . . . . . . . . 10 ⊢ (𝜑 → ({(𝐵‘𝑍)} ∪ 𝑂) ∈ Fin)
54	12, 53	eqeltrid 2919	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ∈ Fin)
55		fvex 6685	. . . . . . . . . . . . . 14 ⊢ (𝐵‘𝑍) ∈ V
56	55	snid 4603	. . . . . . . . . . . . 13 ⊢ (𝐵‘𝑍) ∈ {(𝐵‘𝑍)}
57		elun1 4154	. . . . . . . . . . . . 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 2832	. . . . . . . . . . . 12 ⊢ ({(𝐵‘𝑍)} ∪ 𝑂) = 𝑉
60	58, 59	eleqtri 2913	. . . . . . . . . . 11 ⊢ (𝐵‘𝑍) ∈ 𝑉
61	60	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵‘𝑍) ∈ 𝑉)
62		ne0i 4302	. . . . . . . . . 10 ⊢ ((𝐵‘𝑍) ∈ 𝑉 → 𝑉 ≠ ∅)
63	61, 62	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ≠ ∅)
64		fiinfcl 8967	. . . . . . . . 9 ⊢ (( < Or ℝ ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅ ∧ 𝑉 ⊆ ℝ)) → inf(𝑉, ℝ, < ) ∈ 𝑉)
65	42, 54, 63, 39, 64	syl13anc 1368	. . . . . . . 8 ⊢ (𝜑 → inf(𝑉, ℝ, < ) ∈ 𝑉)
66	40, 65	eqeltrid 2919	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ 𝑉)
67	39, 66	sseldd 3970	. . . . . 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 4058	. . . . . . . . . . . 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 4003	. . . . . . . . . . 11 ⊢ 𝑈 ⊆ ((𝐴‘𝑍)[,](𝐵‘𝑍))
71	70	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ⊆ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
72	9, 11	iccssred 41787	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴‘𝑍)[,](𝐵‘𝑍)) ⊆ ℝ)
73	71, 72	sstrd 3979	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ⊆ ℝ)
74		hoidmvlelem2.su	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ 𝑈)
75	73, 74	sseldd 3970	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℝ)
76	9	rexrd 10693	. . . . . . . . 9 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ*)
77	11	rexrd 10693	. . . . . . . . 9 ⊢ (𝜑 → (𝐵‘𝑍) ∈ ℝ*)
78	70, 74	sseldi 3967	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
79		iccgelb 12796	. . . . . . . . 9 ⊢ (((𝐴‘𝑍) ∈ ℝ* ∧ (𝐵‘𝑍) ∈ ℝ* ∧ 𝑆 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍))) → (𝐴‘𝑍) ≤ 𝑆)
80	76, 77, 78, 79	syl3anc 1367	. . . . . . . 8 ⊢ (𝜑 → (𝐴‘𝑍) ≤ 𝑆)
81		hoidmvlelem2.sb	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 < (𝐵‘𝑍))
82	81	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 = (𝐵‘𝑍)) → 𝑆 < (𝐵‘𝑍))
83		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝐵‘𝑍) → 𝑥 = (𝐵‘𝑍))
84	83	eqcomd 2829	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝐵‘𝑍) → (𝐵‘𝑍) = 𝑥)
85	84	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 = (𝐵‘𝑍)) → (𝐵‘𝑍) = 𝑥)
86	82, 85	breqtrd 5094	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 = (𝐵‘𝑍)) → 𝑆 < 𝑥)
87	86	adantlr 713	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = (𝐵‘𝑍)) → 𝑆 < 𝑥)
88		simpll 765	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ ¬ 𝑥 = (𝐵‘𝑍)) → 𝜑)
89		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝑉 → 𝑥 ∈ 𝑉)
90	89, 12	eleqtrdi 2925	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑉 → 𝑥 ∈ ({(𝐵‘𝑍)} ∪ 𝑂))
91	90	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑉 ∧ ¬ 𝑥 = (𝐵‘𝑍)) → 𝑥 ∈ ({(𝐵‘𝑍)} ∪ 𝑂))
92		elsni 4586	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ {(𝐵‘𝑍)} → 𝑥 = (𝐵‘𝑍))
93	92	con3i 157	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑥 = (𝐵‘𝑍) → ¬ 𝑥 ∈ {(𝐵‘𝑍)})
94	93	adantl 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑉 ∧ ¬ 𝑥 = (𝐵‘𝑍)) → ¬ 𝑥 ∈ {(𝐵‘𝑍)})
95		elunnel1 4128	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ({(𝐵‘𝑍)} ∪ 𝑂) ∧ ¬ 𝑥 ∈ {(𝐵‘𝑍)}) → 𝑥 ∈ 𝑂)
96	91, 94, 95	syl2anc 586	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑉 ∧ ¬ 𝑥 = (𝐵‘𝑍)) → 𝑥 ∈ 𝑂)
97	96	adantll 712	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ ¬ 𝑥 = (𝐵‘𝑍)) → 𝑥 ∈ 𝑂)
98		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝑂 → 𝑥 ∈ 𝑂)
99	98, 14	eleqtrdi 2925	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑂 → 𝑥 ∈ ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ↦ ((𝐷‘𝑖)‘𝑍)))
100		vex 3499	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ V
101	16	elrnmpt 5830	. . . . . . . . . . . . . . . . 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 220	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑂 → ∃𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))}𝑥 = ((𝐷‘𝑖)‘𝑍))
104	103	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑂) → ∃𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))}𝑥 = ((𝐷‘𝑖)‘𝑍))
105		fveq2 6672	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 = 𝑖 → (𝐶‘𝑗) = (𝐶‘𝑖))
106	105	fveq1d 6674	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 = 𝑖 → ((𝐶‘𝑗)‘𝑍) = ((𝐶‘𝑖)‘𝑍))
107	106	eleq1d 2899	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 = 𝑖 → (((𝐶‘𝑗)‘𝑍) ∈ ℝ ↔ ((𝐶‘𝑖)‘𝑍) ∈ ℝ))
108	23, 107	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . 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	ffvelrnda 6853	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ (ℝ ↑m 𝑊))
111		elmapi 8430	. . . . . . . . . . . . . . . . . . . . . . . . 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	ffvelrnd 6854	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)‘𝑍) ∈ ℝ)
114	108, 113	chvarvv 2005	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝐶‘𝑖)‘𝑍) ∈ ℝ)
115	114	rexrd 10693	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝐶‘𝑖)‘𝑍) ∈ ℝ*)
116	17, 21, 115	syl2anc 586	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))}) → ((𝐶‘𝑖)‘𝑍) ∈ ℝ*)
117	34	rexrd 10693	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝐷‘𝑖)‘𝑍) ∈ ℝ*)
118	17, 21, 117	syl2anc 586	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))}) → ((𝐷‘𝑖)‘𝑍) ∈ ℝ*)
119	106, 25	oveq12d 7176	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 = 𝑖 → (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) = (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)))
120	119	eleq2d 2900	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 = 𝑖 → (𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) ↔ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))))
121	120	elrab 3682	. . . . . . . . . . . . . . . . . . . . . . 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 218	. . . . . . . . . . . . . . . . . . . . . 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 498	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} → 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)))
124	123	adantl 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))}) → 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)))
125		icoltub 41791	. . . . . . . . . . . . . . . . . . . 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 1367	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))}) → 𝑆 < ((𝐷‘𝑖)‘𝑍))
127	126	3adant3 1128	. . . . . . . . . . . . . . . . . 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 2829	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ((𝐷‘𝑖)‘𝑍) → ((𝐷‘𝑖)‘𝑍) = 𝑥)
130	129	3ad2ant3 1131	. . . . . . . . . . . . . . . . . 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 5094	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ∧ 𝑥 = ((𝐷‘𝑖)‘𝑍)) → 𝑆 < 𝑥)
132	131	3exp 1115	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} → (𝑥 = ((𝐷‘𝑖)‘𝑍) → 𝑆 < 𝑥)))
133	132	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑂) → (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} → (𝑥 = ((𝐷‘𝑖)‘𝑍) → 𝑆 < 𝑥)))
134	133	rexlimdv 3285	. . . . . . . . . . . . . 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 586	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ ¬ 𝑥 = (𝐵‘𝑍)) → 𝑆 < 𝑥)
137	87, 136	pm2.61dan 811	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑆 < 𝑥)
