Theorem hoidmvlelem2

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)

	Ref	Expression
Hypotheses	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 ) · Σ 𝑗 ∈ ( 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	hoidmvlelem2	⊢ ( 𝜑 → ∃ 𝑢 ∈ 𝑈 𝑆 < 𝑢 )

Proof

Step	Hyp	Ref	Expression
1		hoidmvlelem2.l	⊢ 𝐿 = ( 𝑥 ∈ Fin ↦ ( 𝑎 ∈ ( ℝ ↑m 𝑥 ) , 𝑏 ∈ ( ℝ ↑m 𝑥 ) ↦ if ( 𝑥 = ∅ , 0 , ∏ 𝑘 ∈ 𝑥 ( vol ‘ ( ( 𝑎 ‘ 𝑘 ) [,) ( 𝑏 ‘ 𝑘 ) ) ) ) ) )
2		hoidmvlelem2.x	⊢ ( 𝜑 → 𝑋 ∈ Fin )
3		hoidmvlelem2.y	⊢ ( 𝜑 → 𝑌 ⊆ 𝑋 )
4		hoidmvlelem2.z	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑋 ∖ 𝑌 ) )
5		hoidmvlelem2.w	⊢ 𝑊 = ( 𝑌 ∪ { 𝑍 } )
6		hoidmvlelem2.a	⊢ ( 𝜑 → 𝐴 : 𝑊 ⟶ ℝ )
7		hoidmvlelem2.b	⊢ ( 𝜑 → 𝐵 : 𝑊 ⟶ ℝ )
8		hoidmvlelem2.c	⊢ ( 𝜑 → 𝐶 : ℕ ⟶ ( ℝ ↑m 𝑊 ) )
9		hoidmvlelem2.f	⊢ 𝐹 = ( 𝑦 ∈ 𝑌 ↦ 0 )
10		hoidmvlelem2.j	⊢ 𝐽 = ( 𝑗 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) )
11		hoidmvlelem2.d	⊢ ( 𝜑 → 𝐷 : ℕ ⟶ ( ℝ ↑m 𝑊 ) )
12		hoidmvlelem2.k	⊢ 𝐾 = ( 𝑗 ∈ ℕ ↦ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) )
13		hoidmvlelem2.r	⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( 𝐷 ‘ 𝑗 ) ) ) ) ∈ ℝ )
14		hoidmvlelem2.h	⊢ 𝐻 = ( 𝑥 ∈ ℝ ↦ ( 𝑐 ∈ ( ℝ ↑m 𝑊 ) ↦ ( 𝑗 ∈ 𝑊 ↦ if ( 𝑗 ∈ 𝑌 , ( 𝑐 ‘ 𝑗 ) , if ( ( 𝑐 ‘ 𝑗 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑗 ) , 𝑥 ) ) ) ) )
15		hoidmvlelem2.g	⊢ 𝐺 = ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) )
16		hoidmvlelem2.e	⊢ ( 𝜑 → 𝐸 ∈ ℝ+ )
17		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 ) ) ) ) ) ) }
18		hoidmvlelem2.su	⊢ ( 𝜑 → 𝑆 ∈ 𝑈 )
19		hoidmvlelem2.sb	⊢ ( 𝜑 → 𝑆 < ( 𝐵 ‘ 𝑍 ) )
20		hoidmvlelem2.p	⊢ 𝑃 = ( 𝑗 ∈ ℕ ↦ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) )
21		hoidmvlelem2.m	⊢ ( 𝜑 → 𝑀 ∈ ℕ )
22		hoidmvlelem2.le	⊢ ( 𝜑 → 𝐺 ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( 𝑃 ‘ 𝑗 ) ) )
23		hoidmvlelem2.O	⊢ 𝑂 = ran ( 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) )
24		hoidmvlelem2.v	⊢ 𝑉 = ( { ( 𝐵 ‘ 𝑍 ) } ∪ 𝑂 )
25		hoidmvlelem2.q	⊢ 𝑄 = inf ( 𝑉 , ℝ , < )
26		snidg	⊢ ( 𝑍 ∈ ( 𝑋 ∖ 𝑌 ) → 𝑍 ∈ { 𝑍 } )
27	4 26	syl	⊢ ( 𝜑 → 𝑍 ∈ { 𝑍 } )
28		elun2	⊢ ( 𝑍 ∈ { 𝑍 } → 𝑍 ∈ ( 𝑌 ∪ { 𝑍 } ) )
29	27 28	syl	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑌 ∪ { 𝑍 } ) )
30	29 5	eleqtrrdi	⊢ ( 𝜑 → 𝑍 ∈ 𝑊 )
31	6 30	ffvelcdmd	⊢ ( 𝜑 → ( 𝐴 ‘ 𝑍 ) ∈ ℝ )
32	7 30	ffvelcdmd	⊢ ( 𝜑 → ( 𝐵 ‘ 𝑍 ) ∈ ℝ )
33	32	snssd	⊢ ( 𝜑 → { ( 𝐵 ‘ 𝑍 ) } ⊆ ℝ )
34		nfv	⊢ Ⅎ 𝑖 𝜑
35		eqid	⊢ ( 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 ) )
36		simpl	⊢ ( ( 𝜑 ∧ 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ) → 𝜑 )
37		fz1ssnn	⊢ ( 1 ... 𝑀 ) ⊆ ℕ
38		elrabi	⊢ ( 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } → 𝑖 ∈ ( 1 ... 𝑀 ) )
39	37 38	sselid	⊢ ( 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } → 𝑖 ∈ ℕ )
40	39	adantl	⊢ ( ( 𝜑 ∧ 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ) → 𝑖 ∈ ℕ )
41		eleq1w	⊢ ( 𝑗 = 𝑖 → ( 𝑗 ∈ ℕ ↔ 𝑖 ∈ ℕ ) )
42	41	anbi2d	⊢ ( 𝑗 = 𝑖 → ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ↔ ( 𝜑 ∧ 𝑖 ∈ ℕ ) ) )
43		fveq2	⊢ ( 𝑗 = 𝑖 → ( 𝐷 ‘ 𝑗 ) = ( 𝐷 ‘ 𝑖 ) )
44	43	fveq1d	⊢ ( 𝑗 = 𝑖 → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) )
45	44	eleq1d	⊢ ( 𝑗 = 𝑖 → ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ ↔ ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ∈ ℝ ) )
46	42 45	imbi12d	⊢ ( 𝑗 = 𝑖 → ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ ) ↔ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ∈ ℝ ) ) )
47	11	ffvelcdmda	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐷 ‘ 𝑗 ) ∈ ( ℝ ↑m 𝑊 ) )
48		elmapi	⊢ ( ( 𝐷 ‘ 𝑗 ) ∈ ( ℝ ↑m 𝑊 ) → ( 𝐷 ‘ 𝑗 ) : 𝑊 ⟶ ℝ )
49	47 48	syl	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐷 ‘ 𝑗 ) : 𝑊 ⟶ ℝ )
50	30	adantr	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑍 ∈ 𝑊 )
51	49 50	ffvelcdmd	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ )
52	46 51	chvarvv	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ∈ ℝ )
53	36 40 52	syl2anc	⊢ ( ( 𝜑 ∧ 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ) → ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ∈ ℝ )
54	34 35 53	rnmptssd	⊢ ( 𝜑 → ran ( 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ⊆ ℝ )
55	23 54	eqsstrid	⊢ ( 𝜑 → 𝑂 ⊆ ℝ )
56	33 55	unssd	⊢ ( 𝜑 → ( { ( 𝐵 ‘ 𝑍 ) } ∪ 𝑂 ) ⊆ ℝ )
57	24 56	eqsstrid	⊢ ( 𝜑 → 𝑉 ⊆ ℝ )
58		ltso	⊢ < Or ℝ
59	58	a1i	⊢ ( 𝜑 → < Or ℝ )
60		snfi	⊢ { ( 𝐵 ‘ 𝑍 ) } ∈ Fin
61	60	a1i	⊢ ( 𝜑 → { ( 𝐵 ‘ 𝑍 ) } ∈ Fin )
62		fzfi	⊢ ( 1 ... 𝑀 ) ∈ Fin
63		rabfi	⊢ ( ( 1 ... 𝑀 ) ∈ Fin → { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ∈ Fin )
64	62 63	ax-mp	⊢ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ∈ Fin
65	64	a1i	⊢ ( 𝜑 → { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ∈ Fin )
66	35	rnmptfi	⊢ ( { 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 )
67	65 66	syl	⊢ ( 𝜑 → ran ( 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ∈ Fin )
68	23 67	eqeltrid	⊢ ( 𝜑 → 𝑂 ∈ Fin )
69		unfi	⊢ ( ( { ( 𝐵 ‘ 𝑍 ) } ∈ Fin ∧ 𝑂 ∈ Fin ) → ( { ( 𝐵 ‘ 𝑍 ) } ∪ 𝑂 ) ∈ Fin )
70	61 68 69	syl2anc	⊢ ( 𝜑 → ( { ( 𝐵 ‘ 𝑍 ) } ∪ 𝑂 ) ∈ Fin )
71	24 70	eqeltrid	⊢ ( 𝜑 → 𝑉 ∈ Fin )
72		fvex	⊢ ( 𝐵 ‘ 𝑍 ) ∈ V
73	72	snid	⊢ ( 𝐵 ‘ 𝑍 ) ∈ { ( 𝐵 ‘ 𝑍 ) }
74		elun1	⊢ ( ( 𝐵 ‘ 𝑍 ) ∈ { ( 𝐵 ‘ 𝑍 ) } → ( 𝐵 ‘ 𝑍 ) ∈ ( { ( 𝐵 ‘ 𝑍 ) } ∪ 𝑂 ) )
75	73 74	ax-mp	⊢ ( 𝐵 ‘ 𝑍 ) ∈ ( { ( 𝐵 ‘ 𝑍 ) } ∪ 𝑂 )
76	24	eqcomi	⊢ ( { ( 𝐵 ‘ 𝑍 ) } ∪ 𝑂 ) = 𝑉
77	75 76	eleqtri	⊢ ( 𝐵 ‘ 𝑍 ) ∈ 𝑉
78	77	a1i	⊢ ( 𝜑 → ( 𝐵 ‘ 𝑍 ) ∈ 𝑉 )
79		ne0i	⊢ ( ( 𝐵 ‘ 𝑍 ) ∈ 𝑉 → 𝑉 ≠ ∅ )
80	78 79	syl	⊢ ( 𝜑 → 𝑉 ≠ ∅ )
81		fiinfcl	⊢ ( ( < Or ℝ ∧ ( 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅ ∧ 𝑉 ⊆ ℝ ) ) → inf ( 𝑉 , ℝ , < ) ∈ 𝑉 )
82	59 71 80 57 81	syl13anc	⊢ ( 𝜑 → inf ( 𝑉 , ℝ , < ) ∈ 𝑉 )
83	25 82	eqeltrid	⊢ ( 𝜑 → 𝑄 ∈ 𝑉 )
84	57 83	sseldd	⊢ ( 𝜑 → 𝑄 ∈ ℝ )
85		ssrab2	⊢ { 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 ) )
86	17 85	eqsstri	⊢ 𝑈 ⊆ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) )
87	86	a1i	⊢ ( 𝜑 → 𝑈 ⊆ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) )
88	31 32	iccssred	⊢ ( 𝜑 → ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ⊆ ℝ )
89	87 88	sstrd	⊢ ( 𝜑 → 𝑈 ⊆ ℝ )
90	89 18	sseldd	⊢ ( 𝜑 → 𝑆 ∈ ℝ )
91	31	rexrd	⊢ ( 𝜑 → ( 𝐴 ‘ 𝑍 ) ∈ ℝ* )
92	32	rexrd	⊢ ( 𝜑 → ( 𝐵 ‘ 𝑍 ) ∈ ℝ* )
93	86 18	sselid	⊢ ( 𝜑 → 𝑆 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) )
94		iccgelb	⊢ ( ( ( 𝐴 ‘ 𝑍 ) ∈ ℝ* ∧ ( 𝐵 ‘ 𝑍 ) ∈ ℝ* ∧ 𝑆 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ) → ( 𝐴 ‘ 𝑍 ) ≤ 𝑆 )
95	91 92 93 94	syl3anc	⊢ ( 𝜑 → ( 𝐴 ‘ 𝑍 ) ≤ 𝑆 )
96	19	adantr	⊢ ( ( 𝜑 ∧ 𝑥 = ( 𝐵 ‘ 𝑍 ) ) → 𝑆 < ( 𝐵 ‘ 𝑍 ) )
97		id	⊢ ( 𝑥 = ( 𝐵 ‘ 𝑍 ) → 𝑥 = ( 𝐵 ‘ 𝑍 ) )
98	97	eqcomd	⊢ ( 𝑥 = ( 𝐵 ‘ 𝑍 ) → ( 𝐵 ‘ 𝑍 ) = 𝑥 )
99	98	adantl	⊢ ( ( 𝜑 ∧ 𝑥 = ( 𝐵 ‘ 𝑍 ) ) → ( 𝐵 ‘ 𝑍 ) = 𝑥 )
100	96 99	breqtrd	⊢ ( ( 𝜑 ∧ 𝑥 = ( 𝐵 ‘ 𝑍 ) ) → 𝑆 < 𝑥 )
101	100	adantlr	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) ∧ 𝑥 = ( 𝐵 ‘ 𝑍 ) ) → 𝑆 < 𝑥 )
102		simpll	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) ∧ ¬ 𝑥 = ( 𝐵 ‘ 𝑍 ) ) → 𝜑 )
103		id	⊢ ( 𝑥 ∈ 𝑉 → 𝑥 ∈ 𝑉 )
104	103 24	eleqtrdi	⊢ ( 𝑥 ∈ 𝑉 → 𝑥 ∈ ( { ( 𝐵 ‘ 𝑍 ) } ∪ 𝑂 ) )
105	104	adantr	⊢ ( ( 𝑥 ∈ 𝑉 ∧ ¬ 𝑥 = ( 𝐵 ‘ 𝑍 ) ) → 𝑥 ∈ ( { ( 𝐵 ‘ 𝑍 ) } ∪ 𝑂 ) )
106		elsni	⊢ ( 𝑥 ∈ { ( 𝐵 ‘ 𝑍 ) } → 𝑥 = ( 𝐵 ‘ 𝑍 ) )
107	106	con3i	⊢ ( ¬ 𝑥 = ( 𝐵 ‘ 𝑍 ) → ¬ 𝑥 ∈ { ( 𝐵 ‘ 𝑍 ) } )
108	107	adantl	⊢ ( ( 𝑥 ∈ 𝑉 ∧ ¬ 𝑥 = ( 𝐵 ‘ 𝑍 ) ) → ¬ 𝑥 ∈ { ( 𝐵 ‘ 𝑍 ) } )
109		elunnel1	⊢ ( ( 𝑥 ∈ ( { ( 𝐵 ‘ 𝑍 ) } ∪ 𝑂 ) ∧ ¬ 𝑥 ∈ { ( 𝐵 ‘ 𝑍 ) } ) → 𝑥 ∈ 𝑂 )
110	105 108 109	syl2anc	⊢ ( ( 𝑥 ∈ 𝑉 ∧ ¬ 𝑥 = ( 𝐵 ‘ 𝑍 ) ) → 𝑥 ∈ 𝑂 )
111	110	adantll	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) ∧ ¬ 𝑥 = ( 𝐵 ‘ 𝑍 ) ) → 𝑥 ∈ 𝑂 )
112		id	⊢ ( 𝑥 ∈ 𝑂 → 𝑥 ∈ 𝑂 )
113	112 23	eleqtrdi	⊢ ( 𝑥 ∈ 𝑂 → 𝑥 ∈ ran ( 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ↦ ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) )
114		vex	⊢ 𝑥 ∈ V
115	35	elrnmpt	⊢ ( 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 ) ) )
116	114 115	ax-mp	⊢ ( 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 ) )
117	113 116	sylib	⊢ ( 𝑥 ∈ 𝑂 → ∃ 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } 𝑥 = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) )
118	117	adantl	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑂 ) → ∃ 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } 𝑥 = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) )
119		fveq2	⊢ ( 𝑗 = 𝑖 → ( 𝐶 ‘ 𝑗 ) = ( 𝐶 ‘ 𝑖 ) )
120	119	fveq1d	⊢ ( 𝑗 = 𝑖 → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) = ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) )
121	120	eleq1d	⊢ ( 𝑗 = 𝑖 → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ ↔ ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) ∈ ℝ ) )
122	42 121	imbi12d	⊢ ( 𝑗 = 𝑖 → ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ ) ↔ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) ∈ ℝ ) ) )
