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Theorem cphipval2 24405

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 483	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) → 𝑊 ∈ ℂPreHil)
2	1	3ad2ant1 1132	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝑊 ∈ ℂPreHil)
3		cphngp 24337	. . . . . . . . . . 11 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp)
4	3	adantr 481	. . . . . . . . . 10 ⊢ ((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) → 𝑊 ∈ NrmGrp)
5		ngpgrp 23755	. . . . . . . . . 10 ⊢ (𝑊 ∈ NrmGrp → 𝑊 ∈ Grp)
6	4, 5	syl 17	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) → 𝑊 ∈ Grp)
7		cphipfval.x	. . . . . . . . . 10 ⊢ 𝑋 = (Base‘𝑊)
8		cphipfval.p	. . . . . . . . . 10 ⊢ + = (+_g‘𝑊)
9	7, 8	grpcl 18585	. . . . . . . . 9 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 + 𝐵) ∈ 𝑋)
10	6, 9	syl3an1 1162	. . . . . . . 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 24358	. . . . . . . 8 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 + 𝐵) ∈ 𝑋) → ((𝑁‘(𝐴 + 𝐵))↑2) = ((𝐴 + 𝐵) , (𝐴 + 𝐵)))
14	2, 10, 13	syl2anc 584	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴 + 𝐵))↑2) = ((𝐴 + 𝐵) , (𝐴 + 𝐵)))
15		simp2 1136	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ 𝑋)
16		simp3 1137	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐵 ∈ 𝑋)
17	11, 7, 8, 2, 15, 16, 15, 16	cph2di 24371	. . . . . . 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 2778	. . . . . 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 18655	. . . . . . . . 9 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 − 𝐵) ∈ 𝑋)
21	6, 20	syl3an1 1162	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 − 𝐵) ∈ 𝑋)
22	7, 11, 12	nmsq 24358	. . . . . . . 8 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 − 𝐵) ∈ 𝑋) → ((𝑁‘(𝐴 − 𝐵))↑2) = ((𝐴 − 𝐵) , (𝐴 − 𝐵)))
23	2, 21, 22	syl2anc 584	. . . . . . 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 24374	. . . . . . 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 2778	. . . . . 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 7293	. . . . 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 24361	. . . . . . . . . 10 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑋) → (𝐴 , 𝐴) ∈ ℝ)
28	27	adantlr 712	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋) → (𝐴 , 𝐴) ∈ ℝ)
29	28	recnd 11003	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋) → (𝐴 , 𝐴) ∈ ℂ)
30	29	3adant3 1131	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 , 𝐴) ∈ ℂ)
31	7, 11	reipcl 24361	. . . . . . . . . 10 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐵 ∈ 𝑋) → (𝐵 , 𝐵) ∈ ℝ)
32	31	adantlr 712	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐵 ∈ 𝑋) → (𝐵 , 𝐵) ∈ ℝ)
33	32	recnd 11003	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐵 ∈ 𝑋) → (𝐵 , 𝐵) ∈ ℂ)
34	33	3adant2 1130	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵 , 𝐵) ∈ ℂ)
35	30, 34	addcld 10994	. . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 , 𝐴) + (𝐵 , 𝐵)) ∈ ℂ)
36	7, 11	cphipcl 24355	. . . . . . . 8 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 , 𝐵) ∈ ℂ)
37	1, 36	syl3an1 1162	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 , 𝐵) ∈ ℂ)
38	7, 11	cphipcl 24355	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐵 , 𝐴) ∈ ℂ)
39	1, 38	syl3an1 1162	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐵 , 𝐴) ∈ ℂ)
40	39	3com23 1125	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵 , 𝐴) ∈ ℂ)
41	37, 40	addcld 10994	. . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 , 𝐵) + (𝐵 , 𝐴)) ∈ ℂ)
42	35, 41, 41	pnncand 11371	. . . . 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 2778	. . . 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 1132	. . . . . . . . . 10 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝑊 ∈ Grp)
45		cphlmod 24338	. . . . . . . . . . . . . 14 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LMod)
46	45	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) → 𝑊 ∈ LMod)
47	46	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐵 ∈ 𝑋) → 𝑊 ∈ LMod)
48		simplr 766	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐵 ∈ 𝑋) → i ∈ 𝐾)
49		simpr 485	. . . . . . . . . . . 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 20140	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ i ∈ 𝐾 ∧ 𝐵 ∈ 𝑋) → (i · 𝐵) ∈ 𝑋)
54	47, 48, 49, 53	syl3anc 1370	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐵 ∈ 𝑋) → (i · 𝐵) ∈ 𝑋)
55	54	3adant2 1130	. . . . . . . . . 10 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (i · 𝐵) ∈ 𝑋)
56	7, 8	grpcl 18585	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (i · 𝐵) ∈ 𝑋) → (𝐴 + (i · 𝐵)) ∈ 𝑋)
57	44, 15, 55, 56	syl3anc 1370	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 + (i · 𝐵)) ∈ 𝑋)
58	7, 11, 12	nmsq 24358	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 + (i · 𝐵)) ∈ 𝑋) → ((𝑁‘(𝐴 + (i · 𝐵)))↑2) = ((𝐴 + (i · 𝐵)) , (𝐴 + (i · 𝐵))))
59	2, 57, 58	syl2anc 584	. . . . . . . 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 24371	. . . . . . . 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 2778	. . . . . . 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 18655	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (i · 𝐵) ∈ 𝑋) → (𝐴 − (i · 𝐵)) ∈ 𝑋)
63	44, 15, 55, 62	syl3anc 1370	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 − (i · 𝐵)) ∈ 𝑋)
64	7, 11, 12	nmsq 24358	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 − (i · 𝐵)) ∈ 𝑋) → ((𝑁‘(𝐴 − (i · 𝐵)))↑2) = ((𝐴 − (i · 𝐵)) , (𝐴 − (i · 𝐵))))
65	2, 63, 64	syl2anc 584	. . . . . . . 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 24374	. . . . . . . 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 2778	. . . . . . 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 7293	. . . . . 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 7291	. . . . 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 24355	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℂPreHil ∧ (i · 𝐵) ∈ 𝑋 ∧ (i · 𝐵) ∈ 𝑋) → ((i · 𝐵) , (i · 𝐵)) ∈ ℂ)
71	2, 55, 55, 70	syl3anc 1370	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((i · 𝐵) , (i · 𝐵)) ∈ ℂ)
72	30, 71	addcld 10994	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 , 𝐴) + ((i · 𝐵) , (i · 𝐵))) ∈ ℂ)
73	7, 11	cphipcl 24355	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑋 ∧ (i · 𝐵) ∈ 𝑋) → (𝐴 , (i · 𝐵)) ∈ ℂ)
74	2, 15, 55, 73	syl3anc 1370	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 , (i · 𝐵)) ∈ ℂ)
75	7, 11	cphipcl 24355	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℂPreHil ∧ (i · 𝐵) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((i · 𝐵) , 𝐴) ∈ ℂ)
76	2, 55, 15, 75	syl3anc 1370	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((i · 𝐵) , 𝐴) ∈ ℂ)
77	74, 76	addcld 10994	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 , (i · 𝐵)) + ((i · 𝐵) , 𝐴)) ∈ ℂ)
78	72, 77, 77	pnncand 11371	. . . . . 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 7291	. . . . 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 24379	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 , (i · 𝐵)) = (-i · (𝐴 , 𝐵)))
81	7, 51, 11, 50, 52	cphassi 24378	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((i · 𝐵) , 𝐴) = (i · (𝐵 , 𝐴)))
82	80, 81	oveq12d 7293	. . . . . . . 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 7293	. . . . . . 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 7291	. . . . . 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 10930	. . . . . . . 8 ⊢ i ∈ ℂ
86	85	a1i 11	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → i ∈ ℂ)
87		negicn 11222	. . . . . . . . . 10 ⊢ -i ∈ ℂ
88	87	a1i 11	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → -i ∈ ℂ)
