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Theorem cphipval2 25379

Description: Value of the inner product expressed by the norm defined by it. (Contributed by NM, 31-Jan-2007.) (Revised by AV, 18-Oct-2021.)

**Hypotheses**
Ref	Expression
cphipfval.x	⊢ 𝑋 = (Base‘𝑊)
cphipfval.p	⊢ + = (+_g‘𝑊)
cphipfval.s	⊢ · = ( ·_𝑠 ‘𝑊)
cphipfval.n	⊢ 𝑁 = (norm‘𝑊)
cphipfval.i	⊢ , = (·_𝑖‘𝑊)
cphipval2.m	⊢ − = (-_g‘𝑊)
cphipval2.f	⊢ 𝐹 = (Scalar‘𝑊)
cphipval2.k	⊢ 𝐾 = (Base‘𝐹)

**Assertion**
Ref	Expression
cphipval2	⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 , 𝐵) = (((((𝑁‘(𝐴 + 𝐵))↑2) − ((𝑁‘(𝐴 − 𝐵))↑2)) + (i · (((𝑁‘(𝐴 + (i · 𝐵)))↑2) − ((𝑁‘(𝐴 − (i · 𝐵)))↑2)))) / 4))

**Proof of Theorem cphipval2**
Step	Hyp	Ref	Expression
1		simpl 487	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) → 𝑊 ∈ ℂPreHil)
2	1	3ad2ant1 1149	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝑊 ∈ ℂPreHil)
3		cphngp 25311	. . . . . . . . . . 11 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp)
4	3	adantr 485	. . . . . . . . . 10 ⊢ ((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) → 𝑊 ∈ NrmGrp)
5		ngpgrp 24735	. . . . . . . . . 10 ⊢ (𝑊 ∈ NrmGrp → 𝑊 ∈ Grp)
6	4, 5	syl 18	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) → 𝑊 ∈ Grp)
7		cphipfval.x	. . . . . . . . . 10 ⊢ 𝑋 = (Base‘𝑊)
8		cphipfval.p	. . . . . . . . . 10 ⊢ + = (+_g‘𝑊)
9	7, 8	grpcl 19007	. . . . . . . . 9 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 + 𝐵) ∈ 𝑋)
10	6, 9	syl3an1 1179	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 + 𝐵) ∈ 𝑋)
11		cphipfval.i	. . . . . . . . 9 ⊢ , = (·_𝑖‘𝑊)
12		cphipfval.n	. . . . . . . . 9 ⊢ 𝑁 = (norm‘𝑊)
13	7, 11, 12	nmsq 25332	. . . . . . . 8 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 + 𝐵) ∈ 𝑋) → ((𝑁‘(𝐴 + 𝐵))↑2) = ((𝐴 + 𝐵) , (𝐴 + 𝐵)))
14	2, 10, 13	syl2anc 595	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴 + 𝐵))↑2) = ((𝐴 + 𝐵) , (𝐴 + 𝐵)))
15		simp2 1153	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ 𝑋)
16		simp3 1154	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐵 ∈ 𝑋)
17	11, 7, 8, 2, 15, 16, 15, 16	cph2di 25345	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 + 𝐵) , (𝐴 + 𝐵)) = (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + ((𝐴 , 𝐵) + (𝐵 , 𝐴))))
18	14, 17	eqtrd 2796	. . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴 + 𝐵))↑2) = (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + ((𝐴 , 𝐵) + (𝐵 , 𝐴))))
19		cphipval2.m	. . . . . . . . . 10 ⊢ − = (-_g‘𝑊)
20	7, 19	grpsubcl 19085	. . . . . . . . 9 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 − 𝐵) ∈ 𝑋)
