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Theorem pjthlem1 25370

Description: Lemma for pjth 25372. (Contributed by NM, 10-Oct-1999.) (Revised by Mario Carneiro, 17-Oct-2015.) (Proof shortened by AV, 10-Jul-2022.)

**Hypotheses**
Ref	Expression
pjthlem.v	⊢ 𝑉 = (Base‘𝑊)
pjthlem.n	⊢ 𝑁 = (norm‘𝑊)
pjthlem.p	⊢ + = (+_g‘𝑊)
pjthlem.m	⊢ − = (-_g‘𝑊)
pjthlem.h	⊢ , = (·_𝑖‘𝑊)
pjthlem.l	⊢ 𝐿 = (LSubSp‘𝑊)
pjthlem.1	⊢ (𝜑 → 𝑊 ∈ ℂHil)
pjthlem.2	⊢ (𝜑 → 𝑈 ∈ 𝐿)
pjthlem.4	⊢ (𝜑 → 𝐴 ∈ 𝑉)
pjthlem.5	⊢ (𝜑 → 𝐵 ∈ 𝑈)
pjthlem.7	⊢ (𝜑 → ∀𝑥 ∈ 𝑈 (𝑁‘𝐴) ≤ (𝑁‘(𝐴 − 𝑥)))
pjthlem.8	⊢ 𝑇 = ((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1))

**Assertion**
Ref	Expression
pjthlem1	⊢ (𝜑 → (𝐴 , 𝐵) = 0)

Distinct variable groups: 𝑥, − 𝑥,𝐴 𝑥,𝐵 𝑥,𝑁 𝜑,𝑥 𝑥,𝑈 𝑥,𝑉 𝑥,𝑇 𝑥,𝑊 Allowed substitution hints: + (𝑥) , (𝑥) 𝐿(𝑥)

**Proof of Theorem pjthlem1**
Step	Hyp	Ref	Expression
1		pjthlem.1	. . . 4 ⊢ (𝜑 → 𝑊 ∈ ℂHil)
2		hlcph 25297	. . . 4 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → 𝑊 ∈ ℂPreHil)
4		pjthlem.4	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5		pjthlem.2	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝐿)
6		pjthlem.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑊)
7		pjthlem.l	. . . . . 6 ⊢ 𝐿 = (LSubSp‘𝑊)
8	6, 7	lssss 20875	. . . . 5 ⊢ (𝑈 ∈ 𝐿 → 𝑈 ⊆ 𝑉)
9	5, 8	syl 17	. . . 4 ⊢ (𝜑 → 𝑈 ⊆ 𝑉)
10		pjthlem.5	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑈)
11	9, 10	sseldd 3930	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
12		pjthlem.h	. . . 4 ⊢ , = (·_𝑖‘𝑊)
13	6, 12	cphipcl 25124	. . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 , 𝐵) ∈ ℂ)
14	3, 4, 11, 13	syl3anc 1373	. 2 ⊢ (𝜑 → (𝐴 , 𝐵) ∈ ℂ)
15	14	abscld 15352	. . . 4 ⊢ (𝜑 → (abs‘(𝐴 , 𝐵)) ∈ ℝ)
16	15	recnd 11146	. . 3 ⊢ (𝜑 → (abs‘(𝐴 , 𝐵)) ∈ ℂ)
17	15	resqcld 14038	. . . . . . 7 ⊢ (𝜑 → ((abs‘(𝐴 , 𝐵))↑2) ∈ ℝ)
18	17	renegcld 11550	. . . . . 6 ⊢ (𝜑 → -((abs‘(𝐴 , 𝐵))↑2) ∈ ℝ)
19	6, 12	reipcl 25130	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐵 ∈ 𝑉) → (𝐵 , 𝐵) ∈ ℝ)
20	3, 11, 19	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝐵 , 𝐵) ∈ ℝ)
21		2re 12205	. . . . . . . 8 ⊢ 2 ∈ ℝ
22		readdcl 11095	. . . . . . . 8 ⊢ (((𝐵 , 𝐵) ∈ ℝ ∧ 2 ∈ ℝ) → ((𝐵 , 𝐵) + 2) ∈ ℝ)
23	20, 21, 22	sylancl 586	. . . . . . 7 ⊢ (𝜑 → ((𝐵 , 𝐵) + 2) ∈ ℝ)
24		0red 11121	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
25		peano2re 11292	. . . . . . . . 9 ⊢ ((𝐵 , 𝐵) ∈ ℝ → ((𝐵 , 𝐵) + 1) ∈ ℝ)
26	20, 25	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 , 𝐵) + 1) ∈ ℝ)
27	6, 12	ipge0 25131	. . . . . . . . . 10 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐵 ∈ 𝑉) → 0 ≤ (𝐵 , 𝐵))
28	3, 11, 27	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (𝐵 , 𝐵))
29	20	ltp1d 12058	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 , 𝐵) < ((𝐵 , 𝐵) + 1))
30	24, 20, 26, 28, 29	lelttrd 11277	. . . . . . . 8 ⊢ (𝜑 → 0 < ((𝐵 , 𝐵) + 1))
31	26	ltp1d 12058	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 , 𝐵) + 1) < (((𝐵 , 𝐵) + 1) + 1))
32		df-2 12194	. . . . . . . . . . 11 ⊢ 2 = (1 + 1)
33	32	oveq2i 7363	. . . . . . . . . 10 ⊢ ((𝐵 , 𝐵) + 2) = ((𝐵 , 𝐵) + (1 + 1))
34	20	recnd 11146	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 , 𝐵) ∈ ℂ)
35		ax-1cn 11070	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
36		addass 11099	. . . . . . . . . . . 12 ⊢ (((𝐵 , 𝐵) ∈ ℂ ∧ 1 ∈ ℂ ∧ 1 ∈ ℂ) → (((𝐵 , 𝐵) + 1) + 1) = ((𝐵 , 𝐵) + (1 + 1)))
37	35, 35, 36	mp3an23 1455	. . . . . . . . . . 11 ⊢ ((𝐵 , 𝐵) ∈ ℂ → (((𝐵 , 𝐵) + 1) + 1) = ((𝐵 , 𝐵) + (1 + 1)))
38	34, 37	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐵 , 𝐵) + 1) + 1) = ((𝐵 , 𝐵) + (1 + 1)))
39	33, 38	eqtr4id 2785	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 , 𝐵) + 2) = (((𝐵 , 𝐵) + 1) + 1))
40	31, 39	breqtrrd 5121	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 , 𝐵) + 1) < ((𝐵 , 𝐵) + 2))
41	24, 26, 23, 30, 40	lttrd 11280	. . . . . . 7 ⊢ (𝜑 → 0 < ((𝐵 , 𝐵) + 2))
42	23, 41	elrpd 12937	. . . . . 6 ⊢ (𝜑 → ((𝐵 , 𝐵) + 2) ∈ ℝ⁺)
43		oveq2 7360	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑇( ·_𝑠 ‘𝑊)𝐵) → (𝐴 − 𝑥) = (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)))