138	137	ralrimiva 3184	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 𝑆 < 𝑥)
139		breq2 5072	. . . . . . . . . . 11 ⊢ (𝑥 = inf(𝑉, ℝ, < ) → (𝑆 < 𝑥 ↔ 𝑆 < inf(𝑉, ℝ, < )))
140	139	rspcva 3623	. . . . . . . . . 10 ⊢ ((inf(𝑉, ℝ, < ) ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑉 𝑆 < 𝑥) → 𝑆 < inf(𝑉, ℝ, < ))
141	65, 138, 140	syl2anc 586	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 < inf(𝑉, ℝ, < ))
142	40	eqcomi 2832	. . . . . . . . . 10 ⊢ inf(𝑉, ℝ, < ) = 𝑄
143	142	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → inf(𝑉, ℝ, < ) = 𝑄)
144	141, 143	breqtrd 5094	. . . . . . . 8 ⊢ (𝜑 → 𝑆 < 𝑄)
145	9, 75, 67, 80, 144	lelttrd 10800	. . . . . . 7 ⊢ (𝜑 → (𝐴‘𝑍) < 𝑄)
146	9, 67, 145	ltled 10790	. . . . . 6 ⊢ (𝜑 → (𝐴‘𝑍) ≤ 𝑄)
147		fiminre 11590	. . . . . . . . 9 ⊢ ((𝑉 ⊆ ℝ ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ∃𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 𝑥 ≤ 𝑦)
148	39, 54, 63, 147	syl3anc 1367	. . . . . . . 8 ⊢ (𝜑 → ∃𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 𝑥 ≤ 𝑦)
149		lbinfle 11598	. . . . . . . 8 ⊢ ((𝑉 ⊆ ℝ ∧ ∃𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 𝑥 ≤ 𝑦 ∧ (𝐵‘𝑍) ∈ 𝑉) → inf(𝑉, ℝ, < ) ≤ (𝐵‘𝑍))
150	39, 148, 61, 149	syl3anc 1367	. . . . . . 7 ⊢ (𝜑 → inf(𝑉, ℝ, < ) ≤ (𝐵‘𝑍))
151	40, 150	eqbrtrid 5103	. . . . . 6 ⊢ (𝜑 → 𝑄 ≤ (𝐵‘𝑍))
152	9, 11, 67, 146, 151	eliccd 41786	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
153	67	recnd 10671	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ ℂ)
154	75	recnd 10671	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ ℂ)
155	9	recnd 10671	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴‘𝑍) ∈ ℂ)
156	153, 154, 155	npncand 11023	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄 − 𝑆) + (𝑆 − (𝐴‘𝑍))) = (𝑄 − (𝐴‘𝑍)))
157	156	eqcomd 2829	. . . . . . . 8 ⊢ (𝜑 → (𝑄 − (𝐴‘𝑍)) = ((𝑄 − 𝑆) + (𝑆 − (𝐴‘𝑍))))
158	157	oveq2d 7174	. . . . . . 7 ⊢ (𝜑 → (𝐺 · (𝑄 − (𝐴‘𝑍))) = (𝐺 · ((𝑄 − 𝑆) + (𝑆 − (𝐴‘𝑍)))))
159		rge0ssre 12847	. . . . . . . . . 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 8743	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ Fin)
165		ssun1 4150	. . . . . . . . . . . . . . 15 ⊢ 𝑌 ⊆ (𝑌 ∪ {𝑍})
166	165, 7	sseqtrri 4006	. . . . . . . . . . . . . 14 ⊢ 𝑌 ⊆ 𝑊
167	166	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ⊆ 𝑊)
168	1, 167	fssresd 6547	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ↾ 𝑌):𝑌⟶ℝ)
169	10, 167	fssresd 6547	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 ↾ 𝑌):𝑌⟶ℝ)
170	161, 164, 168, 169	hoidmvcl 42871	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ∈ (0[,)+∞))
171	160, 170	eqeltrid 2919	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ (0[,)+∞))
172	159, 171	sseldi 3967	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ ℝ)
173	172	recnd 10671	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ℂ)
174	153, 154	subcld 10999	. . . . . . . 8 ⊢ (𝜑 → (𝑄 − 𝑆) ∈ ℂ)
175	154, 155	subcld 10999	. . . . . . . 8 ⊢ (𝜑 → (𝑆 − (𝐴‘𝑍)) ∈ ℂ)
176	173, 174, 175	adddid 10667	. . . . . . 7 ⊢ (𝜑 → (𝐺 · ((𝑄 − 𝑆) + (𝑆 − (𝐴‘𝑍)))) = ((𝐺 · (𝑄 − 𝑆)) + (𝐺 · (𝑆 − (𝐴‘𝑍)))))
177	173, 174	mulcld 10663	. . . . . . . 8 ⊢ (𝜑 → (𝐺 · (𝑄 − 𝑆)) ∈ ℂ)
178	173, 175	mulcld 10663	. . . . . . . 8 ⊢ (𝜑 → (𝐺 · (𝑆 − (𝐴‘𝑍))) ∈ ℂ)
179	177, 178	addcomd 10844	. . . . . . 7 ⊢ (𝜑 → ((𝐺 · (𝑄 − 𝑆)) + (𝐺 · (𝑆 − (𝐴‘𝑍)))) = ((𝐺 · (𝑆 − (𝐴‘𝑍))) + (𝐺 · (𝑄 − 𝑆))))
180	158, 176, 179	3eqtrd 2862	. . . . . 6 ⊢ (𝜑 → (𝐺 · (𝑄 − (𝐴‘𝑍))) = ((𝐺 · (𝑆 − (𝐴‘𝑍))) + (𝐺 · (𝑄 − 𝑆))))
181	67, 75	jca 514	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄 ∈ ℝ ∧ 𝑆 ∈ ℝ))
182		resubcl 10952	. . . . . . . . . . . . 13 ⊢ ((𝑄 ∈ ℝ ∧ 𝑆 ∈ ℝ) → (𝑄 − 𝑆) ∈ ℝ)
183	181, 182	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄 − 𝑆) ∈ ℝ)
184	172, 183	jca 514	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 ∈ ℝ ∧ (𝑄 − 𝑆) ∈ ℝ))
185		remulcl 10624	. . . . . . . . . . 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 514	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆 ∈ ℝ ∧ (𝐴‘𝑍) ∈ ℝ))
188		resubcl 10952	. . . . . . . . . . . . 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 514	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 ∈ ℝ ∧ (𝑆 − (𝐴‘𝑍)) ∈ ℝ))
191		remulcl 10624	. . . . . . . . . . 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 514	. . . . . . . . 9 ⊢ (𝜑 → ((𝐺 · (𝑄 − 𝑆)) ∈ ℝ ∧ (𝐺 · (𝑆 − (𝐴‘𝑍))) ∈ ℝ))
194		readdcl 10622	. . . . . . . . 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 2916	. . . . . . 7 ⊢ (𝜑 → ((𝐺 · (𝑆 − (𝐴‘𝑍))) + (𝐺 · (𝑄 − 𝑆))) ∈ ℝ)
197		1red 10644	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℝ)
198		hoidmvlelem2.e	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℝ+)
199	198	rpred 12434	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ ℝ)
200	197, 199	readdcld 10672	. . . . . . . . 9 ⊢ (𝜑 → (1 + 𝐸) ∈ ℝ)
201	2	eldifbd 3951	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝑍 ∈ 𝑌)
202	8, 201	eldifd 3949	. . . . . . . . . 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 42878	. . . . . . . . 9 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) ∈ ℝ)
206	200, 205	remulcld 10673	. . . . . . . 8 ⊢ (𝜑 → ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))) ∈ ℝ)
207		fzfid 13344	. . . . . . . . . 10 ⊢ (𝜑 → (1...𝑀) ∈ Fin)
208	183	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → (𝑄 − 𝑆) ∈ ℝ)
209		simpl 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → 𝜑)
210		elfznn 12939	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (1...𝑀) → 𝑗 ∈ ℕ)
211	210	adantl 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → 𝑗 ∈ ℕ)
212		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℕ)
213		ovexd 7193	. . . . . . . . . . . . . . . 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 6781	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ ℕ ∧ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) ∈ V) → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
216	212, 213, 215	syl2anc 586	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℕ → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
217	216	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
218	164	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑌 ∈ Fin)
219	166	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑌 ⊆ 𝑊)
220	112, 219	fssresd 6547	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗) ↾ 𝑌):𝑌⟶ℝ)
221	220	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐶‘𝑗) ↾ 𝑌):𝑌⟶ℝ)
222		iftrue 4475	. . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . 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 6501	. . . . . . . . . . . . . . . . . 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 259	. . . . . . . . . . . . . . . . 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 10646	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 0 ∈ ℝ)
227		hoidmvlelem2.f	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐹 = (𝑦 ∈ 𝑌 ↦ 0)
228	226, 227	fmptd 6880	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹:𝑌⟶ℝ)
229	228	ad2antrr 724	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → 𝐹:𝑌⟶ℝ)
230		iffalse 4478	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) = 𝐹)
231	230	adantl 484	. . . . . . . . . . . . . . . . . . 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 6501	. . . . . . . . . . . . . . . . . 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 259	. . . . . . . . . . . . . . . . 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 811	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
235		simpr 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