21	6, 20	syl3an1 1179	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 − 𝐵) ∈ 𝑋)
22	7, 11, 12	nmsq 25332	. . . . . . . 8 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 − 𝐵) ∈ 𝑋) → ((𝑁‘(𝐴 − 𝐵))↑2) = ((𝐴 − 𝐵) , (𝐴 − 𝐵)))
23	2, 21, 22	syl2anc 595	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴 − 𝐵))↑2) = ((𝐴 − 𝐵) , (𝐴 − 𝐵)))
24	11, 7, 19, 2, 15, 16, 15, 16	cph2subdi 25348	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 − 𝐵) , (𝐴 − 𝐵)) = (((𝐴 , 𝐴) + (𝐵 , 𝐵)) − ((𝐴 , 𝐵) + (𝐵 , 𝐴))))
25	23, 24	eqtrd 2796	. . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴 − 𝐵))↑2) = (((𝐴 , 𝐴) + (𝐵 , 𝐵)) − ((𝐴 , 𝐵) + (𝐵 , 𝐴))))
26	18, 25	oveq12d 7428	. . . . 5 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴 + 𝐵))↑2) − ((𝑁‘(𝐴 − 𝐵))↑2)) = ((((𝐴 , 𝐴) + (𝐵 , 𝐵)) + ((𝐴 , 𝐵) + (𝐵 , 𝐴))) − (((𝐴 , 𝐴) + (𝐵 , 𝐵)) − ((𝐴 , 𝐵) + (𝐵 , 𝐴)))))
27	7, 11	reipcl 25335	. . . . . . . . . 10 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑋) → (𝐴 , 𝐴) ∈ ℝ)
28	27	adantlr 727	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋) → (𝐴 , 𝐴) ∈ ℝ)
29	28	recnd 11236	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋) → (𝐴 , 𝐴) ∈ ℂ)
30	29	3adant3 1148	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 , 𝐴) ∈ ℂ)
31	7, 11	reipcl 25335	. . . . . . . . . 10 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐵 ∈ 𝑋) → (𝐵 , 𝐵) ∈ ℝ)
32	31	adantlr 727	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐵 ∈ 𝑋) → (𝐵 , 𝐵) ∈ ℝ)
33	32	recnd 11236	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐵 ∈ 𝑋) → (𝐵 , 𝐵) ∈ ℂ)
34	33	3adant2 1147	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵 , 𝐵) ∈ ℂ)
35	30, 34	addcld 11227	. . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 , 𝐴) + (𝐵 , 𝐵)) ∈ ℂ)
36	7, 11	cphipcl 25329	. . . . . . . 8 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 , 𝐵) ∈ ℂ)
37	1, 36	syl3an1 1179	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 , 𝐵) ∈ ℂ)
38	7, 11	cphipcl 25329	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐵 , 𝐴) ∈ ℂ)
39	1, 38	syl3an1 1179	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐵 , 𝐴) ∈ ℂ)
40	39	3com23 1142	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵 , 𝐴) ∈ ℂ)
41	37, 40	addcld 11227	. . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 , 𝐵) + (𝐵 , 𝐴)) ∈ ℂ)
42	35, 41, 41	pnncand 11607	. . . . 5 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((((𝐴 , 𝐴) + (𝐵 , 𝐵)) + ((𝐴 , 𝐵) + (𝐵 , 𝐴))) − (((𝐴 , 𝐴) + (𝐵 , 𝐵)) − ((𝐴 , 𝐵) + (𝐵 , 𝐴)))) = (((𝐴 , 𝐵) + (𝐵 , 𝐴)) + ((𝐴 , 𝐵) + (𝐵 , 𝐴))))
43	26, 42	eqtrd 2796	. . . 4 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴 + 𝐵))↑2) − ((𝑁‘(𝐴 − 𝐵))↑2)) = (((𝐴 , 𝐵) + (𝐵 , 𝐴)) + ((𝐴 , 𝐵) + (𝐵 , 𝐴))))