44	43	fveq2d 6832	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑇( ·_𝑠 ‘𝑊)𝐵) → (𝑁‘(𝐴 − 𝑥)) = (𝑁‘(𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵))))
45	44	breq2d 5105	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑇( ·_𝑠 ‘𝑊)𝐵) → ((𝑁‘𝐴) ≤ (𝑁‘(𝐴 − 𝑥)) ↔ (𝑁‘𝐴) ≤ (𝑁‘(𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)))))
46		pjthlem.7	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ 𝑈 (𝑁‘𝐴) ≤ (𝑁‘(𝐴 − 𝑥)))
47		cphlmod 25107	. . . . . . . . . . . . . 14 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LMod)
48	3, 47	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑊 ∈ LMod)
49		pjthlem.8	. . . . . . . . . . . . . 14 ⊢ 𝑇 = ((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1))
50		hlphl 25298	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ PreHil)
51	1, 50	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑊 ∈ PreHil)
52		eqid 2731	. . . . . . . . . . . . . . . . 17 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
53		eqid 2731	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
54	52, 12, 6, 53	ipcl 21576	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 , 𝐵) ∈ (Base‘(Scalar‘𝑊)))
55	51, 4, 11, 54	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 , 𝐵) ∈ (Base‘(Scalar‘𝑊)))
56	52, 53	hlress 25301	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∈ ℂHil → ℝ ⊆ (Base‘(Scalar‘𝑊)))
57	1, 56	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ℝ ⊆ (Base‘(Scalar‘𝑊)))
58	57, 26	sseldd 3930	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐵 , 𝐵) + 1) ∈ (Base‘(Scalar‘𝑊)))
59	20, 28	ge0p1rpd 12970	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐵 , 𝐵) + 1) ∈ ℝ⁺)
60	59	rpne0d 12945	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐵 , 𝐵) + 1) ≠ 0)
61	52, 53	cphdivcl 25115	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ ℂPreHil ∧ ((𝐴 , 𝐵) ∈ (Base‘(Scalar‘𝑊)) ∧ ((𝐵 , 𝐵) + 1) ∈ (Base‘(Scalar‘𝑊)) ∧ ((𝐵 , 𝐵) + 1) ≠ 0)) → ((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1)) ∈ (Base‘(Scalar‘𝑊)))
62	3, 55, 58, 60, 61	syl13anc 1374	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1)) ∈ (Base‘(Scalar‘𝑊)))
63	49, 62	eqeltrid 2835	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑇 ∈ (Base‘(Scalar‘𝑊)))
64		eqid 2731	. . . . . . . . . . . . . 14 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑊)
65	52, 64, 53, 7	lssvscl 20894	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) ∧ (𝑇 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝐵 ∈ 𝑈)) → (𝑇( ·_𝑠 ‘𝑊)𝐵) ∈ 𝑈)
66	48, 5, 63, 10, 65	syl22anc 838	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑇( ·_𝑠 ‘𝑊)𝐵) ∈ 𝑈)
67	45, 46, 66	rspcdva 3573	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁‘𝐴) ≤ (𝑁‘(𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵))))
68		cphngp 25106	. . . . . . . . . . . . . 14 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp)
69	3, 68	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑊 ∈ NrmGrp)
70		pjthlem.n	. . . . . . . . . . . . . 14 ⊢ 𝑁 = (norm‘𝑊)
71	6, 70	nmcl 24537	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝐴 ∈ 𝑉) → (𝑁‘𝐴) ∈ ℝ)
72	69, 4, 71	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁‘𝐴) ∈ ℝ)
73	6, 52, 64, 53	lmodvscl 20817	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝐵 ∈ 𝑉) → (𝑇( ·_𝑠 ‘𝑊)𝐵) ∈ 𝑉)
74	48, 63, 11, 73	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑇( ·_𝑠 ‘𝑊)𝐵) ∈ 𝑉)
75		pjthlem.m	. . . . . . . . . . . . . . 15 ⊢ − = (-_g‘𝑊)
76	6, 75	lmodvsubcl 20846	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉 ∧ (𝑇( ·_𝑠 ‘𝑊)𝐵) ∈ 𝑉) → (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)) ∈ 𝑉)
77	48, 4, 74, 76	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)) ∈ 𝑉)
78	6, 70	nmcl 24537	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ NrmGrp ∧ (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)) ∈ 𝑉) → (𝑁‘(𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵))) ∈ ℝ)
79	69, 77, 78	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁‘(𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵))) ∈ ℝ)
80	6, 70	nmge0 24538	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝐴 ∈ 𝑉) → 0 ≤ (𝑁‘𝐴))