123	8	ffvelcdmda	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐶 ‘ 𝑗 ) ∈ ( ℝ ↑m 𝑊 ) )
124		elmapi	⊢ ( ( 𝐶 ‘ 𝑗 ) ∈ ( ℝ ↑m 𝑊 ) → ( 𝐶 ‘ 𝑗 ) : 𝑊 ⟶ ℝ )
125	123 124	syl	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐶 ‘ 𝑗 ) : 𝑊 ⟶ ℝ )
126	125 50	ffvelcdmd	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ )
127	122 126	chvarvv	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) ∈ ℝ )
128	127	rexrd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) ∈ ℝ* )
129	36 40 128	syl2anc	⊢ ( ( 𝜑 ∧ 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ) → ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) ∈ ℝ* )
130	52	rexrd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ ) → ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ∈ ℝ* )
131	36 40 130	syl2anc	⊢ ( ( 𝜑 ∧ 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ) → ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ∈ ℝ* )
132	120 44	oveq12d	⊢ ( 𝑗 = 𝑖 → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) = ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) )
133	132	eleq2d	⊢ ( 𝑗 = 𝑖 → ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ↔ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) )
134	133	elrab	⊢ ( 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ↔ ( 𝑖 ∈ ( 1 ... 𝑀 ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) )
135	134	biimpi	⊢ ( 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } → ( 𝑖 ∈ ( 1 ... 𝑀 ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) )
136	135	simprd	⊢ ( 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } → 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) )
137	136	adantl	⊢ ( ( 𝜑 ∧ 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ) → 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) )
138		icoltub	⊢ ( ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) ∈ ℝ* ∧ ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ∈ ℝ* ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑖 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) ) → 𝑆 < ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) )
139	129 131 137 138	syl3anc	⊢ ( ( 𝜑 ∧ 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ) → 𝑆 < ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) )
140	139	3adant3	⊢ ( ( 𝜑 ∧ 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ∧ 𝑥 = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) → 𝑆 < ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) )
141		id	⊢ ( 𝑥 = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) → 𝑥 = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) )
142	141	eqcomd	⊢ ( 𝑥 = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) → ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) = 𝑥 )
143	142	3ad2ant3	⊢ ( ( 𝜑 ∧ 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ∧ 𝑥 = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) → ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) = 𝑥 )
144	140 143	breqtrd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } ∧ 𝑥 = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ) → 𝑆 < 𝑥 )
145	144	3exp	⊢ ( 𝜑 → ( 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } → ( 𝑥 = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) → 𝑆 < 𝑥 ) ) )
146	145	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑂 ) → ( 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } → ( 𝑥 = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) → 𝑆 < 𝑥 ) ) )
147	146	rexlimdv	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑂 ) → ( ∃ 𝑖 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } 𝑥 = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) → 𝑆 < 𝑥 ) )
148	118 147	mpd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑂 ) → 𝑆 < 𝑥 )
149	102 111 148	syl2anc	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) ∧ ¬ 𝑥 = ( 𝐵 ‘ 𝑍 ) ) → 𝑆 < 𝑥 )
150	101 149	pm2.61dan	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) → 𝑆 < 𝑥 )
151	150	ralrimiva	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑉 𝑆 < 𝑥 )
152		breq2	⊢ ( 𝑥 = inf ( 𝑉 , ℝ , < ) → ( 𝑆 < 𝑥 ↔ 𝑆 < inf ( 𝑉 , ℝ , < ) ) )
153	152	rspcva	⊢ ( ( inf ( 𝑉 , ℝ , < ) ∈ 𝑉 ∧ ∀ 𝑥 ∈ 𝑉 𝑆 < 𝑥 ) → 𝑆 < inf ( 𝑉 , ℝ , < ) )
154	82 151 153	syl2anc	⊢ ( 𝜑 → 𝑆 < inf ( 𝑉 , ℝ , < ) )
155	25	eqcomi	⊢ inf ( 𝑉 , ℝ , < ) = 𝑄
156	155	a1i	⊢ ( 𝜑 → inf ( 𝑉 , ℝ , < ) = 𝑄 )
157	154 156	breqtrd	⊢ ( 𝜑 → 𝑆 < 𝑄 )
158	31 90 84 95 157	lelttrd	⊢ ( 𝜑 → ( 𝐴 ‘ 𝑍 ) < 𝑄 )
159	31 84 158	ltled	⊢ ( 𝜑 → ( 𝐴 ‘ 𝑍 ) ≤ 𝑄 )
160		fiminre	⊢ ( ( 𝑉 ⊆ ℝ ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅ ) → ∃ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 𝑥 ≤ 𝑦 )
161	57 71 80 160	syl3anc	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 𝑥 ≤ 𝑦 )
162		lbinfle	⊢ ( ( 𝑉 ⊆ ℝ ∧ ∃ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 𝑥 ≤ 𝑦 ∧ ( 𝐵 ‘ 𝑍 ) ∈ 𝑉 ) → inf ( 𝑉 , ℝ , < ) ≤ ( 𝐵 ‘ 𝑍 ) )
163	57 161 78 162	syl3anc	⊢ ( 𝜑 → inf ( 𝑉 , ℝ , < ) ≤ ( 𝐵 ‘ 𝑍 ) )
164	25 163	eqbrtrid	⊢ ( 𝜑 → 𝑄 ≤ ( 𝐵 ‘ 𝑍 ) )
165	31 32 84 159 164	eliccd	⊢ ( 𝜑 → 𝑄 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) )
166	84	recnd	⊢ ( 𝜑 → 𝑄 ∈ ℂ )
167	90	recnd	⊢ ( 𝜑 → 𝑆 ∈ ℂ )
168	31	recnd	⊢ ( 𝜑 → ( 𝐴 ‘ 𝑍 ) ∈ ℂ )
169	166 167 168	npncand	⊢ ( 𝜑 → ( ( 𝑄 − 𝑆 ) + ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) = ( 𝑄 − ( 𝐴 ‘ 𝑍 ) ) )
170	169	eqcomd	⊢ ( 𝜑 → ( 𝑄 − ( 𝐴 ‘ 𝑍 ) ) = ( ( 𝑄 − 𝑆 ) + ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) )
171	170	oveq2d	⊢ ( 𝜑 → ( 𝐺 · ( 𝑄 − ( 𝐴 ‘ 𝑍 ) ) ) = ( 𝐺 · ( ( 𝑄 − 𝑆 ) + ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) ) )
172		rge0ssre	⊢ ( 0 [,) +∞ ) ⊆ ℝ
173	2 3	ssfid	⊢ ( 𝜑 → 𝑌 ∈ Fin )
174		ssun1	⊢ 𝑌 ⊆ ( 𝑌 ∪ { 𝑍 } )
175	174 5	sseqtrri	⊢ 𝑌 ⊆ 𝑊
176	175	a1i	⊢ ( 𝜑 → 𝑌 ⊆ 𝑊 )
177	6 176	fssresd	⊢ ( 𝜑 → ( 𝐴 ↾ 𝑌 ) : 𝑌 ⟶ ℝ )
178	7 176	fssresd	⊢ ( 𝜑 → ( 𝐵 ↾ 𝑌 ) : 𝑌 ⟶ ℝ )
179	1 173 177 178	hoidmvcl	⊢ ( 𝜑 → ( ( 𝐴 ↾ 𝑌 ) ( 𝐿 ‘ 𝑌 ) ( 𝐵 ↾ 𝑌 ) ) ∈ ( 0 [,) +∞ ) )
180	15 179	eqeltrid	⊢ ( 𝜑 → 𝐺 ∈ ( 0 [,) +∞ ) )
181	172 180	sselid	⊢ ( 𝜑 → 𝐺 ∈ ℝ )
182	181	recnd	⊢ ( 𝜑 → 𝐺 ∈ ℂ )
183	166 167	subcld	⊢ ( 𝜑 → ( 𝑄 − 𝑆 ) ∈ ℂ )
184	167 168	subcld	⊢ ( 𝜑 → ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ∈ ℂ )
185	182 183 184	adddid	⊢ ( 𝜑 → ( 𝐺 · ( ( 𝑄 − 𝑆 ) + ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) ) = ( ( 𝐺 · ( 𝑄 − 𝑆 ) ) + ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) ) )
186	182 183	mulcld	⊢ ( 𝜑 → ( 𝐺 · ( 𝑄 − 𝑆 ) ) ∈ ℂ )
187	182 184	mulcld	⊢ ( 𝜑 → ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) ∈ ℂ )
188	186 187	addcomd	⊢ ( 𝜑 → ( ( 𝐺 · ( 𝑄 − 𝑆 ) ) + ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) ) = ( ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) + ( 𝐺 · ( 𝑄 − 𝑆 ) ) ) )
189	171 185 188	3eqtrd	⊢ ( 𝜑 → ( 𝐺 · ( 𝑄 − ( 𝐴 ‘ 𝑍 ) ) ) = ( ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) + ( 𝐺 · ( 𝑄 − 𝑆 ) ) ) )
190	84 90	jca	⊢ ( 𝜑 → ( 𝑄 ∈ ℝ ∧ 𝑆 ∈ ℝ ) )
191		resubcl	⊢ ( ( 𝑄 ∈ ℝ ∧ 𝑆 ∈ ℝ ) → ( 𝑄 − 𝑆 ) ∈ ℝ )
192	190 191	syl	⊢ ( 𝜑 → ( 𝑄 − 𝑆 ) ∈ ℝ )
193	181 192	jca	⊢ ( 𝜑 → ( 𝐺 ∈ ℝ ∧ ( 𝑄 − 𝑆 ) ∈ ℝ ) )
194		remulcl	⊢ ( ( 𝐺 ∈ ℝ ∧ ( 𝑄 − 𝑆 ) ∈ ℝ ) → ( 𝐺 · ( 𝑄 − 𝑆 ) ) ∈ ℝ )
195	193 194	syl	⊢ ( 𝜑 → ( 𝐺 · ( 𝑄 − 𝑆 ) ) ∈ ℝ )
196	90 31	jca	⊢ ( 𝜑 → ( 𝑆 ∈ ℝ ∧ ( 𝐴 ‘ 𝑍 ) ∈ ℝ ) )
197		resubcl	⊢ ( ( 𝑆 ∈ ℝ ∧ ( 𝐴 ‘ 𝑍 ) ∈ ℝ ) → ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ∈ ℝ )
198	196 197	syl	⊢ ( 𝜑 → ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ∈ ℝ )
199	181 198	jca	⊢ ( 𝜑 → ( 𝐺 ∈ ℝ ∧ ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ∈ ℝ ) )
200		remulcl	⊢ ( ( 𝐺 ∈ ℝ ∧ ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ∈ ℝ ) → ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) ∈ ℝ )
201	199 200	syl	⊢ ( 𝜑 → ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) ∈ ℝ )
202	195 201	jca	⊢ ( 𝜑 → ( ( 𝐺 · ( 𝑄 − 𝑆 ) ) ∈ ℝ ∧ ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) ∈ ℝ ) )
203		readdcl	⊢ ( ( ( 𝐺 · ( 𝑄 − 𝑆 ) ) ∈ ℝ ∧ ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) ∈ ℝ ) → ( ( 𝐺 · ( 𝑄 − 𝑆 ) ) + ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) ) ∈ ℝ )
204	202 203	syl	⊢ ( 𝜑 → ( ( 𝐺 · ( 𝑄 − 𝑆 ) ) + ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) ) ∈ ℝ )
205	188 204	eqeltrrd	⊢ ( 𝜑 → ( ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) + ( 𝐺 · ( 𝑄 − 𝑆 ) ) ) ∈ ℝ )
206		1red	⊢ ( 𝜑 → 1 ∈ ℝ )
207	16	rpred	⊢ ( 𝜑 → 𝐸 ∈ ℝ )
208	206 207	readdcld	⊢ ( 𝜑 → ( 1 + 𝐸 ) ∈ ℝ )
209	4	eldifbd	⊢ ( 𝜑 → ¬ 𝑍 ∈ 𝑌 )
210	30 209	eldifd	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑊 ∖ 𝑌 ) )
211	1 173 210 5 8 11 13 14 90	sge0hsphoire	⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ∈ ℝ )
212	208 211	remulcld	⊢ ( 𝜑 → ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ∈ ℝ )
213		fzfid	⊢ ( 𝜑 → ( 1 ... 𝑀 ) ∈ Fin )