44	6	3ad2ant1 1149	. . . . . . . . . 10 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝑊 ∈ Grp)
45		cphlmod 25312	. . . . . . . . . . . . . 14 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LMod)
46	45	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) → 𝑊 ∈ LMod)
47	46	adantr 485	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐵 ∈ 𝑋) → 𝑊 ∈ LMod)
48		simplr 780	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐵 ∈ 𝑋) → i ∈ 𝐾)
49		simpr 489	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐵 ∈ 𝑋) → 𝐵 ∈ 𝑋)
50		cphipval2.f	. . . . . . . . . . . . 13 ⊢ 𝐹 = (Scalar‘𝑊)
51		cphipfval.s	. . . . . . . . . . . . 13 ⊢ · = ( ·_𝑠 ‘𝑊)
52		cphipval2.k	. . . . . . . . . . . . 13 ⊢ 𝐾 = (Base‘𝐹)
53	7, 50, 51, 52	lmodvscl 20978	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ i ∈ 𝐾 ∧ 𝐵 ∈ 𝑋) → (i · 𝐵) ∈ 𝑋)
54	47, 48, 49, 53	syl3anc 1396	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐵 ∈ 𝑋) → (i · 𝐵) ∈ 𝑋)
55	54	3adant2 1147	. . . . . . . . . 10 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (i · 𝐵) ∈ 𝑋)
56	7, 8	grpcl 19007	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (i · 𝐵) ∈ 𝑋) → (𝐴 + (i · 𝐵)) ∈ 𝑋)
57	44, 15, 55, 56	syl3anc 1396	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 + (i · 𝐵)) ∈ 𝑋)
58	7, 11, 12	nmsq 25332	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 + (i · 𝐵)) ∈ 𝑋) → ((𝑁‘(𝐴 + (i · 𝐵)))↑2) = ((𝐴 + (i · 𝐵)) , (𝐴 + (i · 𝐵))))
59	2, 57, 58	syl2anc 595	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴 + (i · 𝐵)))↑2) = ((𝐴 + (i · 𝐵)) , (𝐴 + (i · 𝐵))))
60	11, 7, 8, 2, 15, 55, 15, 55	cph2di 25345	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 + (i · 𝐵)) , (𝐴 + (i · 𝐵))) = (((𝐴 , 𝐴) + ((i · 𝐵) , (i · 𝐵))) + ((𝐴 , (i · 𝐵)) + ((i · 𝐵) , 𝐴))))
61	59, 60	eqtrd 2796	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴 + (i · 𝐵)))↑2) = (((𝐴 , 𝐴) + ((i · 𝐵) , (i · 𝐵))) + ((𝐴 , (i · 𝐵)) + ((i · 𝐵) , 𝐴))))
62	7, 19	grpsubcl 19085	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (i · 𝐵) ∈ 𝑋) → (𝐴 − (i · 𝐵)) ∈ 𝑋)
63	44, 15, 55, 62	syl3anc 1396	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 − (i · 𝐵)) ∈ 𝑋)
64	7, 11, 12	nmsq 25332	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 − (i · 𝐵)) ∈ 𝑋) → ((𝑁‘(𝐴 − (i · 𝐵)))↑2) = ((𝐴 − (i · 𝐵)) , (𝐴 − (i · 𝐵))))
65	2, 63, 64	syl2anc 595	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴 − (i · 𝐵)))↑2) = ((𝐴 − (i · 𝐵)) , (𝐴 − (i · 𝐵))))
66	11, 7, 19, 2, 15, 55, 15, 55	cph2subdi 25348	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 − (i · 𝐵)) , (𝐴 − (i · 𝐵))) = (((𝐴 , 𝐴) + ((i · 𝐵) , (i · 𝐵))) − ((𝐴 , (i · 𝐵)) + ((i · 𝐵) , 𝐴))))
