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Theorem pjthlem1 25287

Description: Lemma for pjth 25289. (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 25214	. . . 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 20773	. . . . 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 3975	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
12		pjthlem.h	. . . 4 ⊢ , = (·_𝑖‘𝑊)
13	6, 12	cphipcl 25041	. . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 , 𝐵) ∈ ℂ)
14	3, 4, 11, 13	syl3anc 1368	. 2 ⊢ (𝜑 → (𝐴 , 𝐵) ∈ ℂ)
15	14	abscld 15380	. . . 4 ⊢ (𝜑 → (abs‘(𝐴 , 𝐵)) ∈ ℝ)
16	15	recnd 11239	. . 3 ⊢ (𝜑 → (abs‘(𝐴 , 𝐵)) ∈ ℂ)
17	15	resqcld 14087	. . . . . . 7 ⊢ (𝜑 → ((abs‘(𝐴 , 𝐵))↑2) ∈ ℝ)
18	17	renegcld 11638	. . . . . 6 ⊢ (𝜑 → -((abs‘(𝐴 , 𝐵))↑2) ∈ ℝ)
19	6, 12	reipcl 25047	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐵 ∈ 𝑉) → (𝐵 , 𝐵) ∈ ℝ)
20	3, 11, 19	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → (𝐵 , 𝐵) ∈ ℝ)
21		2re 12283	. . . . . . . 8 ⊢ 2 ∈ ℝ
22		readdcl 11189	. . . . . . . 8 ⊢ (((𝐵 , 𝐵) ∈ ℝ ∧ 2 ∈ ℝ) → ((𝐵 , 𝐵) + 2) ∈ ℝ)
23	20, 21, 22	sylancl 585	. . . . . . 7 ⊢ (𝜑 → ((𝐵 , 𝐵) + 2) ∈ ℝ)
24		0red 11214	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
25		peano2re 11384	. . . . . . . . 9 ⊢ ((𝐵 , 𝐵) ∈ ℝ → ((𝐵 , 𝐵) + 1) ∈ ℝ)
26	20, 25	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 , 𝐵) + 1) ∈ ℝ)
27	6, 12	ipge0 25048	. . . . . . . . . 10 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐵 ∈ 𝑉) → 0 ≤ (𝐵 , 𝐵))
28	3, 11, 27	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (𝐵 , 𝐵))
29	20	ltp1d 12141	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 , 𝐵) < ((𝐵 , 𝐵) + 1))
30	24, 20, 26, 28, 29	lelttrd 11369	. . . . . . . 8 ⊢ (𝜑 → 0 < ((𝐵 , 𝐵) + 1))
31	26	ltp1d 12141	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 , 𝐵) + 1) < (((𝐵 , 𝐵) + 1) + 1))
32		df-2 12272	. . . . . . . . . . 11 ⊢ 2 = (1 + 1)
33	32	oveq2i 7412	. . . . . . . . . 10 ⊢ ((𝐵 , 𝐵) + 2) = ((𝐵 , 𝐵) + (1 + 1))
34	20	recnd 11239	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 , 𝐵) ∈ ℂ)
35		ax-1cn 11164	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
36		addass 11193	. . . . . . . . . . . 12 ⊢ (((𝐵 , 𝐵) ∈ ℂ ∧ 1 ∈ ℂ ∧ 1 ∈ ℂ) → (((𝐵 , 𝐵) + 1) + 1) = ((𝐵 , 𝐵) + (1 + 1)))
37	35, 35, 36	mp3an23 1449	. . . . . . . . . . 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 2783	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 , 𝐵) + 2) = (((𝐵 , 𝐵) + 1) + 1))
40	31, 39	breqtrrd 5166	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 , 𝐵) + 1) < ((𝐵 , 𝐵) + 2))
41	24, 26, 23, 30, 40	lttrd 11372	. . . . . . 7 ⊢ (𝜑 → 0 < ((𝐵 , 𝐵) + 2))
42	23, 41	elrpd 13010	. . . . . 6 ⊢ (𝜑 → ((𝐵 , 𝐵) + 2) ∈ ℝ⁺)
43		oveq2 7409	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑇( ·_𝑠 ‘𝑊)𝐵) → (𝐴 − 𝑥) = (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)))
44	43	fveq2d 6885	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑇( ·_𝑠 ‘𝑊)𝐵) → (𝑁‘(𝐴 − 𝑥)) = (𝑁‘(𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵))))
45	44	breq2d 5150	. . . . . . . . . . . 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 25024	. . . . . . . . . . . . . 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 25215	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ PreHil)
51	1, 50	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑊 ∈ PreHil)
52		eqid 2724	. . . . . . . . . . . . . . . . 17 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
53		eqid 2724	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
54	52, 12, 6, 53	ipcl 21494	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 , 𝐵) ∈ (Base‘(Scalar‘𝑊)))
55	51, 4, 11, 54	syl3anc 1368	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 , 𝐵) ∈ (Base‘(Scalar‘𝑊)))