236		fvex 6685	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐶‘𝑗) ∈ V
237	236	resex 5901	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐶‘𝑗) ↾ 𝑌) ∈ V
238	237	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝐶‘𝑗) ↾ 𝑌) ∈ V)
239	162, 163	ssexd 5230	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑌 ∈ V)
240		mptexg 6986	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑌 ∈ V → (𝑦 ∈ 𝑌 ↦ 0) ∈ V)
241	239, 240	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ 0) ∈ V)
242	227, 241	eqeltrid 2919	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐹 ∈ V)
243	238, 242	ifcld 4514	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ∈ V)
244	243	adantr 483	. . . . . . . . . . . . . . . . . 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 6781	. . . . . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹))
248	247	feq1d 6501	. . . . . . . . . . . . . . . 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 259	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗):𝑌⟶ℝ)
250	31, 219	fssresd 6547	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐷‘𝑗) ↾ 𝑌):𝑌⟶ℝ)
251	250	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐷‘𝑗) ↾ 𝑌):𝑌⟶ℝ)
252		iftrue 4475	. . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . 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 6501	. . . . . . . . . . . . . . . . . 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 259	. . . . . . . . . . . . . . . . 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 4478	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) = 𝐹)
257	256	adantl 484	. . . . . . . . . . . . . . . . . . 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 6501	. . . . . . . . . . . . . . . . . 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 259	. . . . . . . . . . . . . . . . 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 811	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
261		fvex 6685	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐷‘𝑗) ∈ V
262	261	resex 5901	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐷‘𝑗) ↾ 𝑌) ∈ V
263	262	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝐷‘𝑗) ↾ 𝑌) ∈ V)
264	263, 242	ifcld 4514	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) ∈ V)
265	264	adantr 483	. . . . . . . . . . . . . . . . . 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 6781	. . . . . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐾‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹))
269	268	feq1d 6501	. . . . . . . . . . . . . . . 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 259	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐾‘𝑗):𝑌⟶ℝ)
271	161, 218, 249, 270	hoidmvcl 42871	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) ∈ (0[,)+∞))
272	217, 271	eqeltrd 2915	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ (0[,)+∞))
273	159, 272	sseldi 3967	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ ℝ)
274	209, 211, 273	syl2anc 586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → (𝑃‘𝑗) ∈ ℝ)
275	208, 274	remulcld 10673	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝑄 − 𝑆) · (𝑃‘𝑗)) ∈ ℝ)
276	207, 275	fsumrecl 15093	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)) ∈ ℝ)
277	200, 276	remulcld 10673	. . . . . . . 8 ⊢ (𝜑 → ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗))) ∈ ℝ)
278	206, 277	readdcld 10672	. . . . . . 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 42878	. . . . . . . 8 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) ∈ ℝ)
280	200, 279	remulcld 10673	. . . . . . 7 ⊢ (𝜑 → ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))) ∈ ℝ)
281	74, 68	eleqtrdi 2925	. . . . . . . . . 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 7165	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑆 → (𝑧 − (𝐴‘𝑍)) = (𝑆 − (𝐴‘𝑍)))
283	282	oveq2d 7174	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑆 → (𝐺 · (𝑧 − (𝐴‘𝑍))) = (𝐺 · (𝑆 − (𝐴‘𝑍))))
284		fveq2 6672	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑆 → (𝐻‘𝑧) = (𝐻‘𝑆))
285	284	fveq1d 6674	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑆 → ((𝐻‘𝑧)‘(𝐷‘𝑗)) = ((𝐻‘𝑆)‘(𝐷‘𝑗)))
286	285	oveq2d 7174	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑆 → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))) = ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))
287	286	mpteq2dv 5164	. . . . . . . . . . . . . 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 6676	. . . . . . . . . . . . 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 7174	. . . . . . . . . . . 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 5081	. . . . . . . . . . 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 3682	. . . . . . . . . 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 220	. . . . . . . . 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 498	. . . . . . . 8 ⊢ (𝜑 → (𝐺 · (𝑆 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗)))))))
294	207, 274	fsumrecl 15093	. . . . . . . . . . 11 ⊢ (𝜑 → Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗) ∈ ℝ)
295	200, 294	remulcld 10673	. . . . . . . . . 10 ⊢ (𝜑 → ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗)) ∈ ℝ)
296		0red 10646	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ∈ ℝ)
297	75, 67	posdifd 11229	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆 < 𝑄 ↔ 0 < (𝑄 − 𝑆)))
298	144, 297	mpbid 234	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < (𝑄 − 𝑆))
299	296, 183, 298	ltled 10790	. . . . . . . . . 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 11581	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 · (𝑄 − 𝑆)) ≤ (((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗)) · (𝑄 − 𝑆)))
302	200	recnd 10671	. . . . . . . . . . 11 ⊢ (𝜑 → (1 + 𝐸) ∈ ℂ)
303	294	recnd 10671	. . . . . . . . . . 11 ⊢ (𝜑 → Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗) ∈ ℂ)
304	302, 303, 174	mulassd 10666	. . . . . . . . . 10 ⊢ (𝜑 → (((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗)) · (𝑄 − 𝑆)) = ((1 + 𝐸) · (Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗) · (𝑄 − 𝑆))))
305	274	recnd 10671	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → (𝑃‘𝑗) ∈ ℂ)
306	207, 174, 305	fsummulc1 15142	. . . . . . . . . . . 12 ⊢ (𝜑 → (Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗) · (𝑄 − 𝑆)) = Σ𝑗 ∈ (1...𝑀)((𝑃‘𝑗) · (𝑄 − 𝑆)))
307	174	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → (𝑄 − 𝑆) ∈ ℂ)
308	305, 307	mulcomd 10664	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝑃‘𝑗) · (𝑄 − 𝑆)) = ((𝑄 − 𝑆) · (𝑃‘𝑗)))
309	308	sumeq2dv 15062	. . . . . . . . . . . 12 ⊢ (𝜑 → Σ𝑗 ∈ (1...𝑀)((𝑃‘𝑗) · (𝑄 − 𝑆)) = Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)))
310	306, 309	eqtrd 2858	. . . . . . . . . . 11 ⊢ (𝜑 → (Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗) · (𝑄 − 𝑆)) = Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)))
311	310	oveq2d 7174	. . . . . . . . . 10 ⊢ (𝜑 → ((1 + 𝐸) · (Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗) · (𝑄 − 𝑆))) = ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗))))
312	304, 311	eqtrd 2858	. . . . . . . . 9 ⊢ (𝜑 → (((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗)) · (𝑄 − 𝑆)) = ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗))))
313	301, 312	breqtrd 5094	. . . . . . . 8 ⊢ (𝜑 → (𝐺 · (𝑄 − 𝑆)) ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗))))