67	65, 66	eqtrd 2796	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴 − (i · 𝐵)))↑2) = (((𝐴 , 𝐴) + ((i · 𝐵) , (i · 𝐵))) − ((𝐴 , (i · 𝐵)) + ((i · 𝐵) , 𝐴))))
68	61, 67	oveq12d 7428	. . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴 + (i · 𝐵)))↑2) − ((𝑁‘(𝐴 − (i · 𝐵)))↑2)) = ((((𝐴 , 𝐴) + ((i · 𝐵) , (i · 𝐵))) + ((𝐴 , (i · 𝐵)) + ((i · 𝐵) , 𝐴))) − (((𝐴 , 𝐴) + ((i · 𝐵) , (i · 𝐵))) − ((𝐴 , (i · 𝐵)) + ((i · 𝐵) , 𝐴)))))
69	68	oveq2d 7426	. . . . 5 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (i · (((𝑁‘(𝐴 + (i · 𝐵)))↑2) − ((𝑁‘(𝐴 − (i · 𝐵)))↑2))) = (i · ((((𝐴 , 𝐴) + ((i · 𝐵) , (i · 𝐵))) + ((𝐴 , (i · 𝐵)) + ((i · 𝐵) , 𝐴))) − (((𝐴 , 𝐴) + ((i · 𝐵) , (i · 𝐵))) − ((𝐴 , (i · 𝐵)) + ((i · 𝐵) , 𝐴))))))
70	7, 11	cphipcl 25329	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℂPreHil ∧ (i · 𝐵) ∈ 𝑋 ∧ (i · 𝐵) ∈ 𝑋) → ((i · 𝐵) , (i · 𝐵)) ∈ ℂ)
71	2, 55, 55, 70	syl3anc 1396	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((i · 𝐵) , (i · 𝐵)) ∈ ℂ)
72	30, 71	addcld 11227	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 , 𝐴) + ((i · 𝐵) , (i · 𝐵))) ∈ ℂ)
73	7, 11	cphipcl 25329	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑋 ∧ (i · 𝐵) ∈ 𝑋) → (𝐴 , (i · 𝐵)) ∈ ℂ)
74	2, 15, 55, 73	syl3anc 1396	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 , (i · 𝐵)) ∈ ℂ)
75	7, 11	cphipcl 25329	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℂPreHil ∧ (i · 𝐵) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((i · 𝐵) , 𝐴) ∈ ℂ)
76	2, 55, 15, 75	syl3anc 1396	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((i · 𝐵) , 𝐴) ∈ ℂ)
77	74, 76	addcld 11227	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 , (i · 𝐵)) + ((i · 𝐵) , 𝐴)) ∈ ℂ)
78	72, 77, 77	pnncand 11607	. . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((((𝐴 , 𝐴) + ((i · 𝐵) , (i · 𝐵))) + ((𝐴 , (i · 𝐵)) + ((i · 𝐵) , 𝐴))) − (((𝐴 , 𝐴) + ((i · 𝐵) , (i · 𝐵))) − ((𝐴 , (i · 𝐵)) + ((i · 𝐵) , 𝐴)))) = (((𝐴 , (i · 𝐵)) + ((i · 𝐵) , 𝐴)) + ((𝐴 , (i · 𝐵)) + ((i · 𝐵) , 𝐴))))
79	78	oveq2d 7426	. . . . 5 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (i · ((((𝐴 , 𝐴) + ((i · 𝐵) , (i · 𝐵))) + ((𝐴 , (i · 𝐵)) + ((i · 𝐵) , 𝐴))) − (((𝐴 , 𝐴) + ((i · 𝐵) , (i · 𝐵))) − ((𝐴 , (i · 𝐵)) + ((i · 𝐵) , 𝐴))))) = (i · (((𝐴 , (i · 𝐵)) + ((i · 𝐵) , 𝐴)) + ((𝐴 , (i · 𝐵)) + ((i · 𝐵) , 𝐴)))))
80	7, 51, 11, 50, 52	cphassir 25353	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 , (i · 𝐵)) = (-i · (𝐴 , 𝐵)))
81	7, 51, 11, 50, 52	cphassi 25352	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((i · 𝐵) , 𝐴) = (i · (𝐵 , 𝐴)))