56	52, 53	hlress 25218	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∈ ℂHil → ℝ ⊆ (Base‘(Scalar‘𝑊)))
57	1, 56	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ℝ ⊆ (Base‘(Scalar‘𝑊)))
58	57, 26	sseldd 3975	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐵 , 𝐵) + 1) ∈ (Base‘(Scalar‘𝑊)))
59	20, 28	ge0p1rpd 13043	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐵 , 𝐵) + 1) ∈ ℝ⁺)
60	59	rpne0d 13018	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐵 , 𝐵) + 1) ≠ 0)
61	52, 53	cphdivcl 25032	. . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1)) ∈ (Base‘(Scalar‘𝑊)))
63	49, 62	eqeltrid 2829	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑇 ∈ (Base‘(Scalar‘𝑊)))
64		eqid 2724	. . . . . . . . . . . . . 14 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑊)
65	52, 64, 53, 7	lssvscl 20792	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) ∧ (𝑇 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝐵 ∈ 𝑈)) → (𝑇( ·_𝑠 ‘𝑊)𝐵) ∈ 𝑈)
66	48, 5, 63, 10, 65	syl22anc 836	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑇( ·_𝑠 ‘𝑊)𝐵) ∈ 𝑈)
67	45, 46, 66	rspcdva 3605	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁‘𝐴) ≤ (𝑁‘(𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵))))
68		cphngp 25023	. . . . . . . . . . . . . 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 24447	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝐴 ∈ 𝑉) → (𝑁‘𝐴) ∈ ℝ)
72	69, 4, 71	syl2anc 583	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁‘𝐴) ∈ ℝ)
73	6, 52, 64, 53	lmodvscl 20714	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝐵 ∈ 𝑉) → (𝑇( ·_𝑠 ‘𝑊)𝐵) ∈ 𝑉)
74	48, 63, 11, 73	syl3anc 1368	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑇( ·_𝑠 ‘𝑊)𝐵) ∈ 𝑉)
75		pjthlem.m	. . . . . . . . . . . . . . 15 ⊢ − = (-_g‘𝑊)
76	6, 75	lmodvsubcl 20743	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉 ∧ (𝑇( ·_𝑠 ‘𝑊)𝐵) ∈ 𝑉) → (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)) ∈ 𝑉)
77	48, 4, 74, 76	syl3anc 1368	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)) ∈ 𝑉)
78	6, 70	nmcl 24447	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ NrmGrp ∧ (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)) ∈ 𝑉) → (𝑁‘(𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵))) ∈ ℝ)
79	69, 77, 78	syl2anc 583	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁‘(𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵))) ∈ ℝ)
80	6, 70	nmge0 24448	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝐴 ∈ 𝑉) → 0 ≤ (𝑁‘𝐴))
81	69, 4, 80	syl2anc 583	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (𝑁‘𝐴))
82	6, 70	nmge0 24448	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ NrmGrp ∧ (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)) ∈ 𝑉) → 0 ≤ (𝑁‘(𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵))))
83	69, 77, 82	syl2anc 583	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (𝑁‘(𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵))))
84	72, 79, 81, 83	le2sqd 14217	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁‘𝐴) ≤ (𝑁‘(𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵))) ↔ ((𝑁‘𝐴)↑2) ≤ ((𝑁‘(𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)))↑2)))
85	67, 84	mpbid 231	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑁‘𝐴)↑2) ≤ ((𝑁‘(𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)))↑2))
86	79	resqcld 14087	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁‘(𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)))↑2) ∈ ℝ)
87	72	resqcld 14087	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁‘𝐴)↑2) ∈ ℝ)
88	86, 87	subge0d 11801	. . . . . . . . . 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 12591	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℤ
91		rpexpcl 14043	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐵 , 𝐵) + 1) ∈ ℝ⁺ ∧ 2 ∈ ℤ) → (((𝐵 , 𝐵) + 1)↑2) ∈ ℝ⁺)
92	59, 90, 91	sylancl 585	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((𝐵 , 𝐵) + 1)↑2) ∈ ℝ⁺)
93	17, 92	rerpdivcld 13044	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) ∈ ℝ)
94	93, 23	remulcld 11241	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2)) ∈ ℝ)
95	94	recnd 11239	. . . . . . . . . . . 12 ⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2)) ∈ ℂ)
96	95	negcld 11555	. . . . . . . . . . 11 ⊢ (𝜑 → -((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 2)) ∈ ℂ)
97	6, 12	cphipcl 25041	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐴 , 𝐴) ∈ ℂ)
98	3, 4, 4, 97	syl3anc 1368	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 , 𝐴) ∈ ℂ)
99	96, 98	pncand 11569	. . . . . . . . . 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 25044	. . . . . . . . . . . . . 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 583	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑁‘(𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)))↑2) = ((𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵)) , (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵))))
102	12, 6, 75	cphsubdir 25058	. . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . 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 25059	. . . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 , (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵))) = ((𝐴 , 𝐴) − (𝐴 , (𝑇( ·_𝑠 ‘𝑊)𝐵))))
106	105	oveq1d 7416	. . . . . . . . . . . . . 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 25041	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ (𝑇( ·_𝑠 ‘𝑊)𝐵) ∈ 𝑉) → (𝐴 , (𝑇( ·_𝑠 ‘𝑊)𝐵)) ∈ ℂ)
108	3, 4, 74, 107	syl3anc 1368	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 , (𝑇( ·_𝑠 ‘𝑊)𝐵)) ∈ ℂ)
109	12, 6, 75	cphsubdi 25059	. . . . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . . . 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 25041	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝑇( ·_𝑠 ‘𝑊)𝐵) ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ((𝑇( ·_𝑠 ‘𝑊)𝐵) , 𝐴) ∈ ℂ)
112	3, 74, 4, 111	syl3anc 1368	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑇( ·_𝑠 ‘𝑊)𝐵) , 𝐴) ∈ ℂ)
113	6, 12	cphipcl 25041	. . . . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑇( ·_𝑠 ‘𝑊)𝐵) , (𝑇( ·_𝑠 ‘𝑊)𝐵)) ∈ ℂ)
115	112, 114	subcld 11568	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((𝑇( ·_𝑠 ‘𝑊)𝐵) , 𝐴) − ((𝑇( ·_𝑠 ‘𝑊)𝐵) , (𝑇( ·_𝑠 ‘𝑊)𝐵))) ∈ ℂ)
116	110, 115	eqeltrd 2825	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑇( ·_𝑠 ‘𝑊)𝐵) , (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵))) ∈ ℂ)
117	98, 108, 116	subsub4d 11599	. . . . . . . . . . . . . 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 11239	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) ∈ ℂ)
119	26	recnd 11239	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐵 , 𝐵) + 1) ∈ ℂ)
120		1cnd 11206	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ∈ ℂ)
121	118, 119, 120	adddid 11235	. . . . . . . . . . . . . . . 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 7417	. . . . . . . . . . . . . . . 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 25062	. . . . . . . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐴 , (𝑇( ·_𝑠 ‘𝑊)𝐵)) = ((∗‘𝑇) · (𝐴 , 𝐵)))
125	14, 119, 60	divcld 11987	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1)) ∈ ℂ)
126	49, 125	eqeltrid 2829	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑇 ∈ ℂ)
127	126	cjcld 15140	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (∗‘𝑇) ∈ ℂ)
128	127, 14	mulcomd 11232	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((∗‘𝑇) · (𝐴 , 𝐵)) = ((𝐴 , 𝐵) · (∗‘𝑇)))