214	192	adantr	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( 𝑄 − 𝑆 ) ∈ ℝ )
215		simpl	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → 𝜑 )
216		elfznn	⊢ ( 𝑗 ∈ ( 1 ... 𝑀 ) → 𝑗 ∈ ℕ )
217	216	adantl	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → 𝑗 ∈ ℕ )
218		id	⊢ ( 𝑗 ∈ ℕ → 𝑗 ∈ ℕ )
219		ovexd	⊢ ( 𝑗 ∈ ℕ → ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ∈ V )
220	20	fvmpt2	⊢ ( ( 𝑗 ∈ ℕ ∧ ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ∈ V ) → ( 𝑃 ‘ 𝑗 ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) )
221	218 219 220	syl2anc	⊢ ( 𝑗 ∈ ℕ → ( 𝑃 ‘ 𝑗 ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) )
222	221	adantl	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑃 ‘ 𝑗 ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) )
223	173	adantr	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑌 ∈ Fin )
224	175	a1i	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑌 ⊆ 𝑊 )
225	125 224	fssresd	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) : 𝑌 ⟶ ℝ )
226	225	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) : 𝑌 ⟶ ℝ )
227		iftrue	⊢ ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) )
228	227	adantl	⊢ ( ( ( 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 ) )
229	228	feq1d	⊢ ( ( ( 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 ) )
230	226 229	mpbird	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) : 𝑌 ⟶ ℝ )
231		0red	⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑌 ) → 0 ∈ ℝ )
232	231 9	fmptd	⊢ ( 𝜑 → 𝐹 : 𝑌 ⟶ ℝ )
233	232	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → 𝐹 : 𝑌 ⟶ ℝ )
234		iffalse	⊢ ( ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = 𝐹 )
235	234	adantl	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = 𝐹 )
236	235	feq1d	⊢ ( ( ( 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 ) )
237	233 236	mpbird	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) : 𝑌 ⟶ ℝ )
238	230 237	pm2.61dan	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) : 𝑌 ⟶ ℝ )
239		simpr	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑗 ∈ ℕ )
240		fvex	⊢ ( 𝐶 ‘ 𝑗 ) ∈ V
241	240	resex	⊢ ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ∈ V
242	241	a1i	⊢ ( 𝜑 → ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ∈ V )
243	2 3	ssexd	⊢ ( 𝜑 → 𝑌 ∈ V )
244		mptexg	⊢ ( 𝑌 ∈ V → ( 𝑦 ∈ 𝑌 ↦ 0 ) ∈ V )
245	243 244	syl	⊢ ( 𝜑 → ( 𝑦 ∈ 𝑌 ↦ 0 ) ∈ V )
246	9 245	eqeltrid	⊢ ( 𝜑 → 𝐹 ∈ V )
247	242 246	ifcld	⊢ ( 𝜑 → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ∈ V )
248	247	adantr	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ∈ V )
249	10	fvmpt2	⊢ ( ( 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 ) )
250	239 248 249	syl2anc	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐽 ‘ 𝑗 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) )
251	250	feq1d	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐽 ‘ 𝑗 ) : 𝑌 ⟶ ℝ ↔ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) : 𝑌 ⟶ ℝ ) )
252	238 251	mpbird	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐽 ‘ 𝑗 ) : 𝑌 ⟶ ℝ )
253	49 224	fssresd	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) : 𝑌 ⟶ ℝ )
254	253	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) : 𝑌 ⟶ ℝ )
255		iftrue	⊢ ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) )
256	255	adantl	⊢ ( ( ( 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 ) )
257	256	feq1d	⊢ ( ( ( 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 ) )
258	254 257	mpbird	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) : 𝑌 ⟶ ℝ )
259		iffalse	⊢ ( ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = 𝐹 )
260	259	adantl	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) = 𝐹 )
261	260	feq1d	⊢ ( ( ( 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 ) )
262	233 261	mpbird	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) : 𝑌 ⟶ ℝ )
263	258 262	pm2.61dan	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) : 𝑌 ⟶ ℝ )
264		fvex	⊢ ( 𝐷 ‘ 𝑗 ) ∈ V
265	264	resex	⊢ ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ∈ V
266	265	a1i	⊢ ( 𝜑 → ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ∈ V )
267	266 246	ifcld	⊢ ( 𝜑 → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ∈ V )
268	267	adantr	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) ∈ V )
269	12	fvmpt2	⊢ ( ( 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 ) )
270	239 268 269	syl2anc	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐾 ‘ 𝑗 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) )
271	270	feq1d	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐾 ‘ 𝑗 ) : 𝑌 ⟶ ℝ ↔ if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) : 𝑌 ⟶ ℝ ) )
272	263 271	mpbird	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐾 ‘ 𝑗 ) : 𝑌 ⟶ ℝ )
273	1 223 252 272	hoidmvcl	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) ∈ ( 0 [,) +∞ ) )
274	222 273	eqeltrd	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑃 ‘ 𝑗 ) ∈ ( 0 [,) +∞ ) )
275	172 274	sselid	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑃 ‘ 𝑗 ) ∈ ℝ )
276	215 217 275	syl2anc	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( 𝑃 ‘ 𝑗 ) ∈ ℝ )
277	214 276	remulcld	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ∈ ℝ )
278	213 277	fsumrecl	⊢ ( 𝜑 → Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ∈ ℝ )
279	208 278	remulcld	⊢ ( 𝜑 → ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ∈ ℝ )
280	212 279	readdcld	⊢ ( 𝜑 → ( ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) + ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) ∈ ℝ )
281	1 173 210 5 8 11 13 14 84	sge0hsphoire	⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ∈ ℝ )
282	208 281	remulcld	⊢ ( 𝜑 → ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ∈ ℝ )
283	18 17	eleqtrdi	⊢ ( 𝜑 → 𝑆 ∈ { 𝑧 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ∣ ( 𝐺 · ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) } )
284		oveq1	⊢ ( 𝑧 = 𝑆 → ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) = ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) )
285	284	oveq2d	⊢ ( 𝑧 = 𝑆 → ( 𝐺 · ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) ) = ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) )
286		fveq2	⊢ ( 𝑧 = 𝑆 → ( 𝐻 ‘ 𝑧 ) = ( 𝐻 ‘ 𝑆 ) )
287	286	fveq1d	⊢ ( 𝑧 = 𝑆 → ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) = ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) )
288	287	oveq2d	⊢ ( 𝑧 = 𝑆 → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) = ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) )
289	288	mpteq2dv	⊢ ( 𝑧 = 𝑆 → ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) )
290	289	fveq2d	⊢ ( 𝑧 = 𝑆 → ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) = ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) )
291	290	oveq2d	⊢ ( 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 ) ) ) ) ) ) )
292	285 291	breq12d	⊢ ( 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 ) ) ) ) ) ) ) )
293	292	elrab	⊢ ( 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 ) ) ) ) ) ) ) )
294	283 293	sylib	⊢ ( 𝜑 → ( 𝑆 ∈ ( ( 𝐴 ‘ 𝑍 ) [,] ( 𝐵 ‘ 𝑍 ) ) ∧ ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ) )
295	294	simprd	⊢ ( 𝜑 → ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) )
296	213 276	fsumrecl	⊢ ( 𝜑 → Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( 𝑃 ‘ 𝑗 ) ∈ ℝ )
297	208 296	remulcld	⊢ ( 𝜑 → ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( 𝑃 ‘ 𝑗 ) ) ∈ ℝ )
298		0red	⊢ ( 𝜑 → 0 ∈ ℝ )
299	90 84	posdifd	⊢ ( 𝜑 → ( 𝑆 < 𝑄 ↔ 0 < ( 𝑄 − 𝑆 ) ) )
300	157 299	mpbid	⊢ ( 𝜑 → 0 < ( 𝑄 − 𝑆 ) )
301	298 192 300	ltled	⊢ ( 𝜑 → 0 ≤ ( 𝑄 − 𝑆 ) )
302	181 297 192 301 22	lemul1ad	⊢ ( 𝜑 → ( 𝐺 · ( 𝑄 − 𝑆 ) ) ≤ ( ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( 𝑃 ‘ 𝑗 ) ) · ( 𝑄 − 𝑆 ) ) )
303	208	recnd	⊢ ( 𝜑 → ( 1 + 𝐸 ) ∈ ℂ )
304	296	recnd	⊢ ( 𝜑 → Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( 𝑃 ‘ 𝑗 ) ∈ ℂ )
305	303 304 183	mulassd	⊢ ( 𝜑 → ( ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( 𝑃 ‘ 𝑗 ) ) · ( 𝑄 − 𝑆 ) ) = ( ( 1 + 𝐸 ) · ( Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( 𝑃 ‘ 𝑗 ) · ( 𝑄 − 𝑆 ) ) ) )
306	276	recnd	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( 𝑃 ‘ 𝑗 ) ∈ ℂ )
307	213 183 306	fsummulc1	⊢ ( 𝜑 → ( Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( 𝑃 ‘ 𝑗 ) · ( 𝑄 − 𝑆 ) ) = Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑃 ‘ 𝑗 ) · ( 𝑄 − 𝑆 ) ) )
308	183	adantr	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( 𝑄 − 𝑆 ) ∈ ℂ )
309	306 308	mulcomd	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( 𝑃 ‘ 𝑗 ) · ( 𝑄 − 𝑆 ) ) = ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) )