82	80, 81	oveq12d 7428	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 , (i · 𝐵)) + ((i · 𝐵) , 𝐴)) = ((-i · (𝐴 , 𝐵)) + (i · (𝐵 , 𝐴))))
83	82, 82	oveq12d 7428	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝐴 , (i · 𝐵)) + ((i · 𝐵) , 𝐴)) + ((𝐴 , (i · 𝐵)) + ((i · 𝐵) , 𝐴))) = (((-i · (𝐴 , 𝐵)) + (i · (𝐵 , 𝐴))) + ((-i · (𝐴 , 𝐵)) + (i · (𝐵 , 𝐴)))))
84	83	oveq2d 7426	. . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (i · (((𝐴 , (i · 𝐵)) + ((i · 𝐵) , 𝐴)) + ((𝐴 , (i · 𝐵)) + ((i · 𝐵) , 𝐴)))) = (i · (((-i · (𝐴 , 𝐵)) + (i · (𝐵 , 𝐴))) + ((-i · (𝐴 , 𝐵)) + (i · (𝐵 , 𝐴))))))
85		ax-icn 11158	. . . . . . . 8 ⊢ i ∈ ℂ
86	85	a1i 11	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → i ∈ ℂ)
87		negicn 11457	. . . . . . . . . 10 ⊢ -i ∈ ℂ
88	87	a1i 11	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → -i ∈ ℂ)
89	88, 37	mulcld 11228	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (-i · (𝐴 , 𝐵)) ∈ ℂ)
90	86, 40	mulcld 11228	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (i · (𝐵 , 𝐴)) ∈ ℂ)
91	89, 90	addcld 11227	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((-i · (𝐴 , 𝐵)) + (i · (𝐵 , 𝐴))) ∈ ℂ)
92	86, 91, 91	adddid 11232	. . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (i · (((-i · (𝐴 , 𝐵)) + (i · (𝐵 , 𝐴))) + ((-i · (𝐴 , 𝐵)) + (i · (𝐵 , 𝐴))))) = ((i · ((-i · (𝐴 , 𝐵)) + (i · (𝐵 , 𝐴)))) + (i · ((-i · (𝐴 , 𝐵)) + (i · (𝐵 , 𝐴))))))
93	86, 89, 90	adddid 11232	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (i · ((-i · (𝐴 , 𝐵)) + (i · (𝐵 , 𝐴)))) = ((i · (-i · (𝐴 , 𝐵))) + (i · (i · (𝐵 , 𝐴)))))
94	86, 88, 37	mulassd 11231	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((i · -i) · (𝐴 , 𝐵)) = (i · (-i · (𝐴 , 𝐵))))
95	85, 85	mulneg2i 11660	. . . . . . . . . . . . 13 ⊢ (i · -i) = -(i · i)
96		ixi 11842	. . . . . . . . . . . . . 14 ⊢ (i · i) = -1
97	96	negeqi 11449	. . . . . . . . . . . . 13 ⊢ -(i · i) = --1
98		negneg1e1 12206	. . . . . . . . . . . . 13 ⊢ --1 = 1
99	95, 97, 98	3eqtri 2788	. . . . . . . . . . . 12 ⊢ (i · -i) = 1
100	99	oveq1i 7420	. . . . . . . . . . 11 ⊢ ((i · -i) · (𝐴 , 𝐵)) = (1 · (𝐴 , 𝐵))
101	94, 100	eqtr3di 2811	. . . . . . . . . 10 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (i · (-i · (𝐴 , 𝐵))) = (1 · (𝐴 , 𝐵)))
102	86, 86, 40	mulassd 11231	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((i · i) · (𝐵 , 𝐴)) = (i · (i · (𝐵 , 𝐴))))
103	96	oveq1i 7420	. . . . . . . . . . 11 ⊢ ((i · i) · (𝐵 , 𝐴)) = (-1 · (𝐵 , 𝐴))
104	102, 103	eqtr3di 2811	. . . . . . . . . 10 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (i · (i · (𝐵 , 𝐴))) = (-1 · (𝐵 , 𝐴)))