81	69, 4, 80	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (𝑁‘𝐴))
82	6, 70	nmge0 24538	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ NrmGrp ∧ (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)) ∈ 𝑉) → 0 ≤ (𝑁‘(𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵))))
83	69, 77, 82	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (𝑁‘(𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵))))
84	72, 79, 81, 83	le2sqd 14170	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁‘𝐴) ≤ (𝑁‘(𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵))) ↔ ((𝑁‘𝐴)↑2) ≤ ((𝑁‘(𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)))↑2)))
85	67, 84	mpbid 232	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑁‘𝐴)↑2) ≤ ((𝑁‘(𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)))↑2))
86	79	resqcld 14038	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁‘(𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)))↑2) ∈ ℝ)
87	72	resqcld 14038	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁‘𝐴)↑2) ∈ ℝ)
88	86, 87	subge0d 11713	. . . . . . . . . 10 ⊢ (𝜑 → (0 ≤ (((𝑁‘(𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)))↑2) − ((𝑁‘𝐴)↑2)) ↔ ((𝑁‘𝐴)↑2) ≤ ((𝑁‘(𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)))↑2)))
89	85, 88	mpbird 257	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (((𝑁‘(𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)))↑2) − ((𝑁‘𝐴)↑2)))
90		2z 12510	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℤ
91		rpexpcl 13993	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐵 , 𝐵) + 1) ∈ ℝ⁺ ∧ 2 ∈ ℤ) → (((𝐵 , 𝐵) + 1)↑2) ∈ ℝ⁺)
92	59, 90, 91	sylancl 586	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((𝐵 , 𝐵) + 1)↑2) ∈ ℝ⁺)
93	17, 92	rerpdivcld 12971	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) ∈ ℝ)
94	93, 23	remulcld 11148	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2)) ∈ ℝ)
95	94	recnd 11146	. . . . . . . . . . . 12 ⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2)) ∈ ℂ)
96	95	negcld 11465	. . . . . . . . . . 11 ⊢ (𝜑 → -((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2)) ∈ ℂ)
97	6, 12	cphipcl 25124	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐴 , 𝐴) ∈ ℂ)
98	3, 4, 4, 97	syl3anc 1373	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 , 𝐴) ∈ ℂ)
99	96, 98	pncand 11479	. . . . . . . . . 10 ⊢ (𝜑 → ((-((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2)) + (𝐴 , 𝐴)) − (𝐴 , 𝐴)) = -((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2)))
100	6, 12, 70	nmsq 25127	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)) ∈ 𝑉) → ((𝑁‘(𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)))↑2) = ((𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)) , (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵))))
101	3, 77, 100	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑁‘(𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)))↑2) = ((𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)) , (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵))))
102	12, 6, 75	cphsubdir 25141	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝑉 ∧ (𝑇( ·_𝑠 ‘𝑊)𝐵) ∈ 𝑉 ∧ (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)) ∈ 𝑉)) → ((𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)) , (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵))) = ((𝐴 , (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵))) − ((𝑇( ·_𝑠 ‘𝑊)𝐵) , (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)))))
103	3, 4, 74, 77, 102	syl13anc 1374	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)) , (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵))) = ((𝐴 , (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵))) − ((𝑇( ·_𝑠 ‘𝑊)𝐵) , (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)))))
104	12, 6, 75	cphsubdi 25142	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ (𝑇( ·_𝑠 ‘𝑊)𝐵) ∈ 𝑉)) → (𝐴 , (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵))) = ((𝐴 , 𝐴) − (𝐴 , (𝑇( ·_𝑠 ‘𝑊)𝐵))))
105	3, 4, 4, 74, 104	syl13anc 1374	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 , (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵))) = ((𝐴 , 𝐴) − (𝐴 , (𝑇( ·_𝑠 ‘𝑊)𝐵))))