314	192, 186, 206, 277, 293, 313	leadd12dd 41591	. . . . . . 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 41633	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℕ → ℕ = ((1...𝑀) ∪ (ℤ≥‘(𝑀 + 1))))
317	315, 316	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ℕ = ((1...𝑀) ∪ (ℤ≥‘(𝑀 + 1))))
318		uncom 4131	. . . . . . . . . . . . . . . . 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 2859	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((ℤ≥‘(𝑀 + 1)) ∪ (1...𝑀)) = ℕ)
321	320	eqcomd 2829	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ℕ = ((ℤ≥‘(𝑀 + 1)) ∪ (1...𝑀)))
322	321	mpteq1d 5157	. . . . . . . . . . . . 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 6676	. . . . . . . . . . . 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 1915	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝜑
325		fvexd 6687	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℤ≥‘(𝑀 + 1)) ∈ V)
326		ovexd 7193	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1...𝑀) ∈ V)
327		incom 4180	. . . . . . . . . . . . . . 15 ⊢ ((ℤ≥‘(𝑀 + 1)) ∩ (1...𝑀)) = ((1...𝑀) ∩ (ℤ≥‘(𝑀 + 1)))
328		nnuzdisj 41630	. . . . . . . . . . . . . . 15 ⊢ ((1...𝑀) ∩ (ℤ≥‘(𝑀 + 1))) = ∅
329	327, 328	eqtri 2846	. . . . . . . . . . . . . 14 ⊢ ((ℤ≥‘(𝑀 + 1)) ∩ (1...𝑀)) = ∅
330	329	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((ℤ≥‘(𝑀 + 1)) ∩ (1...𝑀)) = ∅)
331		icossicc 12827	. . . . . . . . . . . . . 14 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
332		ssid 3991	. . . . . . . . . . . . . . 15 ⊢ (0[,)+∞) ⊆ (0[,)+∞)
333		simpl 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑀 + 1))) → 𝜑)
334	315	peano2nnd 11657	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑀 + 1) ∈ ℕ)
335		uznnssnn 12298	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 + 1) ∈ ℕ → (ℤ≥‘(𝑀 + 1)) ⊆ ℕ)
336	334, 335	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (ℤ≥‘(𝑀 + 1)) ⊆ ℕ)
337	336	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑀 + 1))) → (ℤ≥‘(𝑀 + 1)) ⊆ ℕ)
338		simpr 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑗 ∈ (ℤ≥‘(𝑀 + 1)))
339	337, 338	sseldd 3970	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑗 ∈ ℕ)
340		snfi 8596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {𝑍} ∈ Fin
341	340	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → {𝑍} ∈ Fin)
342		unfi 8787	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑌 ∈ Fin ∧ {𝑍} ∈ Fin) → (𝑌 ∪ {𝑍}) ∈ Fin)
343	164, 341, 342	syl2anc 586	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑌 ∪ {𝑍}) ∈ Fin)
344	7, 343	eqeltrid 2919	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑊 ∈ Fin)
345	344	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑊 ∈ Fin)
346		eleq1w 2897	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = 𝑙 → (𝑗 ∈ 𝑌 ↔ 𝑙 ∈ 𝑌))
347		fveq2 6672	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = 𝑙 → (𝑐‘𝑗) = (𝑐‘𝑙))
348	347	breq1d 5078	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 = 𝑙 → ((𝑐‘𝑗) ≤ 𝑥 ↔ (𝑐‘𝑙) ≤ 𝑥))
349	348, 347	ifbieq1d 4492	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = 𝑙 → if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥) = if((𝑐‘𝑙) ≤ 𝑥, (𝑐‘𝑙), 𝑥))
350	346, 347, 349	ifbieq12d 4496	. . . . . . . . . . . . . . . . . . . . . 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 5171	. . . . . . . . . . . . . . . . . . . . 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 5160	. . . . . . . . . . . . . . . . . . . 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 5160	. . . . . . . . . . . . . . . . . . 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 2846	. . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑆 ∈ ℝ)
356	354, 355, 345, 31	hsphoif 42865	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐻‘𝑆)‘(𝐷‘𝑗)):𝑊⟶ℝ)
357	161, 345, 112, 356	hoidmvcl 42871	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ (0[,)+∞))
358	333, 339, 357	syl2anc 586	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑀 + 1))) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ (0[,)+∞))
359	332, 358	sseldi 3967	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑀 + 1))) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ (0[,)+∞))
360	331, 359	sseldi 3967	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑀 + 1))) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ (0[,]+∞))
361	209, 211, 357	syl2anc 586	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ (0[,)+∞))
362	331, 361	sseldi 3967	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ (0[,]+∞))
363	324, 325, 326, 330, 360, 362	sge0splitmpt 42700	. . . . . . . . . . . 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 11646	. . . . . . . . . . . . . . 15 ⊢ ℕ ∈ V
365	364	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ℕ ∈ V)
366	331, 357	sseldi 3967	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ (0[,]+∞))
367	324, 365, 366, 205, 336	sge0ssrempt 42694	. . . . . . . . . . . . 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 42694	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) ∈ ℝ)
370		rexadd 12628	. . . . . . . . . . . . 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 586	. . . . . . . . . . . 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 2862	. . . . . . . . . . 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 7174	. . . . . . . . . 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 7173	. . . . . . . . 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 2916	. . . . . . . . . . . 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 10671	. . . . . . . . . . 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 10671	. . . . . . . . . . 11 ⊢ (𝜑 → Σ𝑗 ∈ (1...𝑀)((𝑄 − 𝑆) · (𝑃‘𝑗)) ∈ ℂ)
378	302, 376, 377	adddid 10667	. . . . . . . . . 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 2829	. . . . . . . . 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 10671	. . . . . . . . . . . 12 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) ∈ ℂ)
381	369	recnd 10671	. . . . . . . . . . . 12 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) ∈ ℂ)
382	380, 381, 377	addassd 10665	. . . . . . . . . . 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 42679	. . . . . . . . . . . . . 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 7173	. . . . . . . . . . . . 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 10596	. . . . . . . . . . . . . . . . . 18 ⊢ ℝ ⊆ ℂ
386	159, 385	sstri 3978	. . . . . . . . . . . . . . . . 17 ⊢ (0[,)+∞) ⊆ ℂ
387	386, 357	sseldi 3967	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ ℂ)
388	209, 211, 387	syl2anc 586	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ ℂ)
389	183	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑄 − 𝑆) ∈ ℝ)
390	389, 273	remulcld 10673	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑄 − 𝑆) · (𝑃‘𝑗)) ∈ ℝ)
391	390	recnd 10671	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑄 − 𝑆) · (𝑃‘𝑗)) ∈ ℂ)
392	211, 391	syldan 593	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝑄 − 𝑆) · (𝑃‘𝑗)) ∈ ℂ)
393	207, 388, 392	fsumadd 15098	. . . . . . . . . . . . . 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 2829	. . . . . . . . . . . . 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 2858	. . . . . . . . . . . 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 7174	. . . . . . . . . . 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 2858	. . . . . . . . . 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 7174	. . . . . . . . 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 2862	. . . . . . . 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	sseldi 3967	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) ∈ ℝ)