105	101, 104	oveq12d 7428	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((i · (-i · (𝐴 , 𝐵))) + (i · (i · (𝐵 , 𝐴)))) = ((1 · (𝐴 , 𝐵)) + (-1 · (𝐵 , 𝐴))))
106	93, 105	eqtrd 2796	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (i · ((-i · (𝐴 , 𝐵)) + (i · (𝐵 , 𝐴)))) = ((1 · (𝐴 , 𝐵)) + (-1 · (𝐵 , 𝐴))))
107	106, 106	oveq12d 7428	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((i · ((-i · (𝐴 , 𝐵)) + (i · (𝐵 , 𝐴)))) + (i · ((-i · (𝐴 , 𝐵)) + (i · (𝐵 , 𝐴))))) = (((1 · (𝐴 , 𝐵)) + (-1 · (𝐵 , 𝐴))) + ((1 · (𝐴 , 𝐵)) + (-1 · (𝐵 , 𝐴)))))
108	37	mullidd 11226	. . . . . . . . . 10 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (1 · (𝐴 , 𝐵)) = (𝐴 , 𝐵))
109	108	oveq1d 7425	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((1 · (𝐴 , 𝐵)) + (-1 · (𝐵 , 𝐴))) = ((𝐴 , 𝐵) + (-1 · (𝐵 , 𝐴))))
110		addneg1mul 11655	. . . . . . . . . 10 ⊢ (((𝐴 , 𝐵) ∈ ℂ ∧ (𝐵 , 𝐴) ∈ ℂ) → ((𝐴 , 𝐵) + (-1 · (𝐵 , 𝐴))) = ((𝐴 , 𝐵) − (𝐵 , 𝐴)))
111	37, 40, 110	syl2anc 595	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 , 𝐵) + (-1 · (𝐵 , 𝐴))) = ((𝐴 , 𝐵) − (𝐵 , 𝐴)))
112	109, 111	eqtrd 2796	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((1 · (𝐴 , 𝐵)) + (-1 · (𝐵 , 𝐴))) = ((𝐴 , 𝐵) − (𝐵 , 𝐴)))
113	112, 112	oveq12d 7428	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((1 · (𝐴 , 𝐵)) + (-1 · (𝐵 , 𝐴))) + ((1 · (𝐴 , 𝐵)) + (-1 · (𝐵 , 𝐴)))) = (((𝐴 , 𝐵) − (𝐵 , 𝐴)) + ((𝐴 , 𝐵) − (𝐵 , 𝐴))))
114	107, 113	eqtrd 2796	. . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((i · ((-i · (𝐴 , 𝐵)) + (i · (𝐵 , 𝐴)))) + (i · ((-i · (𝐴 , 𝐵)) + (i · (𝐵 , 𝐴))))) = (((𝐴 , 𝐵) − (𝐵 , 𝐴)) + ((𝐴 , 𝐵) − (𝐵 , 𝐴))))
115	84, 92, 114	3eqtrd 2800	. . . . 5 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (i · (((𝐴 , (i · 𝐵)) + ((i · 𝐵) , 𝐴)) + ((𝐴 , (i · 𝐵)) + ((i · 𝐵) , 𝐴)))) = (((𝐴 , 𝐵) − (𝐵 , 𝐴)) + ((𝐴 , 𝐵) − (𝐵 , 𝐴))))
116	69, 79, 115	3eqtrd 2800	. . . 4 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (i · (((𝑁‘(𝐴 + (i · 𝐵)))↑2) − ((𝑁‘(𝐴 − (i · 𝐵)))↑2))) = (((𝐴 , 𝐵) − (𝐵 , 𝐴)) + ((𝐴 , 𝐵) − (𝐵 , 𝐴))))
117	43, 116	oveq12d 7428	. . 3 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((((𝑁‘(𝐴 + 𝐵))↑2) − ((𝑁‘(𝐴 − 𝐵))↑2)) + (i · (((𝑁‘(𝐴 + (i · 𝐵)))↑2) − ((𝑁‘(𝐴 − (i · 𝐵)))↑2)))) = ((((𝐴 , 𝐵) + (𝐵 , 𝐴)) + ((𝐴 , 𝐵) + (𝐵 , 𝐴))) + (((𝐴 , 𝐵) − (𝐵 , 𝐴)) + ((𝐴 , 𝐵) − (𝐵 , 𝐴)))))
118	117	oveq1d 7425	. 2 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((((𝑁‘(𝐴 + 𝐵))↑2) − ((𝑁‘(𝐴 − 𝐵))↑2)) + (i · (((𝑁‘(𝐴 + (i · 𝐵)))↑2) − ((𝑁‘(𝐴 − (i · 𝐵)))↑2)))) / 4) = (((((𝐴 , 𝐵) + (𝐵 , 𝐴)) + ((𝐴 , 𝐵) + (𝐵 , 𝐴))) + (((𝐴 , 𝐵) − (𝐵 , 𝐴)) + ((𝐴 , 𝐵) − (𝐵 , 𝐴)))) / 4))
119	37, 40	subcld 11568	. . . . 5 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 , 𝐵) − (𝐵 , 𝐴)) ∈ ℂ)