106	105	oveq1d 7367	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐴 , (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵))) − ((𝑇( ·_𝑠 ‘𝑊)𝐵) , (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)))) = (((𝐴 , 𝐴) − (𝐴 , (𝑇( ·_𝑠 ‘𝑊)𝐵))) − ((𝑇( ·_𝑠 ‘𝑊)𝐵) , (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)))))
107	6, 12	cphipcl 25124	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ (𝑇( ·_𝑠 ‘𝑊)𝐵) ∈ 𝑉) → (𝐴 , (𝑇( ·_𝑠 ‘𝑊)𝐵)) ∈ ℂ)
108	3, 4, 74, 107	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 , (𝑇( ·_𝑠 ‘𝑊)𝐵)) ∈ ℂ)
109	12, 6, 75	cphsubdi 25142	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑊 ∈ ℂPreHil ∧ ((𝑇( ·_𝑠 ‘𝑊)𝐵) ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ (𝑇( ·_𝑠 ‘𝑊)𝐵) ∈ 𝑉)) → ((𝑇( ·_𝑠 ‘𝑊)𝐵) , (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵))) = (((𝑇( ·_𝑠 ‘𝑊)𝐵) , 𝐴) − ((𝑇( ·_𝑠 ‘𝑊)𝐵) , (𝑇( ·_𝑠 ‘𝑊)𝐵))))
110	3, 74, 4, 74, 109	syl13anc 1374	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑇( ·_𝑠 ‘𝑊)𝐵) , (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵))) = (((𝑇( ·_𝑠 ‘𝑊)𝐵) , 𝐴) − ((𝑇( ·_𝑠 ‘𝑊)𝐵) , (𝑇( ·_𝑠 ‘𝑊)𝐵))))
111	6, 12	cphipcl 25124	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝑇( ·_𝑠 ‘𝑊)𝐵) ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ((𝑇( ·_𝑠 ‘𝑊)𝐵) , 𝐴) ∈ ℂ)
112	3, 74, 4, 111	syl3anc 1373	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑇( ·_𝑠 ‘𝑊)𝐵) , 𝐴) ∈ ℂ)
113	6, 12	cphipcl 25124	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝑇( ·_𝑠 ‘𝑊)𝐵) ∈ 𝑉 ∧ (𝑇( ·_𝑠 ‘𝑊)𝐵) ∈ 𝑉) → ((𝑇( ·_𝑠 ‘𝑊)𝐵) , (𝑇( ·_𝑠 ‘𝑊)𝐵)) ∈ ℂ)
114	3, 74, 74, 113	syl3anc 1373	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑇( ·_𝑠 ‘𝑊)𝐵) , (𝑇( ·_𝑠 ‘𝑊)𝐵)) ∈ ℂ)
115	112, 114	subcld 11478	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((𝑇( ·_𝑠 ‘𝑊)𝐵) , 𝐴) − ((𝑇( ·_𝑠 ‘𝑊)𝐵) , (𝑇( ·_𝑠 ‘𝑊)𝐵))) ∈ ℂ)
116	110, 115	eqeltrd 2831	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑇( ·_𝑠 ‘𝑊)𝐵) , (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵))) ∈ ℂ)
117	98, 108, 116	subsub4d 11509	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((𝐴 , 𝐴) − (𝐴 , (𝑇( ·_𝑠 ‘𝑊)𝐵))) − ((𝑇( ·_𝑠 ‘𝑊)𝐵) , (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)))) = ((𝐴 , 𝐴) − ((𝐴 , (𝑇( ·_𝑠 ‘𝑊)𝐵)) + ((𝑇( ·_𝑠 ‘𝑊)𝐵) , (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵))))))
118	93	recnd 11146	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) ∈ ℂ)
119	26	recnd 11146	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐵 , 𝐵) + 1) ∈ ℂ)
120		1cnd 11113	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ∈ ℂ)
121	118, 119, 120	adddid 11142	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · (((𝐵 , 𝐵) + 1) + 1)) = (((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 1)) + ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · 1)))
122	39	oveq2d 7368	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2)) = ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · (((𝐵 , 𝐵) + 1) + 1)))
123	12, 6, 52, 53, 64	cphassr 25145	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝑇 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴 , (𝑇( ·_𝑠 ‘𝑊)𝐵)) = ((∗‘𝑇) · (𝐴 , 𝐵)))
124	3, 63, 4, 11, 123	syl13anc 1374	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐴 , (𝑇( ·_𝑠 ‘𝑊)𝐵)) = ((∗‘𝑇) · (𝐴 , 𝐵)))
125	14, 119, 60	divcld 11903	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1)) ∈ ℂ)
126	49, 125	eqeltrid 2835	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑇 ∈ ℂ)
127	126	cjcld 15109	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (∗‘𝑇) ∈ ℂ)
128	127, 14	mulcomd 11139	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((∗‘𝑇) · (𝐴 , 𝐵)) = ((𝐴 , 𝐵) · (∗‘𝑇)))
129	14	cjcld 15109	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (∗‘(𝐴 , 𝐵)) ∈ ℂ)
130	14, 129, 119, 60	divassd 11938	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (((𝐴 , 𝐵) · (∗‘(𝐴 , 𝐵))) / ((𝐵 , 𝐵) + 1)) = ((𝐴 , 𝐵) · ((∗‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1))))
131	14	absvalsqd 15358	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((abs‘(𝐴 , 𝐵))↑2) = ((𝐴 , 𝐵) · (∗‘(𝐴 , 𝐵))))