401	400, 390	readdcld 10672	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) ∈ ℝ)
402	209, 211, 401	syl2anc 586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) ∈ ℝ)
403	207, 402	fsumrecl 15093	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑗 ∈ (1...𝑀)(((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) ∈ ℝ)
404	367, 403	readdcld 10672	. . . . . . . . 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 11165	. . . . . . . . . . 11 ⊢ 0 ≤ 1
406	405	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ 1)
407	198	rpge0d 12438	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ 𝐸)
408	197, 199, 406, 407	addge0d 11218	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (1 + 𝐸))
409	67	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑄 ∈ ℝ)
410	354, 409, 345, 31	hsphoif 42865	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐻‘𝑄)‘(𝐷‘𝑗)):𝑊⟶ℝ)
411	161, 345, 112, 410	hoidmvcl 42871	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))) ∈ (0[,)+∞))
412	331, 411	sseldi 3967	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))) ∈ (0[,]+∞))
413	324, 365, 412, 279, 336	sge0ssrempt 42694	. . . . . . . . . . 11 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ (ℤ≥‘(𝑀 + 1)) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) ∈ ℝ)
414	159, 411	sseldi 3967	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))) ∈ ℝ)
415	209, 211, 414	syl2anc 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))) ∈ ℝ)
416	207, 415	fsumrecl 15093	. . . . . . . . . . 11 ⊢ (𝜑 → Σ𝑗 ∈ (1...𝑀)((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))) ∈ ℝ)
417	333, 339, 412	syl2anc 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝑀 + 1))) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))) ∈ (0[,]+∞))
418	202	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑍 ∈ (𝑊 ∖ 𝑌))
419	75, 67, 144	ltled 10790	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ≤ 𝑄)
420	419	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑆 ≤ 𝑄)
421	161, 345, 418, 7, 355, 409, 420, 354, 112, 31	hsphoidmvle2 42874	. . . . . . . . . . . . 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 586	. . . . . . . . . . . 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 42699	. . . . . . . . . . 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 483	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) = 0) → 𝜑)
425	211	adantr 483	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) = 0) → 𝑗 ∈ ℕ)
426		simpr 487	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) = 0) → (𝑃‘𝑗) = 0)
427		oveq2 7166	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃‘𝑗) = 0 → ((𝑄 − 𝑆) · (𝑃‘𝑗)) = ((𝑄 − 𝑆) · 0))
428	427	adantl 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) = 0) → ((𝑄 − 𝑆) · (𝑃‘𝑗)) = ((𝑄 − 𝑆) · 0))
429	174	mul01d 10841	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑄 − 𝑆) · 0) = 0)
430	429	ad2antrr 724	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) = 0) → ((𝑄 − 𝑆) · 0) = 0)
431	428, 430	eqtrd 2858	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) = 0) → ((𝑄 − 𝑆) · (𝑃‘𝑗)) = 0)
432	431	oveq2d 7174	. . . . . . . . . . . . . . . 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	addid1d 10842	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + 0) = ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))
434	433	adantr 483	. . . . . . . . . . . . . . . 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 2858	. . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . 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 5090	. . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ ¬ (𝑃‘𝑗) = 0) → (𝜑 ∧ 𝑗 ∈ (1...𝑀)))
440		neqne 3026	. . . . . . . . . . . . . . 15 ⊢ (¬ (𝑃‘𝑗) = 0 → (𝑃‘𝑗) ≠ 0)
441	440	adantl 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ ¬ (𝑃‘𝑗) = 0) → (𝑃‘𝑗) ≠ 0)
442	402	adantr 483	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → (((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) + ((𝑄 − 𝑆) · (𝑃‘𝑗))) ∈ ℝ)
443	209	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → 𝜑)
444	211	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → 𝑗 ∈ ℕ)
445		simpr 487	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → (𝑃‘𝑗) ≠ 0)
446	2	adantr 483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑍 ∈ (𝑋 ∖ 𝑌))
447	201	adantr 483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ¬ 𝑍 ∈ 𝑌)
448		eqid 2823	. . . . . . . . . . . . . . . . . . . . . 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 42877	. . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
452	218	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → 𝑌 ∈ Fin)
453	217	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
454		fveq2 6672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑌 = ∅ → (𝐿‘𝑌) = (𝐿‘∅))
455	454	oveqd 7175	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑌 = ∅ → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) = ((𝐽‘𝑗)(𝐿‘∅)(𝐾‘𝑗)))
456	455	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) = ((𝐽‘𝑗)(𝐿‘∅)(𝐾‘𝑗)))
457	249	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → (𝐽‘𝑗):𝑌⟶ℝ)
458		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑌 = ∅ → 𝑌 = ∅)
459	458	eqcomd 2829	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑌 = ∅ → ∅ = 𝑌)
460	459	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → ∅ = 𝑌)
461	460	feq2d 6502	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → ((𝐽‘𝑗):∅⟶ℝ ↔ (𝐽‘𝑗):𝑌⟶ℝ))
462	457, 461	mpbird 259	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → (𝐽‘𝑗):∅⟶ℝ)
463	270	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → (𝐾‘𝑗):𝑌⟶ℝ)
464	460	feq2d 6502	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → ((𝐾‘𝑗):∅⟶ℝ ↔ (𝐾‘𝑗):𝑌⟶ℝ))
465	463, 464	mpbird 259	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → (𝐾‘𝑗):∅⟶ℝ)
466	161, 462, 465	hoidmv0val 42872	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → ((𝐽‘𝑗)(𝐿‘∅)(𝐾‘𝑗)) = 0)
467	453, 456, 466	3eqtrd 2862	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑌 = ∅) → (𝑃‘𝑗) = 0)
468	467	adantlr 713	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ 𝑌 = ∅) → (𝑃‘𝑗) = 0)
469		neneq 3024	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑃‘𝑗) ≠ 0 → ¬ (𝑃‘𝑗) = 0)
470	469	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ 𝑌 = ∅) → ¬ (𝑃‘𝑗) = 0)
471	468, 470	pm2.65da 815	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ¬ 𝑌 = ∅)
472	471	neqned 3025	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → 𝑌 ≠ ∅)
473	249	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (𝐽‘𝑗):𝑌⟶ℝ)
474	270	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (𝐾‘𝑗):𝑌⟶ℝ)
475	161, 452, 472, 473, 474	hoidmvn0val 42873	. . . . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (𝐽‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹))
477	217	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
478	247	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2858	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝐽‘𝑗) = 𝐹)
480	268	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2858	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝐾‘𝑗) = 𝐹)
482	479, 481	oveq12d 7176	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 42879	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝜑 → (𝐹(𝐿‘𝑌)𝐹) = 0)
484	483	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝐹(𝐿‘𝑌)𝐹) = 0)
485	477, 482, 484	3eqtrd 2862	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝑃‘𝑗) = 0)