129	14	cjcld 15140	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (∗‘(𝐴 , 𝐵)) ∈ ℂ)
130	14, 129, 119, 60	divassd 12022	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (((𝐴 , 𝐵) · (∗‘(𝐴 , 𝐵))) / ((𝐵 , 𝐵) + 1)) = ((𝐴 , 𝐵) · ((∗‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1))))
131	14	absvalsqd 15386	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((abs‘(𝐴 , 𝐵))↑2) = ((𝐴 , 𝐵) · (∗‘(𝐴 , 𝐵))))
132	131	oveq1d 7416	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) / ((𝐵 , 𝐵) + 1)) = (((𝐴 , 𝐵) · (∗‘(𝐴 , 𝐵))) / ((𝐵 , 𝐵) + 1)))
133	49	fveq2i 6884	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∗‘𝑇) = (∗‘((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1)))
134	14, 119, 60	cjdivd 15167	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (∗‘((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1))) = ((∗‘(𝐴 , 𝐵)) / (∗‘((𝐵 , 𝐵) + 1))))
135	26	cjred 15170	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (∗‘((𝐵 , 𝐵) + 1)) = ((𝐵 , 𝐵) + 1))
136	135	oveq2d 7417	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((∗‘(𝐴 , 𝐵)) / (∗‘((𝐵 , 𝐵) + 1))) = ((∗‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1)))
137	134, 136	eqtrd 2764	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (∗‘((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1))) = ((∗‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1)))
138	133, 137	eqtrid 2776	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (∗‘𝑇) = ((∗‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1)))
139	138	oveq2d 7417	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝐴 , 𝐵) · (∗‘𝑇)) = ((𝐴 , 𝐵) · ((∗‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1))))
140	130, 132, 139	3eqtr4rd 2775	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝐴 , 𝐵) · (∗‘𝑇)) = (((abs‘(𝐴 , 𝐵))↑2) / ((𝐵 , 𝐵) + 1)))
141	124, 128, 140	3eqtrd 2768	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐴 , (𝑇( ·_𝑠 ‘𝑊)𝐵)) = (((abs‘(𝐴 , 𝐵))↑2) / ((𝐵 , 𝐵) + 1)))
142	17	recnd 11239	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((abs‘(𝐴 , 𝐵))↑2) ∈ ℂ)
143	142, 119	mulcomd 11232	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) · ((𝐵 , 𝐵) + 1)) = (((𝐵 , 𝐵) + 1) · ((abs‘(𝐴 , 𝐵))↑2)))
144	119	sqvald 14105	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (((𝐵 , 𝐵) + 1)↑2) = (((𝐵 , 𝐵) + 1) · ((𝐵 , 𝐵) + 1)))
145	143, 144	oveq12d 7419	. . . . . . . . . . . . . . . . . . 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 12013	. . . . . . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) / ((𝐵 , 𝐵) + 1)) = ((((abs‘(𝐴 , 𝐵))↑2) · ((𝐵 , 𝐵) + 1)) / (((𝐵 , 𝐵) + 1)↑2)))
148	92	rpcnd 13015	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (((𝐵 , 𝐵) + 1)↑2) ∈ ℂ)
149	92	rpne0d 13018	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (((𝐵 , 𝐵) + 1)↑2) ≠ 0)
150	142, 119, 148, 149	div23d 12024	. . . . . . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴 , (𝑇( ·_𝑠 ‘𝑊)𝐵)) = ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 1)))
152	93, 26	remulcld 11241	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · ((𝐵 , 𝐵) + 1)) ∈ ℝ)
153	151, 152	eqeltrd 2825	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐴 , (𝑇( ·_𝑠 ‘𝑊)𝐵)) ∈ ℝ)
154	153	cjred 15170	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (∗‘(𝐴 , (𝑇( ·_𝑠 ‘𝑊)𝐵))) = (𝐴 , (𝑇( ·_𝑠 ‘𝑊)𝐵)))
155	12, 6	cphipcj 25049	. . . . . . . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (∗‘(𝐴 , (𝑇( ·_𝑠 ‘𝑊)𝐵))) = ((𝑇( ·_𝑠 ‘𝑊)𝐵) , 𝐴))
157	154, 156, 151	3eqtr3d 2772	. . . . . . . . . . . . . . . . . . 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 25063	. . . . . . . . . . . . . . . . . . . . 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 1376	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝑇( ·_𝑠 ‘𝑊)𝐵) , (𝑇( ·_𝑠 ‘𝑊)𝐵)) = ((𝑇 · (∗‘𝑇)) · (𝐵 , 𝐵)))