132	131	oveq1d 7367	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) / ((𝐵 , 𝐵) + 1)) = (((𝐴 , 𝐵) · (∗‘(𝐴 , 𝐵))) / ((𝐵 , 𝐵) + 1)))
133	49	fveq2i 6831	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∗‘𝑇) = (∗‘((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1)))
134	14, 119, 60	cjdivd 15136	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (∗‘((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1))) = ((∗‘(𝐴 , 𝐵)) / (∗‘((𝐵 , 𝐵) + 1))))
135	26	cjred 15139	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (∗‘((𝐵 , 𝐵) + 1)) = ((𝐵 , 𝐵) + 1))
136	135	oveq2d 7368	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((∗‘(𝐴 , 𝐵)) / (∗‘((𝐵 , 𝐵) + 1))) = ((∗‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1)))
137	134, 136	eqtrd 2766	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (∗‘((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1))) = ((∗‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1)))
138	133, 137	eqtrid 2778	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (∗‘𝑇) = ((∗‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1)))
139	138	oveq2d 7368	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝐴 , 𝐵) · (∗‘𝑇)) = ((𝐴 , 𝐵) · ((∗‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1))))
140	130, 132, 139	3eqtr4rd 2777	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝐴 , 𝐵) · (∗‘𝑇)) = (((abs‘(𝐴 , 𝐵))↑2) / ((𝐵 , 𝐵) + 1)))
141	124, 128, 140	3eqtrd 2770	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐴 , (𝑇( ·_𝑠 ‘𝑊)𝐵)) = (((abs‘(𝐴 , 𝐵))↑2) / ((𝐵 , 𝐵) + 1)))
142	17	recnd 11146	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((abs‘(𝐴 , 𝐵))↑2) ∈ ℂ)
143	142, 119	mulcomd 11139	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) · ((𝐵 , 𝐵) + 1)) = (((𝐵 , 𝐵) + 1) · ((abs‘(𝐴 , 𝐵))↑2)))
144	119	sqvald 14056	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (((𝐵 , 𝐵) + 1)↑2) = (((𝐵 , 𝐵) + 1) · ((𝐵 , 𝐵) + 1)))
145	143, 144	oveq12d 7370	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) · ((𝐵 , 𝐵) + 1)) / (((𝐵 , 𝐵) + 1)↑2)) = ((((𝐵 , 𝐵) + 1) · ((abs‘(𝐴 , 𝐵))↑2)) / (((𝐵 , 𝐵) + 1) · ((𝐵 , 𝐵) + 1))))
146	142, 119, 119, 60, 60	divcan5d 11929	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((((𝐵 , 𝐵) + 1) · ((abs‘(𝐴 , 𝐵))↑2)) / (((𝐵 , 𝐵) + 1) · ((𝐵 , 𝐵) + 1))) = (((abs‘(𝐴 , 𝐵))↑2) / ((𝐵 , 𝐵) + 1)))
147	145, 146	eqtr2d 2767	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) / ((𝐵 , 𝐵) + 1)) = ((((abs‘(𝐴 , 𝐵))↑2) · ((𝐵 , 𝐵) + 1)) / (((𝐵 , 𝐵) + 1)↑2)))
148	92	rpcnd 12942	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (((𝐵 , 𝐵) + 1)↑2) ∈ ℂ)
149	92	rpne0d 12945	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (((𝐵 , 𝐵) + 1)↑2) ≠ 0)
150	142, 119, 148, 149	div23d 11940	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) · ((𝐵 , 𝐵) + 1)) / (((𝐵 , 𝐵) + 1)↑2)) = ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 1)))
151	141, 147, 150	3eqtrd 2770	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴 , (𝑇( ·_𝑠 ‘𝑊)𝐵)) = ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 1)))
152	93, 26	remulcld 11148	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 1)) ∈ ℝ)
153	151, 152	eqeltrd 2831	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐴 , (𝑇( ·_𝑠 ‘𝑊)𝐵)) ∈ ℝ)
154	153	cjred 15139	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (∗‘(𝐴 , (𝑇( ·_𝑠 ‘𝑊)𝐵))) = (𝐴 , (𝑇( ·_𝑠 ‘𝑊)𝐵)))
155	12, 6	cphipcj 25132	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ (𝑇( ·_𝑠 ‘𝑊)𝐵) ∈ 𝑉) → (∗‘(𝐴 , (𝑇( ·_𝑠 ‘𝑊)𝐵))) = ((𝑇( ·_𝑠 ‘𝑊)𝐵) , 𝐴))
156	3, 4, 74, 155	syl3anc 1373	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (∗‘(𝐴 , (𝑇( ·_𝑠 ‘𝑊)𝐵))) = ((𝑇( ·_𝑠 ‘𝑊)𝐵) , 𝐴))
157	154, 156, 151	3eqtr3d 2774	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑇( ·_𝑠 ‘𝑊)𝐵) , 𝐴) = ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 1)))
158	12, 6, 52, 53, 64	cph2ass 25146	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝑇 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑇 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝐵 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → ((𝑇( ·_𝑠 ‘𝑊)𝐵) , (𝑇( ·_𝑠 ‘𝑊)𝐵)) = ((𝑇 · (∗‘𝑇)) · (𝐵 , 𝐵)))