486	485	adantlr 713	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝑃‘𝑗) = 0)
487	469	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ¬ (𝑃‘𝑗) = 0)
488	486, 487	condan 816	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)))
489	488	iftrued 4477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2858	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (𝐽‘𝑗) = ((𝐶‘𝑗) ↾ 𝑌))
491	490	fveq1d 6674	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐽‘𝑗)‘𝑘) = (((𝐶‘𝑗) ↾ 𝑌)‘𝑘))
492	491	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ 𝑘 ∈ 𝑌) → ((𝐽‘𝑗)‘𝑘) = (((𝐶‘𝑗) ↾ 𝑌)‘𝑘))
493		fvres 6691	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 ∈ 𝑌 → (((𝐶‘𝑗) ↾ 𝑌)‘𝑘) = ((𝐶‘𝑗)‘𝑘))
494	493	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ 𝑘 ∈ 𝑌) → (((𝐶‘𝑗) ↾ 𝑌)‘𝑘) = ((𝐶‘𝑗)‘𝑘))
495	492, 494	eqtrd 2858	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ 𝑘 ∈ 𝑌) → ((𝐽‘𝑗)‘𝑘) = ((𝐶‘𝑗)‘𝑘))
496	268	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2858	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (𝐾‘𝑗) = ((𝐷‘𝑗) ↾ 𝑌))
499	498	fveq1d 6674	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐾‘𝑗)‘𝑘) = (((𝐷‘𝑗) ↾ 𝑌)‘𝑘))
500	499	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ 𝑘 ∈ 𝑌) → ((𝐾‘𝑗)‘𝑘) = (((𝐷‘𝑗) ↾ 𝑌)‘𝑘))
501		fvres 6691	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 ∈ 𝑌 → (((𝐷‘𝑗) ↾ 𝑌)‘𝑘) = ((𝐷‘𝑗)‘𝑘))
502	501	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ 𝑘 ∈ 𝑌) → (((𝐷‘𝑗) ↾ 𝑌)‘𝑘) = ((𝐷‘𝑗)‘𝑘))
503	500, 502	eqtrd 2858	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ 𝑘 ∈ 𝑌) → ((𝐾‘𝑗)‘𝑘) = ((𝐷‘𝑗)‘𝑘))
504	495, 503	oveq12d 7176	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ 𝑘 ∈ 𝑌) → (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)) = (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
505	504	fveq2d 6676	. . . . . . . . . . . . . . . . . . . . . . . 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 15279	. . . . . . . . . . . . . . . . . . . . . . 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 2858	. . . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → 𝑆 ∈ ℝ)
509	345	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → 𝑊 ∈ Fin)
510	31	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → (𝐷‘𝑗):𝑊⟶ℝ)
511		elun1 4154	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑘 ∈ 𝑌 → 𝑘 ∈ (𝑌 ∪ {𝑍}))
512	511, 7	eleqtrrdi 2926	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑘 ∈ 𝑌 → 𝑘 ∈ 𝑊)
513	512	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → 𝑘 ∈ 𝑊)
514	354, 508, 509, 510, 513	hsphoival 42868	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4475	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 ∈ 𝑌 → if(𝑘 ∈ 𝑌, ((𝐷‘𝑗)‘𝑘), if(((𝐷‘𝑗)‘𝑘) ≤ 𝑆, ((𝐷‘𝑗)‘𝑘), 𝑆)) = ((𝐷‘𝑗)‘𝑘))
516	515	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2858	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → (((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑘) = ((𝐷‘𝑗)‘𝑘))
518	517	oveq2d 7174	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → (((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑘)) = (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
519	518	fveq2d 6676	. . . . . . . . . . . . . . . . . . . . . . . . 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 15279	. . . . . . . . . . . . . . . . . . . . . . . 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 2829	. . . . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . 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 2863	. . . . . . . . . . . . . . . . . . . . 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 42868	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 4480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2858	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑍) = if(((𝐷‘𝑗)‘𝑍) ≤ 𝑆, ((𝐷‘𝑗)‘𝑍), 𝑆))
528	527	oveq2d 7174	. . . . . . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . . . 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 10693	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)‘𝑍) ∈ ℝ*)
531	530	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐶‘𝑗)‘𝑍) ∈ ℝ*)
532	33	rexrd 10693	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐷‘𝑗)‘𝑍) ∈ ℝ*)
533	532	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐷‘𝑗)‘𝑍) ∈ ℝ*)
534		icoltub 41791	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1367	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → 𝑆 < ((𝐷‘𝑗)‘𝑍))
536	355	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → 𝑆 ∈ ℝ)
537	33	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐷‘𝑗)‘𝑍) ∈ ℝ)
538	536, 537	ltnled 10789	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (𝑆 < ((𝐷‘𝑗)‘𝑍) ↔ ¬ ((𝐷‘𝑗)‘𝑍) ≤ 𝑆))
539	535, 538	mpbid 234	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ¬ ((𝐷‘𝑗)‘𝑍) ≤ 𝑆)
540	539	iffalsed 4480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → if(((𝐷‘𝑗)‘𝑍) ≤ 𝑆, ((𝐷‘𝑗)‘𝑍), 𝑆) = 𝑆)
541	540	oveq2d 7174	. . . . . . . . . . . . . . . . . . . . . . . 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 2858	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (((𝐶‘𝑗)‘𝑍)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑍)) = (((𝐶‘𝑗)‘𝑍)[,)𝑆))
543	542	fveq2d 6676	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (vol‘(((𝐶‘𝑗)‘𝑍)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑍))) = (vol‘(((𝐶‘𝑗)‘𝑍)[,)𝑆)))
544		volico 42275	. . . . . . . . . . . . . . . . . . . . . . . 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 597	. . . . . . . . . . . . . . . . . . . . . . 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 666	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (vol‘(((𝐶‘𝑗)‘𝑍)[,)𝑆)) = if(((𝐶‘𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶‘𝑗)‘𝑍)), 0))
547		iftrue 4475	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐶‘𝑗)‘𝑍) < 𝑆 → if(((𝐶‘𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶‘𝑗)‘𝑍)), 0) = (𝑆 − ((𝐶‘𝑗)‘𝑍)))
548	547	adantl 484	. . . . . . . . . . . . . . . . . . . . . . 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 4478	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (¬ ((𝐶‘𝑗)‘𝑍) < 𝑆 → if(((𝐶‘𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶‘𝑗)‘𝑍)), 0) = 0)
550	549	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐶‘𝑗)‘𝑍) < 𝑆) → if(((𝐶‘𝑗)‘𝑍) < 𝑆, (𝑆 − ((𝐶‘𝑗)‘𝑍)), 0) = 0)
551		simpll 765	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐶‘𝑗)‘𝑍) < 𝑆) → (𝜑 ∧ 𝑗 ∈ ℕ))
552		icogelb 12791	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1367	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐶‘𝑗)‘𝑍) ≤ 𝑆)
554	553	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐶‘𝑗)‘𝑍) < 𝑆) → ((𝐶‘𝑗)‘𝑍) ≤ 𝑆)
555		simpr 487	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐶‘𝑗)‘𝑍) < 𝑆) → ¬ ((𝐶‘𝑗)‘𝑍) < 𝑆)
556	554, 555	jca 514	. . . . . . . . . . . . . . . . . . . . . . . . . 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 10786	. . . . . . . . . . . . . . . . . . . . . . . . . 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 259	. . . . . . . . . . . . . . . . . . . . . . . . 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 2829	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝐶‘𝑗)‘𝑍) = 𝑆 → 𝑆 = ((𝐶‘𝑗)‘𝑍))
563	562	oveq1d 7173	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐶‘𝑗)‘𝑍) = 𝑆 → (𝑆 − ((𝐶‘𝑗)‘𝑍)) = (((𝐶‘𝑗)‘𝑍) − ((𝐶‘𝑗)‘𝑍)))
564	563	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ((𝐶‘𝑗)‘𝑍) = 𝑆) → (𝑆 − ((𝐶‘𝑗)‘𝑍)) = (((𝐶‘𝑗)‘𝑍) − ((𝐶‘𝑗)‘𝑍)))
565	385, 113	sseldi 3967	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)‘𝑍) ∈ ℂ)