310	309	sumeq2dv	⊢ ( 𝜑 → Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑃 ‘ 𝑗 ) · ( 𝑄 − 𝑆 ) ) = Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) )
311	307 310	eqtrd	⊢ ( 𝜑 → ( Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( 𝑃 ‘ 𝑗 ) · ( 𝑄 − 𝑆 ) ) = Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) )
312	311	oveq2d	⊢ ( 𝜑 → ( ( 1 + 𝐸 ) · ( Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( 𝑃 ‘ 𝑗 ) · ( 𝑄 − 𝑆 ) ) ) = ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) )
313	305 312	eqtrd	⊢ ( 𝜑 → ( ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( 𝑃 ‘ 𝑗 ) ) · ( 𝑄 − 𝑆 ) ) = ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) )
314	302 313	breqtrd	⊢ ( 𝜑 → ( 𝐺 · ( 𝑄 − 𝑆 ) ) ≤ ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) )
315	201 195 212 279 295 314	le2addd	⊢ ( 𝜑 → ( ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) + ( 𝐺 · ( 𝑄 − 𝑆 ) ) ) ≤ ( ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) + ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) )
316		nnsplit	⊢ ( 𝑀 ∈ ℕ → ℕ = ( ( 1 ... 𝑀 ) ∪ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) )
317	21 316	syl	⊢ ( 𝜑 → ℕ = ( ( 1 ... 𝑀 ) ∪ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) )
318		uncom	⊢ ( ( 1 ... 𝑀 ) ∪ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) = ( ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∪ ( 1 ... 𝑀 ) )
319	318	a1i	⊢ ( 𝜑 → ( ( 1 ... 𝑀 ) ∪ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) = ( ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∪ ( 1 ... 𝑀 ) ) )
320	317 319	eqtr2d	⊢ ( 𝜑 → ( ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∪ ( 1 ... 𝑀 ) ) = ℕ )
321	320	eqcomd	⊢ ( 𝜑 → ℕ = ( ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∪ ( 1 ... 𝑀 ) ) )
322	321	mpteq1d	⊢ ( 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	⊢ ( 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	⊢ Ⅎ 𝑗 𝜑
325		fvexd	⊢ ( 𝜑 → ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∈ V )
326		ovexd	⊢ ( 𝜑 → ( 1 ... 𝑀 ) ∈ V )
327		incom	⊢ ( ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∩ ( 1 ... 𝑀 ) ) = ( ( 1 ... 𝑀 ) ∩ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) )
328		nnuzdisj	⊢ ( ( 1 ... 𝑀 ) ∩ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) = ∅
329	327 328	eqtri	⊢ ( ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∩ ( 1 ... 𝑀 ) ) = ∅
330	329	a1i	⊢ ( 𝜑 → ( ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ∩ ( 1 ... 𝑀 ) ) = ∅ )
331		icossicc	⊢ ( 0 [,) +∞ ) ⊆ ( 0 [,] +∞ )
332		ssid	⊢ ( 0 [,) +∞ ) ⊆ ( 0 [,) +∞ )
333		simpl	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) → 𝜑 )
334	21	peano2nnd	⊢ ( 𝜑 → ( 𝑀 + 1 ) ∈ ℕ )
335		uznnssnn	⊢ ( ( 𝑀 + 1 ) ∈ ℕ → ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ⊆ ℕ )
336	334 335	syl	⊢ ( 𝜑 → ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ⊆ ℕ )
337	336	adantr	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) → ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ⊆ ℕ )
338		simpr	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) → 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) )
339	337 338	sseldd	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) → 𝑗 ∈ ℕ )
340		snfi	⊢ { 𝑍 } ∈ Fin
341	340	a1i	⊢ ( 𝜑 → { 𝑍 } ∈ Fin )
342		unfi	⊢ ( ( 𝑌 ∈ Fin ∧ { 𝑍 } ∈ Fin ) → ( 𝑌 ∪ { 𝑍 } ) ∈ Fin )
343	173 341 342	syl2anc	⊢ ( 𝜑 → ( 𝑌 ∪ { 𝑍 } ) ∈ Fin )
344	5 343	eqeltrid	⊢ ( 𝜑 → 𝑊 ∈ Fin )
345	344	adantr	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑊 ∈ Fin )
346		eleq1w	⊢ ( 𝑗 = 𝑙 → ( 𝑗 ∈ 𝑌 ↔ 𝑙 ∈ 𝑌 ) )
347		fveq2	⊢ ( 𝑗 = 𝑙 → ( 𝑐 ‘ 𝑗 ) = ( 𝑐 ‘ 𝑙 ) )
348	347	breq1d	⊢ ( 𝑗 = 𝑙 → ( ( 𝑐 ‘ 𝑗 ) ≤ 𝑥 ↔ ( 𝑐 ‘ 𝑙 ) ≤ 𝑥 ) )
349	348 347	ifbieq1d	⊢ ( 𝑗 = 𝑙 → if ( ( 𝑐 ‘ 𝑗 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑗 ) , 𝑥 ) = if ( ( 𝑐 ‘ 𝑙 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑙 ) , 𝑥 ) )
350	346 347 349	ifbieq12d	⊢ ( 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	⊢ ( 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	⊢ ( 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	⊢ ( 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	14 353	eqtri	⊢ 𝐻 = ( 𝑥 ∈ ℝ ↦ ( 𝑐 ∈ ( ℝ ↑m 𝑊 ) ↦ ( 𝑙 ∈ 𝑊 ↦ if ( 𝑙 ∈ 𝑌 , ( 𝑐 ‘ 𝑙 ) , if ( ( 𝑐 ‘ 𝑙 ) ≤ 𝑥 , ( 𝑐 ‘ 𝑙 ) , 𝑥 ) ) ) ) )
355	90	adantr	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑆 ∈ ℝ )
356	354 355 345 49	hsphoif	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) : 𝑊 ⟶ ℝ )
357	1 345 125 356	hoidmvcl	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ( 0 [,) +∞ ) )
358	333 339 357	syl2anc	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ( 0 [,) +∞ ) )
359	332 358	sselid	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ( 0 [,) +∞ ) )
360	331 359	sselid	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ( 0 [,] +∞ ) )
361	215 217 357	syl2anc	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ( 0 [,) +∞ ) )
362	331 361	sselid	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ( 0 [,] +∞ ) )
363	324 325 326 330 360 362	sge0splitmpt	⊢ ( 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	⊢ ℕ ∈ V
365	364	a1i	⊢ ( 𝜑 → ℕ ∈ V )
366	331 357	sselid	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ( 0 [,] +∞ ) )
367	324 365 366 211 336	sge0ssrempt	⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ∈ ℝ )
368	37	a1i	⊢ ( 𝜑 → ( 1 ... 𝑀 ) ⊆ ℕ )
369	324 365 366 211 368	sge0ssrempt	⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ∈ ℝ )
370		rexadd	⊢ ( ( ( 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	⊢ ( 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	⊢ ( 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	⊢ ( 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	⊢ ( 𝜑 → ( ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) + ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 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 ) · Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) )
375	372 211	eqeltrrd	⊢ ( 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	⊢ ( 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	278	recnd	⊢ ( 𝜑 → Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ∈ ℂ )
378	303 376 377	adddid	⊢ ( 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 ) ) ) ) ) ) + Σ 𝑗 ∈ ( 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 ) · Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) )
379	378	eqcomd	⊢ ( 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 ) · Σ 𝑗 ∈ ( 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 ) ) ) ) ) ) + Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) )
380	367	recnd	⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ∈ ℂ )
381	369	recnd	⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ∈ ℂ )
382	380 381 377	addassd	⊢ ( 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 ) ) ) ) ) ) + Σ 𝑗 ∈ ( 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 ) ) ) ) ) + Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) )
383	213 361	sge0fsummpt	⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) = Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) )
384	383	oveq1d	⊢ ( 𝜑 → ( ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) = ( Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) )
385		ax-resscn	⊢ ℝ ⊆ ℂ
386	172 385	sstri	⊢ ( 0 [,) +∞ ) ⊆ ℂ
387	386 357	sselid	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ℂ )
388	215 217 387	syl2anc	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ℂ )
389	192	adantr	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑄 − 𝑆 ) ∈ ℝ )
390	389 275	remulcld	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ∈ ℝ )
391	390	recnd	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ∈ ℂ )
392	217 391	syldan	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ∈ ℂ )
393	213 388 392	fsumadd	⊢ ( 𝜑 → Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) = ( Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) )
394	393	eqcomd	⊢ ( 𝜑 → ( Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) = Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) )
395	384 394	eqtrd	⊢ ( 𝜑 → ( ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) + Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) = Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) )
396	395	oveq2d	⊢ ( 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 ) ) ) ) ) + Σ 𝑗 ∈ ( 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 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) )
397	382 396	eqtrd	⊢ ( 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 ) ) ) ) ) ) + Σ 𝑗 ∈ ( 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 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) )