159	3, 63, 63, 11, 11, 158	syl122anc 1381	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝑇( ·_𝑠 ‘𝑊)𝐵) , (𝑇( ·_𝑠 ‘𝑊)𝐵)) = ((𝑇 · (∗‘𝑇)) · (𝐵 , 𝐵)))
160	49	fveq2i 6831	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (abs‘𝑇) = (abs‘((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1)))
161	14, 119, 60	absdivd 15371	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (abs‘((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1))) = ((abs‘(𝐴 , 𝐵)) / (abs‘((𝐵 , 𝐵) + 1))))
162	59	rpge0d 12944	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 0 ≤ ((𝐵 , 𝐵) + 1))
163	26, 162	absidd 15336	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (abs‘((𝐵 , 𝐵) + 1)) = ((𝐵 , 𝐵) + 1))
164	163	oveq2d 7368	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ((abs‘(𝐴 , 𝐵)) / (abs‘((𝐵 , 𝐵) + 1))) = ((abs‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1)))
165	161, 164	eqtrd 2766	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (abs‘((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1))) = ((abs‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1)))
166	160, 165	eqtrid 2778	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (abs‘𝑇) = ((abs‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1)))
167	166	oveq1d 7367	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((abs‘𝑇)↑2) = (((abs‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1))↑2))
168	126	absvalsqd 15358	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((abs‘𝑇)↑2) = (𝑇 · (∗‘𝑇)))
169	16, 119, 60	sqdivd 14072	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (((abs‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1))↑2) = (((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)))
170	167, 168, 169	3eqtr3d 2774	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑇 · (∗‘𝑇)) = (((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)))
171	170	oveq1d 7367	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝑇 · (∗‘𝑇)) · (𝐵 , 𝐵)) = ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · (𝐵 , 𝐵)))
172	159, 171	eqtrd 2766	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑇( ·_𝑠 ‘𝑊)𝐵) , (𝑇( ·_𝑠 ‘𝑊)𝐵)) = ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · (𝐵 , 𝐵)))
173	157, 172	oveq12d 7370	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (((𝑇( ·_𝑠 ‘𝑊)𝐵) , 𝐴) − ((𝑇( ·_𝑠 ‘𝑊)𝐵) , (𝑇( ·_𝑠 ‘𝑊)𝐵))) = (((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 1)) − ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · (𝐵 , 𝐵))))
174		pncan2 11373	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐵 , 𝐵) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝐵 , 𝐵) + 1) − (𝐵 , 𝐵)) = 1)
175	34, 35, 174	sylancl 586	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (((𝐵 , 𝐵) + 1) − (𝐵 , 𝐵)) = 1)
176	175	oveq2d 7368	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · (((𝐵 , 𝐵) + 1) − (𝐵 , 𝐵))) = ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · 1))
177	118, 119, 34	subdid 11579	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · (((𝐵 , 𝐵) + 1) − (𝐵 , 𝐵))) = (((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 1)) − ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · (𝐵 , 𝐵))))
178	176, 177	eqtr3d 2768	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · 1) = (((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 1)) − ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · (𝐵 , 𝐵))))
179	173, 110, 178	3eqtr4d 2776	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑇( ·_𝑠 ‘𝑊)𝐵) , (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵))) = ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · 1))
180	151, 179	oveq12d 7370	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐴 , (𝑇( ·_𝑠 ‘𝑊)𝐵)) + ((𝑇( ·_𝑠 ‘𝑊)𝐵) , (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)))) = (((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 1)) + ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · 1)))
181	121, 122, 180	3eqtr4rd 2777	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐴 , (𝑇( ·_𝑠 ‘𝑊)𝐵)) + ((𝑇( ·_𝑠 ‘𝑊)𝐵) , (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)))) = ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2)))