120	41, 41, 119, 119	add4d 11438	. . . 4 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((((𝐴 , 𝐵) + (𝐵 , 𝐴)) + ((𝐴 , 𝐵) + (𝐵 , 𝐴))) + (((𝐴 , 𝐵) − (𝐵 , 𝐴)) + ((𝐴 , 𝐵) − (𝐵 , 𝐴)))) = ((((𝐴 , 𝐵) + (𝐵 , 𝐴)) + ((𝐴 , 𝐵) − (𝐵 , 𝐴))) + (((𝐴 , 𝐵) + (𝐵 , 𝐴)) + ((𝐴 , 𝐵) − (𝐵 , 𝐴)))))
121	37, 40, 37	ppncand 11608	. . . . 5 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝐴 , 𝐵) + (𝐵 , 𝐴)) + ((𝐴 , 𝐵) − (𝐵 , 𝐴))) = ((𝐴 , 𝐵) + (𝐴 , 𝐵)))
122	121, 121	oveq12d 7428	. . . 4 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((((𝐴 , 𝐵) + (𝐵 , 𝐴)) + ((𝐴 , 𝐵) − (𝐵 , 𝐴))) + (((𝐴 , 𝐵) + (𝐵 , 𝐴)) + ((𝐴 , 𝐵) − (𝐵 , 𝐴)))) = (((𝐴 , 𝐵) + (𝐴 , 𝐵)) + ((𝐴 , 𝐵) + (𝐴 , 𝐵))))
123	120, 122	eqtrd 2796	. . 3 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((((𝐴 , 𝐵) + (𝐵 , 𝐴)) + ((𝐴 , 𝐵) + (𝐵 , 𝐴))) + (((𝐴 , 𝐵) − (𝐵 , 𝐴)) + ((𝐴 , 𝐵) − (𝐵 , 𝐴)))) = (((𝐴 , 𝐵) + (𝐴 , 𝐵)) + ((𝐴 , 𝐵) + (𝐴 , 𝐵))))
124	123	oveq1d 7425	. 2 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((((𝐴 , 𝐵) + (𝐵 , 𝐴)) + ((𝐴 , 𝐵) + (𝐵 , 𝐴))) + (((𝐴 , 𝐵) − (𝐵 , 𝐴)) + ((𝐴 , 𝐵) − (𝐵 , 𝐴)))) / 4) = ((((𝐴 , 𝐵) + (𝐴 , 𝐵)) + ((𝐴 , 𝐵) + (𝐴 , 𝐵))) / 4))
125	37	2timesd 12486	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (2 · (𝐴 , 𝐵)) = ((𝐴 , 𝐵) + (𝐴 , 𝐵)))
126	125	eqcomd 2767	. . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 , 𝐵) + (𝐴 , 𝐵)) = (2 · (𝐴 , 𝐵)))
127	126, 126	oveq12d 7428	. . . . 5 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝐴 , 𝐵) + (𝐴 , 𝐵)) + ((𝐴 , 𝐵) + (𝐴 , 𝐵))) = ((2 · (𝐴 , 𝐵)) + (2 · (𝐴 , 𝐵))))
128		2cnd 12318	. . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 2 ∈ ℂ)
129	128, 128, 37	adddird 11233	. . . . 5 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((2 + 2) · (𝐴 , 𝐵)) = ((2 · (𝐴 , 𝐵)) + (2 · (𝐴 , 𝐵))))
130		2p2e4 12374	. . . . . . 7 ⊢ (2 + 2) = 4
131	130	a1i 11	. . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (2 + 2) = 4)
132	131	oveq1d 7425	. . . . 5 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((2 + 2) · (𝐴 , 𝐵)) = (4 · (𝐴 , 𝐵)))
133	127, 129, 132	3eqtr2d 2802	. . . 4 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝐴 , 𝐵) + (𝐴 , 𝐵)) + ((𝐴 , 𝐵) + (𝐴 , 𝐵))) = (4 · (𝐴 , 𝐵)))
134	133	oveq1d 7425	. . 3 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((((𝐴 , 𝐵) + (𝐴 , 𝐵)) + ((𝐴 , 𝐵) + (𝐴 , 𝐵))) / 4) = ((4 · (𝐴 , 𝐵)) / 4))
135		4cn 12325	. . . . 5 ⊢ 4 ∈ ℂ
136	135	a1i 11	. . . 4 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 4 ∈ ℂ)
137		4ne0 12351	. . . . 5 ⊢ 4 ≠ 0
138	137	a1i 11	. . . 4 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 4 ≠ 0)