398	397	oveq2d	⊢ ( 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 ) ) ) ) ) ) + Σ 𝑗 ∈ ( 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 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) ) )
399	374 379 398	3eqtrd	⊢ ( 𝜑 → ( ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) + ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 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 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) ) )
400	172 357	sselid	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ℝ )
401	400 390	readdcld	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ∈ ℝ )
402	215 217 401	syl2anc	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ∈ ℝ )
403	213 402	fsumrecl	⊢ ( 𝜑 → Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ∈ ℝ )
404	367 403	readdcld	⊢ ( 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 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) ∈ ℝ )
405		0le1	⊢ 0 ≤ 1
406	405	a1i	⊢ ( 𝜑 → 0 ≤ 1 )
407	16	rpge0d	⊢ ( 𝜑 → 0 ≤ 𝐸 )
408	206 207 406 407	addge0d	⊢ ( 𝜑 → 0 ≤ ( 1 + 𝐸 ) )
409	84	adantr	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑄 ∈ ℝ )
410	354 409 345 49	hsphoif	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) : 𝑊 ⟶ ℝ )
411	1 345 125 410	hoidmvcl	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ( 0 [,) +∞ ) )
412	331 411	sselid	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ( 0 [,] +∞ ) )
413	324 365 412 281 336	sge0ssrempt	⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ∈ ℝ )
414	172 411	sselid	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ℝ )
415	215 217 414	syl2anc	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ℝ )
416	213 415	fsumrecl	⊢ ( 𝜑 → Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ℝ )
417	333 339 412	syl2anc	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ( 0 [,] +∞ ) )
418	210	adantr	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑍 ∈ ( 𝑊 ∖ 𝑌 ) )
419	90 84 157	ltled	⊢ ( 𝜑 → 𝑆 ≤ 𝑄 )
420	419	adantr	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑆 ≤ 𝑄 )
421	1 345 418 5 355 409 420 354 125 49	hsphoidmvle2	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ≤ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) )
422	333 339 421	syl2anc	⊢ ( ( 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	⊢ ( 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	215	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) = 0 ) → 𝜑 )
425	217	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) = 0 ) → 𝑗 ∈ ℕ )
426		simpr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) = 0 ) → ( 𝑃 ‘ 𝑗 ) = 0 )
427		oveq2	⊢ ( ( 𝑃 ‘ 𝑗 ) = 0 → ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) = ( ( 𝑄 − 𝑆 ) · 0 ) )
428	427	adantl	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) = 0 ) → ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) = ( ( 𝑄 − 𝑆 ) · 0 ) )
429	183	mul01d	⊢ ( 𝜑 → ( ( 𝑄 − 𝑆 ) · 0 ) = 0 )
430	429	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) = 0 ) → ( ( 𝑄 − 𝑆 ) · 0 ) = 0 )
431	428 430	eqtrd	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) = 0 ) → ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) = 0 )
432	431	oveq2d	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) = 0 ) → ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + 0 ) )
433	387	addridd	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + 0 ) = ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) )
434	433	adantr	⊢ ( ( ( 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	⊢ ( ( ( 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	⊢ ( ( ( 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	⊢ ( ( ( 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	⊢ ( ( ( 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	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ¬ ( 𝑃 ‘ 𝑗 ) = 0 ) → ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) )
440		neqne	⊢ ( ¬ ( 𝑃 ‘ 𝑗 ) = 0 → ( 𝑃 ‘ 𝑗 ) ≠ 0 )
441	440	adantl	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ¬ ( 𝑃 ‘ 𝑗 ) = 0 ) → ( 𝑃 ‘ 𝑗 ) ≠ 0 )
442	402	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ∈ ℝ )
443	215	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → 𝜑 )
444	217	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → 𝑗 ∈ ℕ )
445		simpr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( 𝑃 ‘ 𝑗 ) ≠ 0 )
446	4	adantr	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑍 ∈ ( 𝑋 ∖ 𝑌 ) )
447	209	adantr	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ¬ 𝑍 ∈ 𝑌 )
448		eqid	⊢ ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) = ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) ) )
449	1 223 446 447 5 125 356 448	hoiprodp1	⊢ ( ( 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	⊢ ( ( ( 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	222	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( 𝑃 ‘ 𝑗 ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) )
452	223	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → 𝑌 ∈ Fin )
453	222	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑌 = ∅ ) → ( 𝑃 ‘ 𝑗 ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) )
454		fveq2	⊢ ( 𝑌 = ∅ → ( 𝐿 ‘ 𝑌 ) = ( 𝐿 ‘ ∅ ) )
455	454	oveqd	⊢ ( 𝑌 = ∅ → ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ ∅ ) ( 𝐾 ‘ 𝑗 ) ) )
456	455	adantl	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑌 = ∅ ) → ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ ∅ ) ( 𝐾 ‘ 𝑗 ) ) )
457	252	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑌 = ∅ ) → ( 𝐽 ‘ 𝑗 ) : 𝑌 ⟶ ℝ )
458		id	⊢ ( 𝑌 = ∅ → 𝑌 = ∅ )
459	458	eqcomd	⊢ ( 𝑌 = ∅ → ∅ = 𝑌 )
460	459	adantl	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑌 = ∅ ) → ∅ = 𝑌 )
461	460	feq2d	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑌 = ∅ ) → ( ( 𝐽 ‘ 𝑗 ) : ∅ ⟶ ℝ ↔ ( 𝐽 ‘ 𝑗 ) : 𝑌 ⟶ ℝ ) )
462	457 461	mpbird	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑌 = ∅ ) → ( 𝐽 ‘ 𝑗 ) : ∅ ⟶ ℝ )
463	272	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑌 = ∅ ) → ( 𝐾 ‘ 𝑗 ) : 𝑌 ⟶ ℝ )
464	460	feq2d	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑌 = ∅ ) → ( ( 𝐾 ‘ 𝑗 ) : ∅ ⟶ ℝ ↔ ( 𝐾 ‘ 𝑗 ) : 𝑌 ⟶ ℝ ) )
465	463 464	mpbird	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑌 = ∅ ) → ( 𝐾 ‘ 𝑗 ) : ∅ ⟶ ℝ )
466	1 462 465	hoidmv0val	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑌 = ∅ ) → ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ ∅ ) ( 𝐾 ‘ 𝑗 ) ) = 0 )
467	453 456 466	3eqtrd	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑌 = ∅ ) → ( 𝑃 ‘ 𝑗 ) = 0 )
468	467	adantlr	⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ 𝑌 = ∅ ) → ( 𝑃 ‘ 𝑗 ) = 0 )
469		neneq	⊢ ( ( 𝑃 ‘ 𝑗 ) ≠ 0 → ¬ ( 𝑃 ‘ 𝑗 ) = 0 )
470	469	ad2antlr	⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ 𝑌 = ∅ ) → ¬ ( 𝑃 ‘ 𝑗 ) = 0 )
471	468 470	pm2.65da	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ¬ 𝑌 = ∅ )
472	471	neqned	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → 𝑌 ≠ ∅ )
473	252	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( 𝐽 ‘ 𝑗 ) : 𝑌 ⟶ ℝ )
474	272	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( 𝐾 ‘ 𝑗 ) : 𝑌 ⟶ ℝ )
475	1 452 472 473 474	hoidmvn0val	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) = ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) ) )
476	250	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( 𝐽 ‘ 𝑗 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) )
477	222	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( 𝑃 ‘ 𝑗 ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) )
478	250	adantr	⊢ ( ( ( 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	478 235	eqtrd	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( 𝐽 ‘ 𝑗 ) = 𝐹 )
480	270	adantr	⊢ ( ( ( 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	480 260	eqtrd	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( 𝐾 ‘ 𝑗 ) = 𝐹 )
482	479 481	oveq12d	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) = ( 𝐹 ( 𝐿 ‘ 𝑌 ) 𝐹 ) )
483	1 173 232	hoidmvval0b	⊢ ( 𝜑 → ( 𝐹 ( 𝐿 ‘ 𝑌 ) 𝐹 ) = 0 )
484	483	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( 𝐹 ( 𝐿 ‘ 𝑌 ) 𝐹 ) = 0 )
485	477 482 484	3eqtrd	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( 𝑃 ‘ 𝑗 ) = 0 )
486	485	adantlr	⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ( 𝑃 ‘ 𝑗 ) = 0 )
487	469	ad2antlr	⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) → ¬ ( 𝑃 ‘ 𝑗 ) = 0 )
488	486 487	condan	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) )
489	488	iftrued	⊢ ( ( ( 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	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( 𝐽 ‘ 𝑗 ) = ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) )