182	181	oveq2d 7368	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐴 , 𝐴) − ((𝐴 , (𝑇( ·_𝑠 ‘𝑊)𝐵)) + ((𝑇( ·_𝑠 ‘𝑊)𝐵) , (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵))))) = ((𝐴 , 𝐴) − ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2))))
183	106, 117, 182	3eqtrd 2770	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐴 , (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵))) − ((𝑇( ·_𝑠 ‘𝑊)𝐵) , (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)))) = ((𝐴 , 𝐴) − ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2))))
184	101, 103, 183	3eqtrd 2770	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁‘(𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)))↑2) = ((𝐴 , 𝐴) − ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2))))
185	98, 95	negsubd 11484	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐴 , 𝐴) + -((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2))) = ((𝐴 , 𝐴) − ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2))))
186	98, 96	addcomd 11321	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐴 , 𝐴) + -((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2))) = (-((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2)) + (𝐴 , 𝐴)))
187	184, 185, 186	3eqtr2d 2772	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁‘(𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)))↑2) = (-((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2)) + (𝐴 , 𝐴)))
188	6, 12, 70	nmsq 25127	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → ((𝑁‘𝐴)↑2) = (𝐴 , 𝐴))
189	3, 4, 188	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁‘𝐴)↑2) = (𝐴 , 𝐴))
190	187, 189	oveq12d 7370	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑁‘(𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)))↑2) − ((𝑁‘𝐴)↑2)) = ((-((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2)) + (𝐴 , 𝐴)) − (𝐴 , 𝐴)))
191	23	renegcld 11550	. . . . . . . . . . . . 13 ⊢ (𝜑 → -((𝐵 , 𝐵) + 2) ∈ ℝ)
192	191	recnd 11146	. . . . . . . . . . . 12 ⊢ (𝜑 → -((𝐵 , 𝐵) + 2) ∈ ℂ)
193	142, 192, 148, 149	div23d 11940	. . . . . . . . . . 11 ⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) · -((𝐵 , 𝐵) + 2)) / (((𝐵 , 𝐵) + 1)↑2)) = ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · -((𝐵 , 𝐵) + 2)))
194	23	recnd 11146	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐵 , 𝐵) + 2) ∈ ℂ)
195	118, 194	mulneg2d 11577	. . . . . . . . . . 11 ⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · -((𝐵 , 𝐵) + 2)) = -((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2)))
196	193, 195	eqtrd 2766	. . . . . . . . . 10 ⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) · -((𝐵 , 𝐵) + 2)) / (((𝐵 , 𝐵) + 1)↑2)) = -((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2)))
197	99, 190, 196	3eqtr4rd 2777	. . . . . . . . 9 ⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) · -((𝐵 , 𝐵) + 2)) / (((𝐵 , 𝐵) + 1)↑2)) = (((𝑁‘(𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)))↑2) − ((𝑁‘𝐴)↑2)))
198	89, 197	breqtrrd 5121	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ ((((abs‘(𝐴 , 𝐵))↑2) · -((𝐵 , 𝐵) + 2)) / (((𝐵 , 𝐵) + 1)↑2)))
199	17, 191	remulcld 11148	. . . . . . . . 9 ⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) · -((𝐵 , 𝐵) + 2)) ∈ ℝ)
200	199, 92	ge0divd 12978	. . . . . . . 8 ⊢ (𝜑 → (0 ≤ (((abs‘(𝐴 , 𝐵))↑2) · -((𝐵 , 𝐵) + 2)) ↔ 0 ≤ ((((abs‘(𝐴 , 𝐵))↑2) · -((𝐵 , 𝐵) + 2)) / (((𝐵 , 𝐵) + 1)↑2))))
201	198, 200	mpbird 257	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (((abs‘(𝐴 , 𝐵))↑2) · -((𝐵 , 𝐵) + 2)))
202		mulneg12 11561	. . . . . . . 8 ⊢ ((((abs‘(𝐴 , 𝐵))↑2) ∈ ℂ ∧ ((𝐵 , 𝐵) + 2) ∈ ℂ) → (-((abs‘(𝐴 , 𝐵))↑2) · ((𝐵 , 𝐵) + 2)) = (((abs‘(𝐴 , 𝐵))↑2) · -((𝐵 , 𝐵) + 2)))
203	142, 194, 202	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (-((abs‘(𝐴 , 𝐵))↑2) · ((𝐵 , 𝐵) + 2)) = (((abs‘(𝐴 , 𝐵))↑2) · -((𝐵 , 𝐵) + 2)))
204	201, 203	breqtrrd 5121	. . . . . 6 ⊢ (𝜑 → 0 ≤ (-((abs‘(𝐴 , 𝐵))↑2) · ((𝐵 , 𝐵) + 2)))
205	18, 42, 204	prodge0ld 13006	. . . . 5 ⊢ (𝜑 → 0 ≤ -((abs‘(𝐴 , 𝐵))↑2))
206	17	le0neg1d 11694	. . . . 5 ⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) ≤ 0 ↔ 0 ≤ -((abs‘(𝐴 , 𝐵))↑2)))