566	565	subidd 10987	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝐶‘𝑗)‘𝑍) − ((𝐶‘𝑗)‘𝑍)) = 0)
567	566	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ((𝐶‘𝑗)‘𝑍) = 𝑆) → (((𝐶‘𝑗)‘𝑍) − ((𝐶‘𝑗)‘𝑍)) = 0)
568	564, 567	eqtr2d 2859	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ((𝐶‘𝑗)‘𝑍) = 𝑆) → 0 = (𝑆 − ((𝐶‘𝑗)‘𝑍)))
569	551, 560, 568	syl2anc 586	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐶‘𝑗)‘𝑍) < 𝑆) → 0 = (𝑆 − ((𝐶‘𝑗)‘𝑍)))
570	550, 569	eqtrd 2858	. . . . . . . . . . . . . . . . . . . . . . 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 811	. . . . . . . . . . . . . . . . . . . . . 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 2862	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → (vol‘(((𝐶‘𝑗)‘𝑍)[,)(((𝐻‘𝑆)‘(𝐷‘𝑗))‘𝑍))) = (𝑆 − ((𝐶‘𝑗)‘𝑍)))
573	523, 572	oveq12d 7176	. . . . . . . . . . . . . . . . . . . 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	sseldi 3967	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ ℂ)
575	355, 113	resubcld 11070	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑆 − ((𝐶‘𝑗)‘𝑍)) ∈ ℝ)
576	575	recnd 10671	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑆 − ((𝐶‘𝑗)‘𝑍)) ∈ ℂ)
577	574, 576	mulcomd 10664	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑃‘𝑗) · (𝑆 − ((𝐶‘𝑗)‘𝑍))) = ((𝑆 − ((𝐶‘𝑗)‘𝑍)) · (𝑃‘𝑗)))
578	577	adantr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ((𝑃‘𝑗) · (𝑆 − ((𝐶‘𝑗)‘𝑍))) = ((𝑆 − ((𝐶‘𝑗)‘𝑍)) · (𝑃‘𝑗)))
579	450, 573, 578	3eqtrd 2862	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))) = ((𝑆 − ((𝐶‘𝑗)‘𝑍)) · (𝑃‘𝑗)))
580	579	oveq1d 7173	. . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑄 − 𝑆) ∈ ℂ)
582	576, 581, 574	adddird 10668	. . . . . . . . . . . . . . . . . . . 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 2829	. . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . 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 10844	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑆 − ((𝐶‘𝑗)‘𝑍)) + (𝑄 − 𝑆)) = ((𝑄 − 𝑆) + (𝑆 − ((𝐶‘𝑗)‘𝑍))))
586	153	adantr 483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑄 ∈ ℂ)
587	154	adantr 483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑆 ∈ ℂ)
588	586, 587, 565	npncand 11023	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑄 − 𝑆) + (𝑆 − ((𝐶‘𝑗)‘𝑍))) = (𝑄 − ((𝐶‘𝑗)‘𝑍)))
589	585, 588	eqtrd 2858	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑆 − ((𝐶‘𝑗)‘𝑍)) + (𝑄 − 𝑆)) = (𝑄 − ((𝐶‘𝑗)‘𝑍)))
590	589	oveq1d 7173	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝑆 − ((𝐶‘𝑗)‘𝑍)) + (𝑄 − 𝑆)) · (𝑃‘𝑗)) = ((𝑄 − ((𝐶‘𝑗)‘𝑍)) · (𝑃‘𝑗)))
591	590	adantr 483	. . . . . . . . . . . . . . . . . 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 2862	. . . . . . . . . . . . . . . . 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 2823	. . . . . . . . . . . . . . . . . . . 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 42877	. . . . . . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . 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 2829	. . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → 𝑄 ∈ ℝ)
600	354, 599, 509, 510, 513	hsphoival 42868	. . . . . . . . . . . . . . . . . . . . . . . . 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 4475	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 ∈ 𝑌 → if(𝑘 ∈ 𝑌, ((𝐷‘𝑗)‘𝑘), if(((𝐷‘𝑗)‘𝑘) ≤ 𝑄, ((𝐷‘𝑗)‘𝑘), 𝑄)) = ((𝐷‘𝑗)‘𝑘))
602	601	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . 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 2858	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → (((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑘) = ((𝐷‘𝑗)‘𝑘))
604	603	oveq2d 7174	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑌) → (((𝐶‘𝑗)‘𝑘)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑘)) = (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
605	604	fveq2d 6676	. . . . . . . . . . . . . . . . . . . . . 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 15279	. . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . 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 2868	. . . . . . . . . . . . . . . . . . 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 42868	. . . . . . . . . . . . . . . . . . . . . . . 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 593	. . . . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . 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 4480	. . . . . . . . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . . . . . 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 593	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝐷‘𝑗)‘𝑍) ∈ ℝ)
616	615	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ ((𝐷‘𝑗)‘𝑍) = 𝑄) → ((𝐷‘𝑗)‘𝑍) ∈ ℝ)
617		simpr 487	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ ((𝐷‘𝑗)‘𝑍) = 𝑄) → ((𝐷‘𝑗)‘𝑍) = 𝑄)
618	616, 617	eqled 10745	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ ((𝐷‘𝑗)‘𝑍) = 𝑄) → ((𝐷‘𝑗)‘𝑍) ≤ 𝑄)
619	618	iftrued 4477	. . . . . . . . . . . . . . . . . . . . . . . . 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 2858	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ ((𝐷‘𝑗)‘𝑍) = 𝑄) → if(((𝐷‘𝑗)‘𝑍) ≤ 𝑄, ((𝐷‘𝑗)‘𝑍), 𝑄) = 𝑄)
621	620	adantlr 713	. . . . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → 𝑄 ∈ ℝ)
623	622	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ ¬ ((𝐷‘𝑗)‘𝑍) = 𝑄) → 𝑄 ∈ ℝ)
624	623	adantlr 713	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐷‘𝑗)‘𝑍) = 𝑄) → 𝑄 ∈ ℝ)
625	615	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ ¬ ((𝐷‘𝑗)‘𝑍) = 𝑄) → ((𝐷‘𝑗)‘𝑍) ∈ ℝ)
626	625	adantlr 713	. . . . . . . . . . . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ∃𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 𝑥 ≤ 𝑦)
630		simplr 767	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → 𝑗 ∈ (1...𝑀))
631	210, 488	sylanl2 679	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)))
632	630, 631	jca 514	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → (𝑗 ∈ (1...𝑀) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))))
633		rabid 3380	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 236	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → 𝑗 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))})
635		eqidd 2824	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐷‘𝑗)‘𝑍) = ((𝐷‘𝑗)‘𝑍))
636		fveq2 6672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑖 = 𝑗 → (𝐷‘𝑖) = (𝐷‘𝑗))
637	636	fveq1d 6674	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑖 = 𝑗 → ((𝐷‘𝑖)‘𝑍) = ((𝐷‘𝑗)‘𝑍))
638	637	eqeq2d 2834	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑖 = 𝑗 → (((𝐷‘𝑗)‘𝑍) = ((𝐷‘𝑖)‘𝑍) ↔ ((𝐷‘𝑗)‘𝑍) = ((𝐷‘𝑗)‘𝑍)))
639	638	rspcev 3625	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6687	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐷‘𝑗)‘𝑍) ∈ V)
642	16, 640, 641	elrnmptd 41447	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2926	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐷‘𝑗)‘𝑍) ∈ 𝑂)
644		elun2 4155	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2917	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐷‘𝑗)‘𝑍) ∈ 𝑉)
648		lbinfle 11598	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1367	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → inf(𝑉, ℝ, < ) ≤ ((𝐷‘𝑗)‘𝑍))
650	627, 649	eqbrtrd 5090	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → 𝑄 ≤ ((𝐷‘𝑗)‘𝑍))
651	650	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐷‘𝑗)‘𝑍) = 𝑄) → 𝑄 ≤ ((𝐷‘𝑗)‘𝑍))
652		neqne 3026	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (¬ ((𝐷‘𝑗)‘𝑍) = 𝑄 → ((𝐷‘𝑗)‘𝑍) ≠ 𝑄)