139	37, 136, 138	divcan3d 11995	. . 3 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((4 · (𝐴 , 𝐵)) / 4) = (𝐴 , 𝐵))
140	134, 139	eqtrd 2796	. 2 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((((𝐴 , 𝐵) + (𝐴 , 𝐵)) + ((𝐴 , 𝐵) + (𝐴 , 𝐵))) / 4) = (𝐴 , 𝐵))
141	118, 124, 140	3eqtrrd 2801	1 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 , 𝐵) = (((((𝑁‘(𝐴 + 𝐵))↑2) − ((𝑁‘(𝐴 − 𝐵))↑2)) + (i · (((𝑁‘(𝐴 + (i · 𝐵)))↑2) − ((𝑁‘(𝐴 − (i · 𝐵)))↑2)))) / 4))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1101 = wceq 1568 ∈ wcel 2141 ≠ wne 2956 ‘cfv 6536 (class class class)co 7410 ℂcc 11097 ℝcr 11098 0cc0 11099 1c1 11100 ici 11101 + caddc 11102 · cmul 11104 − cmin 11440 -cneg 11441 / cdiv 11870 2c2 12294 4c4 12296 ↑cexp 14097 Basecbs 17268 +_gcplusg 17309 Scalarcsca 17312 ·_𝑠 cvsca 17313 ·_𝑖cip 17314 Grpcgrp 18999 -_gcsg 19001 LModclmod 20960 normcnm 24712 NrmGrpcngp 24713 ℂPreHilccph 25304

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176 ax-pre-sup 11177 ax-addf 11178 ax-mulf 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7862 df-1st 7985 df-2nd 7986 df-tpos 8221 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-er 8693 df-map 8825 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-sup 9401 df-inf 9402 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-div 11871 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-5 12305 df-6 12306 df-7 12307 df-8 12308 df-9 12309 df-n0 12504 df-z 12591 df-dec 12711 df-uz 12862 df-q 12972 df-rp 13016 df-xneg 13136 df-xadd 13137 df-xmul 13138 df-fz 13535 df-seq 14038 df-exp 14098 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-struct 17206 df-sets 17223 df-slot 17241 df-ndx 17253 df-base 17269 df-ress 17290 df-plusg 17322 df-mulr 17323 df-starv 17324 df-sca 17325 df-vsca 17326 df-ip 17327 df-tset 17328 df-ple 17329 df-ds 17331 df-unif 17332 df-0g 17493 df-topgen 17495 df-mgm 18697 df-sgrp 18776 df-mnd 18792 df-mhm 18840 df-grp 19002 df-minusg 19003 df-sbg 19004 df-subg 19188 df-ghm 19283 df-cmn 19851 df-abl 19852 df-mgp 20216 df-rng 20230 df-ur 20263 df-ring 20316 df-cring 20317 df-oppr 20418 df-dvdsr 20438 df-unit 20439 df-rhm 20553 df-subrg 20654 df-drng 20814 df-staf 20921 df-srng 20922 df-lmod 20962 df-lmhm 21122 df-lvec 21203 df-sra 21273 df-rgmod 21274 df-psmet 21493 df-xmet 21494 df-met 21495 df-bl 21496 df-mopn 21497 df-cnfld 21502 df-phl 21755 df-top 23030 df-topon 23047 df-topsp 23069 df-bases 23082 df-xms 24456 df-ms 24457 df-nm 24718 df-ngp 24719 df-nlm 24722 df-clm 25201 df-cph 25306

This theorem is referenced by: 4cphipval2 25380 cphipval 25381

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