207	205, 206	mpbird 257	. . . 4 ⊢ (𝜑 → ((abs‘(𝐴 , 𝐵))↑2) ≤ 0)
208	15	sqge0d 14050	. . . 4 ⊢ (𝜑 → 0 ≤ ((abs‘(𝐴 , 𝐵))↑2))
209		0re 11120	. . . . 5 ⊢ 0 ∈ ℝ
210		letri3 11204	. . . . 5 ⊢ ((((abs‘(𝐴 , 𝐵))↑2) ∈ ℝ ∧ 0 ∈ ℝ) → (((abs‘(𝐴 , 𝐵))↑2) = 0 ↔ (((abs‘(𝐴 , 𝐵))↑2) ≤ 0 ∧ 0 ≤ ((abs‘(𝐴 , 𝐵))↑2))))
211	17, 209, 210	sylancl 586	. . . 4 ⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) = 0 ↔ (((abs‘(𝐴 , 𝐵))↑2) ≤ 0 ∧ 0 ≤ ((abs‘(𝐴 , 𝐵))↑2))))
212	207, 208, 211	mpbir2and 713	. . 3 ⊢ (𝜑 → ((abs‘(𝐴 , 𝐵))↑2) = 0)
213	16, 212	sqeq0d 14058	. 2 ⊢ (𝜑 → (abs‘(𝐴 , 𝐵)) = 0)
214	14, 213	abs00d 15362	1 ⊢ (𝜑 → (𝐴 , 𝐵) = 0)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 ⊆ wss 3897 class class class wbr 5093 ‘cfv 6487 (class class class)co 7352 ℂcc 11010 ℝcr 11011 0cc0 11012 1c1 11013 + caddc 11015 · cmul 11017 < clt 11152 ≤ cle 11153 − cmin 11350 -cneg 11351 / cdiv 11780 2c2 12186 ℤcz 12474 ℝ⁺crp 12896 ↑cexp 13974 ∗ccj 15009 abscabs 15147 Basecbs 17126 +_gcplusg 17167 Scalarcsca 17170 ·_𝑠 cvsca 17171 ·_𝑖cip 17172 -_gcsg 18854 LModclmod 20799 LSubSpclss 20870 PreHilcphl 21567 normcnm 24497 NrmGrpcngp 24498 ℂPreHilccph 25099 ℂHilchl 25267

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 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11068 ax-resscn 11069 ax-1cn 11070 ax-icn 11071 ax-addcl 11072 ax-addrcl 11073 ax-mulcl 11074 ax-mulrcl 11075 ax-mulcom 11076 ax-addass 11077 ax-mulass 11078 ax-distr 11079 ax-i2m1 11080 ax-1ne0 11081 ax-1rid 11082 ax-rnegex 11083 ax-rrecex 11084 ax-cnre 11085 ax-pre-lttri 11086 ax-pre-lttrn 11087 ax-pre-ltadd 11088 ax-pre-mulgt0 11089 ax-pre-sup 11090 ax-addf 11091 ax-mulf 11092

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-iin 4944 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-isom 6496 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7616 df-om 7803 df-1st 7927 df-2nd 7928 df-supp 8097 df-tpos 8162 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-2o 8392 df-er 8628 df-map 8758 df-ixp 8828 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-card 9838 df-pnf 11154 df-mnf 11155 df-xr 11156 df-ltxr 11157 df-le 11158 df-sub 11352 df-neg 11353 df-div 11781 df-nn 12132 df-2 12194 df-3 12195 df-4 12196 df-5 12197 df-6 12198 df-7 12199 df-8 12200 df-9 12201 df-n0 12388 df-z 12475 df-dec 12595 df-uz 12739 df-q 12853 df-rp 12897 df-xneg 13017 df-xadd 13018 df-xmul 13019 df-ioo 13255 df-ico 13257 df-icc 13258 df-fz 13414 df-fzo 13561 df-seq 13915 df-exp 13975 df-hash 14244 df-cj 15012 df-re 15013 df-im 15014 df-sqrt 15148 df-abs 15149 df-struct 17064 df-sets 17081 df-slot 17099 df-ndx 17111 df-base 17127 df-ress 17148 df-plusg 17180 df-mulr 17181 df-starv 17182 df-sca 17183 df-vsca 17184 df-ip 17185 df-tset 17186 df-ple 17187 df-ds 17189 df-unif 17190 df-hom 17191 df-cco 17192 df-rest 17332 df-topn 17333 df-0g 17351 df-gsum 17352 df-topgen 17353 df-pt 17354 df-prds 17357 df-xrs 17412 df-qtop 17417 df-imas 17418 df-xps 17420 df-mre 17494 df-mrc 17495 df-acs 17497 df-mgm 18554 df-sgrp 18633 df-mnd 18649 df-mhm 18697 df-submnd 18698 df-grp 18855 df-minusg 18856 df-sbg 18857 df-mulg 18987 df-subg 19042 df-ghm 19131 df-cntz 19235 df-cmn 19700 df-abl 19701 df-mgp 20065 df-rng 20077 df-ur 20106 df-ring 20159 df-cring 20160 df-oppr 20261 df-dvdsr 20281 df-unit 20282 df-invr 20312 df-dvr 20325 df-rhm 20396 df-subrng 20467 df-subrg 20491 df-drng 20652 df-staf 20760 df-srng 20761 df-lmod 20801 df-lss 20871 df-lmhm 20962 df-lvec 21043 df-sra 21113 df-rgmod 21114 df-psmet 21289 df-xmet 21290 df-met 21291 df-bl 21292 df-mopn 21293 df-fbas 21294 df-fg 21295 df-cnfld 21298 df-phl 21569 df-top 22815 df-topon 22832 df-topsp 22854 df-bases 22867 df-cld 22940 df-ntr 22941 df-cls 22942 df-nei 23019 df-cn 23148 df-cnp 23149 df-haus 23236 df-cmp 23308 df-tx 23483 df-hmeo 23676 df-fil 23767 df-flim 23860 df-fcls 23862 df-xms 24241 df-ms 24242 df-tms 24243 df-nm 24503 df-ngp 24504 df-nlm 24507 df-cncf 24804 df-clm 24996 df-cph 25101 df-cfil 25188 df-cmet 25190 df-cms 25268 df-bn 25269 df-hl 25270

This theorem is referenced by: pjthlem2 25371

W3C validator