653	652	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐷‘𝑗)‘𝑍) = 𝑄) → ((𝐷‘𝑗)‘𝑍) ≠ 𝑄)
654	624, 626, 651, 653	leneltd 10796	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐷‘𝑗)‘𝑍) = 𝑄) → 𝑄 < ((𝐷‘𝑗)‘𝑍))
655	624, 626	ltnled 10789	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐷‘𝑗)‘𝑍) = 𝑄) → (𝑄 < ((𝐷‘𝑗)‘𝑍) ↔ ¬ ((𝐷‘𝑗)‘𝑍) ≤ 𝑄))
656	654, 655	mpbid 234	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) ∧ ¬ ((𝐷‘𝑗)‘𝑍) = 𝑄) → ¬ ((𝐷‘𝑗)‘𝑍) ≤ 𝑄)
657	656	iffalsed 4480	. . . . . . . . . . . . . . . . . . . . . . 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 811	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → if(((𝐷‘𝑗)‘𝑍) ≤ 𝑄, ((𝐷‘𝑗)‘𝑍), 𝑄) = 𝑄)
659	612, 614, 658	3eqtrd 2862	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → (((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑍) = 𝑄)
660	659	oveq2d 7174	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → (((𝐶‘𝑗)‘𝑍)[,)(((𝐻‘𝑄)‘(𝐷‘𝑗))‘𝑍)) = (((𝐶‘𝑗)‘𝑍)[,)𝑄))
661	660	fveq2d 6676	. . . . . . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝐶‘𝑗)‘𝑍) ∈ ℝ)
663	662	adantr 483	. . . . . . . . . . . . . . . . . . . 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 42275	. . . . . . . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . . . . . . 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 10800	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑀)) ∧ (𝑃‘𝑗) ≠ 0) → ((𝐶‘𝑗)‘𝑍) < 𝑄)
671	670	iftrued 4477	. . . . . . . . . . . . . . . . . . 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 2862	. . . . . . . . . . . . . . . . . 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 7176	. . . . . . . . . . . . . . . . 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	sseldi 3967	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝐶‘𝑗)‘𝑍) ∈ ℂ)
676	674, 675	subcld 10999	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → (𝑄 − ((𝐶‘𝑗)‘𝑍)) ∈ ℂ)
677	305, 676	mulcomd 10664	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝑃‘𝑗) · (𝑄 − ((𝐶‘𝑗)‘𝑍))) = ((𝑄 − ((𝐶‘𝑗)‘𝑍)) · (𝑃‘𝑗)))
678	677	adantr 483	. . . . . . . . . . . . . . . . 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 2862	. . . . . . . . . . . . . . . 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 2861	. . . . . . . . . . . . . . 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 10745	. . . . . . . . . . . . . 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 586	. . . . . . . . . . . . 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 811	. . . . . . . . . . . 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 15156	. . . . . . . . . . 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 41591	. . . . . . . . . 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 5157	. . . . . . . . . . . . 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 6676	. . . . . . . . . . . 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 593	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))) ∈ (0[,]+∞))
689	324, 325, 326, 330, 417, 688	sge0splitmpt 42700	. . . . . . . . . . . 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 2858	. . . . . . . . . . 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 586	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))) ∈ (0[,)+∞))
692	207, 691	sge0fsummpt 42679	. . . . . . . . . . . . 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 2915	. . . . . . . . . . . 12 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ (1...𝑀) ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))) ∈ ℝ)
694		rexadd 12628	. . . . . . . . . . . 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 586	. . . . . . . . . . 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 7174	. . . . . . . . . . 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 2863	. . . . . . . . . 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 5094	. . . . . . . . 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 11582	. . . . . . . 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 5090	. . . . . . 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 10799	. . . . . 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 5090	. . . . 5 ⊢ (𝜑 → (𝐺 · (𝑄 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗)))))))
703	152, 702	jca 514	. . . 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 7165	. . . . . . 7 ⊢ (𝑧 = 𝑄 → (𝑧 − (𝐴‘𝑍)) = (𝑄 − (𝐴‘𝑍)))
705	704	oveq2d 7174	. . . . . 6 ⊢ (𝑧 = 𝑄 → (𝐺 · (𝑧 − (𝐴‘𝑍))) = (𝐺 · (𝑄 − (𝐴‘𝑍))))
706		fveq2 6672	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑄 → (𝐻‘𝑧) = (𝐻‘𝑄))
707	706	fveq1d 6674	. . . . . . . . . 10 ⊢ (𝑧 = 𝑄 → ((𝐻‘𝑧)‘(𝐷‘𝑗)) = ((𝐻‘𝑄)‘(𝐷‘𝑗)))
708	707	oveq2d 7174	. . . . . . . . 9 ⊢ (𝑧 = 𝑄 → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))) = ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑄)‘(𝐷‘𝑗))))
709	708	mpteq2dv 5164	. . . . . . . 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 6676	. . . . . . 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 7174	. . . . . 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 5081	. . . . 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 3682	. . . 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 236	. . 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 2926	. 2 ⊢ (𝜑 → 𝑄 ∈ 𝑈)
716		breq2 5072	. . 3 ⊢ (𝑢 = 𝑄 → (𝑆 < 𝑢 ↔ 𝑆 < 𝑄))
717	716	rspcev 3625	. 2 ⊢ ((𝑄 ∈ 𝑈 ∧ 𝑆 < 𝑄) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
718	715, 144, 717	syl2anc 586	1 ⊢ (𝜑 → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  ∀wral 3140  ∃wrex 3141  {crab 3144  Vcvv 3496   ∖ cdif 3935   ∪ cun 3936   ∩ cin 3937   ⊆ wss 3938  ∅c0 4293  ifcif 4469  {csn 4569   class class class wbr 5068   ↦ cmpt 5148   Or wor 5475  ran crn 5558   ↾ cres 5559  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158   ∈ cmpo 7160   ↑_m cmap 8408  Fincfn 8511  infcinf 8907  ℂcc 10537  ℝcr 10538  0cc0 10539  1c1 10540   + caddc 10542   · cmul 10544  +∞cpnf 10674  ℝ^*cxr 10676   < clt 10677   ≤ cle 10678   − cmin 10872  ℕcn 11640  ℤ_≥cuz 12246  ℝ⁺crp 12392   +_𝑒 cxad 12508  [,)cico 12743  [,]cicc 12744  ...cfz 12895  Σcsu 15044  ∏cprod 15261  volcvol 24066  Σ^{^}csumge0 42651

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-inf2 9106  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616  ax-pre-sup 10617

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-se 5517  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-isom 6366  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-of 7411  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-2o 8105  df-oadd 8108  df-er 8291  df-map 8410  df-pm 8411  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-fi 8877  df-sup 8908  df-inf 8909  df-oi 8976  df-dju 9332  df-card 9370  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-div 11300  df-nn 11641  df-2 11703  df-3 11704  df-n0 11901  df-z 11985  df-uz 12247  df-q 12352  df-rp 12393  df-xneg 12510  df-xadd 12511  df-xmul 12512  df-ioo 12745  df-ico 12747  df-icc 12748  df-fz 12896  df-fzo 13037  df-fl 13165  df-seq 13373  df-exp 13433  df-hash 13694  df-cj 14460  df-re 14461  df-im 14462  df-sqrt 14596  df-abs 14597  df-clim 14847  df-rlim 14848  df-sum 15045  df-prod 15262  df-rest 16698  df-topgen 16719  df-psmet 20539  df-xmet 20540  df-met 20541  df-bl 20542  df-mopn 20543  df-top 21504  df-topon 21521  df-bases 21556  df-cmp 21997  df-ovol 24067  df-vol 24068  df-sumge0 42652

This theorem is referenced by:  hoidmvlelem3  42886