491	490	fveq1d	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) )
492	491	adantr	⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) )
493		fvres	⊢ ( 𝑘 ∈ 𝑌 → ( ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) = ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) )
494	493	adantl	⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ 𝑘 ∈ 𝑌 ) → ( ( ( 𝐶 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) = ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) )
495	492 494	eqtrd	⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) )
496	270	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( 𝐾 ‘ 𝑗 ) = if ( 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) , ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) , 𝐹 ) )
497	488 255	syl	⊢ ( ( ( 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	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( 𝐾 ‘ 𝑗 ) = ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) )
499	498	fveq1d	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) )
500	499	adantr	⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) )
501		fvres	⊢ ( 𝑘 ∈ 𝑌 → ( ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) )
502	501	adantl	⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ 𝑘 ∈ 𝑌 ) → ( ( ( 𝐷 ‘ 𝑗 ) ↾ 𝑌 ) ‘ 𝑘 ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) )
503	500 502	eqtrd	⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) )
504	495 503	oveq12d	⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ 𝑘 ∈ 𝑌 ) → ( ( ( 𝐽 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐾 ‘ 𝑗 ) ‘ 𝑘 ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) )
505	504	fveq2d	⊢ ( ( ( ( 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	⊢ ( ( ( 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	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) = ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) )
508	355	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑌 ) → 𝑆 ∈ ℝ )
509	345	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑌 ) → 𝑊 ∈ Fin )
510	49	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑌 ) → ( 𝐷 ‘ 𝑗 ) : 𝑊 ⟶ ℝ )
511		elun1	⊢ ( 𝑘 ∈ 𝑌 → 𝑘 ∈ ( 𝑌 ∪ { 𝑍 } ) )
512	511 5	eleqtrrdi	⊢ ( 𝑘 ∈ 𝑌 → 𝑘 ∈ 𝑊 )
513	512	adantl	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑌 ) → 𝑘 ∈ 𝑊 )
514	354 508 509 510 513	hsphoival	⊢ ( ( ( 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	⊢ ( 𝑘 ∈ 𝑌 → if ( 𝑘 ∈ 𝑌 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) , if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ≤ 𝑆 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) , 𝑆 ) ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) )
516	515	adantl	⊢ ( ( ( 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	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑌 ) → ( ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) )
518	517	oveq2d	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑌 ) → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) )
519	518	fveq2d	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑌 ) → ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) = ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) )
520	519	prodeq2dv	⊢ ( ( 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	⊢ ( ( 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	⊢ ( ( ( 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	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) = ( 𝑃 ‘ 𝑗 ) )
524	354 355 345 49 50	hsphoival	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑍 ) = if ( 𝑍 ∈ 𝑌 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑆 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , 𝑆 ) ) )
525	209	iffalsed	⊢ ( 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	⊢ ( ( 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	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑍 ) = if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑆 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , 𝑆 ) )
528	527	oveq2d	⊢ ( ( 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	⊢ ( ( ( 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	126	rexrd	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ* )
531	530	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ* )
532	51	rexrd	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ* )
533	532	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ* )
534		icoltub	⊢ ( ( ( ( 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	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → 𝑆 < ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) )
536	355	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → 𝑆 ∈ ℝ )
537	51	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ )
538	536 537	ltnled	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( 𝑆 < ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ↔ ¬ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑆 ) )
539	535 538	mpbid	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ¬ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑆 )
540	539	iffalsed	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑆 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , 𝑆 ) = 𝑆 )
541	540	oveq2d	⊢ ( ( ( 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	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑍 ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) 𝑆 ) )
543	542	fveq2d	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑍 ) ) ) = ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) 𝑆 ) ) )
544		volico	⊢ ( ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ ∧ 𝑆 ∈ ℝ ) → ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) 𝑆 ) ) = if ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 , ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) , 0 ) )
545	126 536 544	syl2an	⊢ ( ( ( 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	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) 𝑆 ) ) = if ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 , ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) , 0 ) )
547		iftrue	⊢ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 → if ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 , ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) , 0 ) = ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) )
548	547	adantl	⊢ ( ( ( ( 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	⊢ ( ¬ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 → if ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 , ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) , 0 ) = 0 )
550	549	adantl	⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 ) → if ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 , ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) , 0 ) = 0 )
551		simpll	⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 ) → ( 𝜑 ∧ 𝑗 ∈ ℕ ) )
552		icogelb	⊢ ( ( ( ( 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	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑆 )
554	553	adantr	⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 ) → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑆 )
555		simpr	⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 ) → ¬ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 )
556	554 555	jca	⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 ) → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑆 ∧ ¬ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 ) )
557	551 126	syl	⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 ) → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ )
558	551 355	syl	⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 ) → 𝑆 ∈ ℝ )
559	557 558	eqleltd	⊢ ( ( ( ( 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	⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 ) → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑆 )
561		id	⊢ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑆 → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑆 )
562	561	eqcomd	⊢ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑆 → 𝑆 = ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) )
563	562	oveq1d	⊢ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑆 → ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) )
564	563	adantl	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑆 ) → ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) )
565	385 126	sselid	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℂ )
566	565	subidd	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) = 0 )
567	566	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑆 ) → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) = 0 )
568	564 567	eqtr2d	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑆 ) → 0 = ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) )
569	551 560 568	syl2anc	⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑆 ) → 0 = ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) )
570	550 569	eqtrd	⊢ ( ( ( ( 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	⊢ ( ( ( 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	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑍 ) ) ) = ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) )
573	523 572	oveq12d	⊢ ( ( ( 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	386 274	sselid	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑃 ‘ 𝑗 ) ∈ ℂ )
575	355 126	resubcld	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ∈ ℝ )
576	575	recnd	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ∈ ℂ )
577	574 576	mulcomd	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝑃 ‘ 𝑗 ) · ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ) = ( ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) · ( 𝑃 ‘ 𝑗 ) ) )
578	577	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝑃 ‘ 𝑗 ) · ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ) = ( ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) · ( 𝑃 ‘ 𝑗 ) ) )
579	450 573 578	3eqtrd	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) = ( ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) · ( 𝑃 ‘ 𝑗 ) ) )
580	579	oveq1d	⊢ ( ( ( 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	183	adantr	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝑄 − 𝑆 ) ∈ ℂ )
582	576 581 574	adddird	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) + ( 𝑄 − 𝑆 ) ) · ( 𝑃 ‘ 𝑗 ) ) = ( ( ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) · ( 𝑃 ‘ 𝑗 ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) )
583	582	eqcomd	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) · ( 𝑃 ‘ 𝑗 ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) = ( ( ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) + ( 𝑄 − 𝑆 ) ) · ( 𝑃 ‘ 𝑗 ) ) )
584	583	adantr	⊢ ( ( ( 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	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) + ( 𝑄 − 𝑆 ) ) = ( ( 𝑄 − 𝑆 ) + ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ) )
586	166	adantr	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑄 ∈ ℂ )
587	167	adantr	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → 𝑆 ∈ ℂ )
588	586 587 565	npncand	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝑄 − 𝑆 ) + ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ) = ( 𝑄 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) )
589	585 588	eqtrd	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) + ( 𝑄 − 𝑆 ) ) = ( 𝑄 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) )
590	589	oveq1d	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( ( 𝑆 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) + ( 𝑄 − 𝑆 ) ) · ( 𝑃 ‘ 𝑗 ) ) = ( ( 𝑄 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) · ( 𝑃 ‘ 𝑗 ) ) )
591	590	adantr	⊢ ( ( ( 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	⊢ ( ( ( 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	⊢ ( ( ( 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	⊢ ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) = ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) ) )
595	1 223 50 447 5 125 410 594	hoiprodp1	⊢ ( ( 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	215 217 595	syl2anc	⊢ ( ( 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	⊢ ( ( ( 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	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) = ( ( 𝐽 ‘ 𝑗 ) ( 𝐿 ‘ 𝑌 ) ( 𝐾 ‘ 𝑗 ) ) )
599	409	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑌 ) → 𝑄 ∈ ℝ )
600	354 599 509 510 513	hsphoival	⊢ ( ( ( 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	⊢ ( 𝑘 ∈ 𝑌 → if ( 𝑘 ∈ 𝑌 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) , if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ≤ 𝑄 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) , 𝑄 ) ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) )
602	601	adantl	⊢ ( ( ( 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	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑌 ) → ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) )
604	603	oveq2d	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑌 ) → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) )
605	604	fveq2d	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) ∧ 𝑘 ∈ 𝑌 ) → ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) = ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑘 ) ) ) )
606	605	prodeq2dv	⊢ ( ( 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	⊢ ( ( ( 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	⊢ ( ( ( 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	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ∏ 𝑘 ∈ 𝑌 ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑘 ) [,) ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑘 ) ) ) = ( 𝑃 ‘ 𝑗 ) )
610	354 409 345 49 50	hsphoival	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑍 ) = if ( 𝑍 ∈ 𝑌 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑄 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , 𝑄 ) ) )
611	217 610	syldan	⊢ ( ( 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	⊢ ( ( ( 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	209	iffalsed	⊢ ( 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	⊢ ( ( ( 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	217 51	syldan	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ )
616	615	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ )
617		simpr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 )
618	616 617	eqled	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑄 )
619	618	iftrued	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 ) → if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑄 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , 𝑄 ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) )
620	619 617	eqtrd	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 ) → if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑄 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , 𝑄 ) = 𝑄 )
621	620	adantlr	⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 ) → if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑄 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , 𝑄 ) = 𝑄 )
622	84	adantr	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → 𝑄 ∈ ℝ )
623	622	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ¬ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 ) → 𝑄 ∈ ℝ )
624	623	adantlr	⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 ) → 𝑄 ∈ ℝ )
625	615	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ¬ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ )
626	625	adantlr	⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ )
627	25	a1i	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → 𝑄 = inf ( 𝑉 , ℝ , < ) )
628	443 57	syl	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → 𝑉 ⊆ ℝ )
629	161	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ∃ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 𝑥 ≤ 𝑦 )
630		simplr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → 𝑗 ∈ ( 1 ... 𝑀 ) )
631	216 488	sylanl2	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) )
632	630 631	jca	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( 𝑗 ∈ ( 1 ... 𝑀 ) ∧ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) ) )
633		rabid	⊢ ( 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	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → 𝑗 ∈ { 𝑗 ∈ ( 1 ... 𝑀 ) ∣ 𝑆 ∈ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) } )
635		eqidd	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) )
636		fveq2	⊢ ( 𝑖 = 𝑗 → ( 𝐷 ‘ 𝑖 ) = ( 𝐷 ‘ 𝑗 ) )
637	636	fveq1d	⊢ ( 𝑖 = 𝑗 → ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) )
638	637	eqeq2d	⊢ ( 𝑖 = 𝑗 → ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = ( ( 𝐷 ‘ 𝑖 ) ‘ 𝑍 ) ↔ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ) )
639	638	rspcev	⊢ ( ( 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	⊢ ( ( ( 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	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ V )