160	49	fveq2i 6884	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (abs‘𝑇) = (abs‘((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1)))
161	14, 119, 60	absdivd 15399	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (abs‘((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1))) = ((abs‘(𝐴 , 𝐵)) / (abs‘((𝐵 , 𝐵) + 1))))
162	59	rpge0d 13017	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 0 ≤ ((𝐵 , 𝐵) + 1))
163	26, 162	absidd 15366	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (abs‘((𝐵 , 𝐵) + 1)) = ((𝐵 , 𝐵) + 1))
164	163	oveq2d 7417	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ((abs‘(𝐴 , 𝐵)) / (abs‘((𝐵 , 𝐵) + 1))) = ((abs‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1)))
165	161, 164	eqtrd 2764	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (abs‘((𝐴 , 𝐵) / ((𝐵 , 𝐵) + 1))) = ((abs‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1)))
166	160, 165	eqtrid 2776	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (abs‘𝑇) = ((abs‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1)))
167	166	oveq1d 7416	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((abs‘𝑇)↑2) = (((abs‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1))↑2))
168	126	absvalsqd 15386	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((abs‘𝑇)↑2) = (𝑇 · (∗‘𝑇)))
169	16, 119, 60	sqdivd 14121	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (((abs‘(𝐴 , 𝐵)) / ((𝐵 , 𝐵) + 1))↑2) = (((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)))
170	167, 168, 169	3eqtr3d 2772	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑇 · (∗‘𝑇)) = (((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)))
171	170	oveq1d 7416	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝑇 · (∗‘𝑇)) · (𝐵 , 𝐵)) = ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · (𝐵 , 𝐵)))
172	159, 171	eqtrd 2764	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑇( ·_𝑠 ‘𝑊)𝐵) , (𝑇( ·_𝑠 ‘𝑊)𝐵)) = ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · (𝐵 , 𝐵)))
173	157, 172	oveq12d 7419	. . . . . . . . . . . . . . . . . 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 11464	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐵 , 𝐵) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝐵 , 𝐵) + 1) − (𝐵 , 𝐵)) = 1)
175	34, 35, 174	sylancl 585	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (((𝐵 , 𝐵) + 1) − (𝐵 , 𝐵)) = 1)
176	175	oveq2d 7417	. . . . . . . . . . . . . . . . . . 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 11667	. . . . . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑇( ·_𝑠 ‘𝑊)𝐵) , (𝐴 − (𝑇( ·_𝑠 ‘𝑊)𝐵))) = ((((abs‘(𝐴 , 𝐵))↑2) / (((𝐵 , 𝐵) + 1)↑2)) · 1))
180	151, 179	oveq12d 7419	. . . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . . 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 7417	. . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . 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 2768	. . . . . . . . . . . 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 11574	. . . . . . . . . . . 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 11413	. . . . . . . . . . . 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 2770	. . . . . . . . . . 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 25044	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → ((𝑁‘𝐴)↑2) = (𝐴 , 𝐴))
189	3, 4, 188	syl2anc 583	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁‘𝐴)↑2) = (𝐴 , 𝐴))
190	187, 189	oveq12d 7419	. . . . . . . . . 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 11638	. . . . . . . . . . . . 13 ⊢ (𝜑 → -((𝐵 , 𝐵) + 2) ∈ ℝ)
192	191	recnd 11239	. . . . . . . . . . . 12 ⊢ (𝜑 → -((𝐵 , 𝐵) + 2) ∈ ℂ)