642	35 640 641	elrnmptd	⊢ ( ( ( 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 23	eleqtrrdi	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ 𝑂 )
644		elun2	⊢ ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ 𝑂 → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ( { ( 𝐵 ‘ 𝑍 ) } ∪ 𝑂 ) )
645	643 644	syl	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ( { ( 𝐵 ‘ 𝑍 ) } ∪ 𝑂 ) )
646	76	a1i	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( { ( 𝐵 ‘ 𝑍 ) } ∪ 𝑂 ) = 𝑉 )
647	645 646	eleqtrd	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ 𝑉 )
648		lbinfle	⊢ ( ( 𝑉 ⊆ ℝ ∧ ∃ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 𝑥 ≤ 𝑦 ∧ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ∈ 𝑉 ) → inf ( 𝑉 , ℝ , < ) ≤ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) )
649	628 629 647 648	syl3anc	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → inf ( 𝑉 , ℝ , < ) ≤ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) )
650	627 649	eqbrtrd	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → 𝑄 ≤ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) )
651	650	adantr	⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 ) → 𝑄 ≤ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) )
652		neqne	⊢ ( ¬ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≠ 𝑄 )
653	652	adantl	⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 ) → ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≠ 𝑄 )
654	624 626 651 653	leneltd	⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 ) → 𝑄 < ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) )
655	624 626	ltnled	⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 ) → ( 𝑄 < ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ↔ ¬ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑄 ) )
656	654 655	mpbid	⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) ∧ ¬ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) = 𝑄 ) → ¬ ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑄 )
657	656	iffalsed	⊢ ( ( ( ( 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	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → if ( ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑄 , ( ( 𝐷 ‘ 𝑗 ) ‘ 𝑍 ) , 𝑄 ) = 𝑄 )
659	612 614 658	3eqtrd	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑍 ) = 𝑄 )
660	659	oveq2d	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑍 ) ) = ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) 𝑄 ) )
661	660	fveq2d	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑍 ) ) ) = ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) 𝑄 ) ) )
662	215 217 126	syl2anc	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ )
663	662	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℝ )
664	443 84	syl	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → 𝑄 ∈ ℝ )
665		volico	⊢ ( ( ( ( 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	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) 𝑄 ) ) = if ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑄 , ( 𝑄 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) , 0 ) )
667	443 90	syl	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → 𝑆 ∈ ℝ )
668	443 444 445 553	syl21anc	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ≤ 𝑆 )
669	443 157	syl	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → 𝑆 < 𝑄 )
670	663 667 664 668 669	lelttrd	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) < 𝑄 )
671	670	iftrued	⊢ ( ( ( 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	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( vol ‘ ( ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) [,) ( ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ‘ 𝑍 ) ) ) = ( 𝑄 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) )
673	609 672	oveq12d	⊢ ( ( ( 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	215 166	syl	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → 𝑄 ∈ ℂ )
675	385 662	sselid	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ∈ ℂ )
676	674 675	subcld	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( 𝑄 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ∈ ℂ )
677	306 676	mulcomd	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( 𝑃 ‘ 𝑗 ) · ( 𝑄 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ) = ( ( 𝑄 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) · ( 𝑃 ‘ 𝑗 ) ) )
678	677	adantr	⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) ∧ ( 𝑃 ‘ 𝑗 ) ≠ 0 ) → ( ( 𝑃 ‘ 𝑗 ) · ( 𝑄 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) ) = ( ( 𝑄 − ( ( 𝐶 ‘ 𝑗 ) ‘ 𝑍 ) ) · ( 𝑃 ‘ 𝑗 ) ) )
679	597 673 678	3eqtrd	⊢ ( ( ( 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	⊢ ( ( ( 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	⊢ ( ( ( 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	⊢ ( ( ( 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	⊢ ( ( 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	213 402 415 683	fsumle	⊢ ( 𝜑 → Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ≤ Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) )
685	367 403 413 416 423 684	le2addd	⊢ ( 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 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 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 ) ) ) ) )
686	321	mpteq1d	⊢ ( 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	⊢ ( 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	217 412	syldan	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ( 0 [,] +∞ ) )
689	324 325 326 330 417 688	sge0splitmpt	⊢ ( 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	⊢ ( 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	215 217 411	syl2anc	⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ∈ ( 0 [,) +∞ ) )
692	213 691	sge0fsummpt	⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) = Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) )
693	692 416	eqeltrd	⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑗 ∈ ( 1 ... 𝑀 ) ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ∈ ℝ )
694		rexadd	⊢ ( ( ( 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	⊢ ( 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	⊢ ( 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 ) ) ) ) ) + Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) )
697	690 695 696	3eqtrrd	⊢ ( 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 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) )
698	685 697	breqtrd	⊢ ( 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 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) ≤ ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) )
699	404 281 208 408 698	lemul2ad	⊢ ( 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 ) ) ) + ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) )
700	399 699	eqbrtrd	⊢ ( 𝜑 → ( ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑆 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) + ( ( 1 + 𝐸 ) · Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑄 − 𝑆 ) · ( 𝑃 ‘ 𝑗 ) ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) )
701	205 280 282 315 700	letrd	⊢ ( 𝜑 → ( ( 𝐺 · ( 𝑆 − ( 𝐴 ‘ 𝑍 ) ) ) + ( 𝐺 · ( 𝑄 − 𝑆 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) )
702	189 701	eqbrtrd	⊢ ( 𝜑 → ( 𝐺 · ( 𝑄 − ( 𝐴 ‘ 𝑍 ) ) ) ≤ ( ( 1 + 𝐸 ) · ( Σ^ ‘ ( 𝑗 ∈ ℕ ↦ ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) )
703	165 702	jca	⊢ ( 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	⊢ ( 𝑧 = 𝑄 → ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) = ( 𝑄 − ( 𝐴 ‘ 𝑍 ) ) )
705	704	oveq2d	⊢ ( 𝑧 = 𝑄 → ( 𝐺 · ( 𝑧 − ( 𝐴 ‘ 𝑍 ) ) ) = ( 𝐺 · ( 𝑄 − ( 𝐴 ‘ 𝑍 ) ) ) )
706		fveq2	⊢ ( 𝑧 = 𝑄 → ( 𝐻 ‘ 𝑧 ) = ( 𝐻 ‘ 𝑄 ) )
707	706	fveq1d	⊢ ( 𝑧 = 𝑄 → ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) = ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) )
708	707	oveq2d	⊢ ( 𝑧 = 𝑄 → ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑧 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) = ( ( 𝐶 ‘ 𝑗 ) ( 𝐿 ‘ 𝑊 ) ( ( 𝐻 ‘ 𝑄 ) ‘ ( 𝐷 ‘ 𝑗 ) ) ) )
709	708	mpteq2dv	⊢ ( 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	⊢ ( 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	⊢ ( 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	⊢ ( 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	⊢ ( 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	⊢ ( 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 17	eleqtrrdi	⊢ ( 𝜑 → 𝑄 ∈ 𝑈 )
716		breq2	⊢ ( 𝑢 = 𝑄 → ( 𝑆 < 𝑢 ↔ 𝑆 < 𝑄 ) )
717	716	rspcev	⊢ ( ( 𝑄 ∈ 𝑈 ∧ 𝑆 < 𝑄 ) → ∃ 𝑢 ∈ 𝑈 𝑆 < 𝑢 )
718	715 157 717	syl2anc	⊢ ( 𝜑 → ∃ 𝑢 ∈ 𝑈 𝑆 < 𝑢 )