193	142, 192, 148, 149	div23d 12024	. . . . . . . . . . 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 11239	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐵 , 𝐵) + 2) ∈ ℂ)
195	118, 194	mulneg2d 11665	. . . . . . . . . . 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 2764	. . . . . . . . . 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 2775	. . . . . . . . 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 5166	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ ((((abs‘(𝐴 , 𝐵))↑2) · -((𝐵 , 𝐵) + 2)) / (((𝐵 , 𝐵) + 1)↑2)))
199	17, 191	remulcld 11241	. . . . . . . . 9 ⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) · -((𝐵 , 𝐵) + 2)) ∈ ℝ)
200	199, 92	ge0divd 13051	. . . . . . . 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 11649	. . . . . . . 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 583	. . . . . . 7 ⊢ (𝜑 → (-((abs‘(𝐴 , 𝐵))↑2) · ((𝐵 , 𝐵) + 2)) = (((abs‘(𝐴 , 𝐵))↑2) · -((𝐵 , 𝐵) + 2)))
204	201, 203	breqtrrd 5166	. . . . . 6 ⊢ (𝜑 → 0 ≤ (-((abs‘(𝐴 , 𝐵))↑2) · ((𝐵 , 𝐵) + 2)))
205	18, 42, 204	prodge0ld 13079	. . . . 5 ⊢ (𝜑 → 0 ≤ -((abs‘(𝐴 , 𝐵))↑2))
206	17	le0neg1d 11782	. . . . 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 14099	. . . 4 ⊢ (𝜑 → 0 ≤ ((abs‘(𝐴 , 𝐵))↑2))
209		0re 11213	. . . . 5 ⊢ 0 ∈ ℝ
210		letri3 11296	. . . . 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 585	. . . 4 ⊢ (𝜑 → (((abs‘(𝐴 , 𝐵))↑2) = 0 ↔ (((abs‘(𝐴 , 𝐵))↑2) ≤ 0 ∧ 0 ≤ ((abs‘(𝐴 , 𝐵))↑2))))
212	207, 208, 211	mpbir2and 710	. . 3 ⊢ (𝜑 → ((abs‘(𝐴 , 𝐵))↑2) = 0)
213	16, 212	sqeq0d 14107	. 2 ⊢ (𝜑 → (abs‘(𝐴 , 𝐵)) = 0)
214	14, 213	abs00d 15390	1 ⊢ (𝜑 → (𝐴 , 𝐵) = 0)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∀wral 3053 ⊆ wss 3940 class class class wbr 5138 ‘cfv 6533 (class class class)co 7401 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 < clt 11245 ≤ cle 11246 − cmin 11441 -cneg 11442 / cdiv 11868 2c2 12264 ℤcz 12555 ℝ⁺crp 12971 ↑cexp 14024 ∗ccj 15040 abscabs 15178 Basecbs 17143 +_gcplusg 17196 Scalarcsca 17199 ·_𝑠 cvsca 17200 ·_𝑖cip 17201 -_gcsg 18855 LModclmod 20696 LSubSpclss 20768 PreHilcphl 21485 normcnm 24407 NrmGrpcngp 24408 ℂPreHilccph 25016 ℂHilchl 25184

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-tpos 8206 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8699 df-map 8818 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-ioo 13325 df-ico 13327 df-icc 13328 df-fz 13482 df-fzo 13625 df-seq 13964 df-exp 14025 df-hash 14288 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17367 df-topn 17368 df-0g 17386 df-gsum 17387 df-topgen 17388 df-pt 17389 df-prds 17392 df-xrs 17447 df-qtop 17452 df-imas 17453 df-xps 17455 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-mhm 18703 df-submnd 18704 df-grp 18856 df-minusg 18857 df-sbg 18858 df-mulg 18986 df-subg 19040 df-ghm 19129 df-cntz 19223 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-cring 20131 df-oppr 20226 df-dvdsr 20249 df-unit 20250 df-invr 20280 df-dvr 20293 df-rhm 20364 df-subrng 20436 df-subrg 20461 df-drng 20579 df-staf 20678 df-srng 20679 df-lmod 20698 df-lss 20769 df-lmhm 20860 df-lvec 20941 df-sra 21011 df-rgmod 21012 df-psmet 21220 df-xmet 21221 df-met 21222 df-bl 21223 df-mopn 21224 df-fbas 21225 df-fg 21226 df-cnfld 21229 df-phl 21487 df-top 22718 df-topon 22735 df-topsp 22757 df-bases 22771 df-cld 22845 df-ntr 22846 df-cls 22847 df-nei 22924 df-cn 23053 df-cnp 23054 df-haus 23141 df-cmp 23213 df-tx 23388 df-hmeo 23581 df-fil 23672 df-flim 23765 df-fcls 23767 df-xms 24148 df-ms 24149 df-tms 24150 df-nm 24413 df-ngp 24414 df-nlm 24417 df-cncf 24720 df-clm 24912 df-cph 25018 df-cfil 25105 df-cmet 25107 df-cms 25185 df-bn 25186 df-hl 25187

This theorem is referenced by: pjthlem2 25288

W3C validator