89	88, 37	mulcld 10995	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (-i · (𝐴 , 𝐵)) ∈ ℂ)
90	86, 40	mulcld 10995	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (i · (𝐵 , 𝐴)) ∈ ℂ)
91	89, 90	addcld 10994	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((-i · (𝐴 , 𝐵)) + (i · (𝐵 , 𝐴))) ∈ ℂ)
92	86, 91, 91	adddid 10999	. . . . . 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 10999	. . . . . . . . 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 10998	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((i · -i) · (𝐴 , 𝐵)) = (i · (-i · (𝐴 , 𝐵))))
95	85, 85	mulneg2i 11422	. . . . . . . . . . . . 13 ⊢ (i · -i) = -(i · i)
96		ixi 11604	. . . . . . . . . . . . . 14 ⊢ (i · i) = -1
97	96	negeqi 11214	. . . . . . . . . . . . 13 ⊢ -(i · i) = --1
98		negneg1e1 12091	. . . . . . . . . . . . 13 ⊢ --1 = 1
99	95, 97, 98	3eqtri 2770	. . . . . . . . . . . 12 ⊢ (i · -i) = 1
100	99	oveq1i 7285	. . . . . . . . . . 11 ⊢ ((i · -i) · (𝐴 , 𝐵)) = (1 · (𝐴 , 𝐵))
101	94, 100	eqtr3di 2793	. . . . . . . . . 10 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (i · (-i · (𝐴 , 𝐵))) = (1 · (𝐴 , 𝐵)))
102	86, 86, 40	mulassd 10998	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((i · i) · (𝐵 , 𝐴)) = (i · (i · (𝐵 , 𝐴))))
103	96	oveq1i 7285	. . . . . . . . . . 11 ⊢ ((i · i) · (𝐵 , 𝐴)) = (-1 · (𝐵 , 𝐴))
104	102, 103	eqtr3di 2793	. . . . . . . . . 10 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (i · (i · (𝐵 , 𝐴))) = (-1 · (𝐵 , 𝐴)))
105	101, 104	oveq12d 7293	. . . . . . . . 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 2778	. . . . . . . 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 7293	. . . . . . 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	mulid2d 10993	. . . . . . . . . 10 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (1 · (𝐴 , 𝐵)) = (𝐴 , 𝐵))
109	108	oveq1d 7290	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((1 · (𝐴 , 𝐵)) + (-1 · (𝐵 , 𝐴))) = ((𝐴 , 𝐵) + (-1 · (𝐵 , 𝐴))))
110		addneg1mul 11417	. . . . . . . . . 10 ⊢ (((𝐴 , 𝐵) ∈ ℂ ∧ (𝐵 , 𝐴) ∈ ℂ) → ((𝐴 , 𝐵) + (-1 · (𝐵 , 𝐴))) = ((𝐴 , 𝐵) − (𝐵 , 𝐴)))
111	37, 40, 110	syl2anc 584	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 , 𝐵) + (-1 · (𝐵 , 𝐴))) = ((𝐴 , 𝐵) − (𝐵 , 𝐴)))
112	109, 111	eqtrd 2778	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((1 · (𝐴 , 𝐵)) + (-1 · (𝐵 , 𝐴))) = ((𝐴 , 𝐵) − (𝐵 , 𝐴)))
113	112, 112	oveq12d 7293	. . . . . . 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 2778	. . . . . 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 2782	. . . . 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 2782	. . . 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 7293	. . 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 7290	. 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 11332	. . . . 5 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 , 𝐵) − (𝐵 , 𝐴)) ∈ ℂ)
120	41, 41, 119, 119	add4d 11203	. . . 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 11372	. . . . 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 7293	. . . 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 2778	. . 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 7290	. 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 12216	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (2 · (𝐴 , 𝐵)) = ((𝐴 , 𝐵) + (𝐴 , 𝐵)))
126	125	eqcomd 2744	. . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 , 𝐵) + (𝐴 , 𝐵)) = (2 · (𝐴 , 𝐵)))
127	126, 126	oveq12d 7293	. . . . 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 12051	. . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 2 ∈ ℂ)
129	128, 128, 37	adddird 11000	. . . . 5 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((2 + 2) · (𝐴 , 𝐵)) = ((2 · (𝐴 , 𝐵)) + (2 · (𝐴 , 𝐵))))
130		2p2e4 12108	. . . . . . 7 ⊢ (2 + 2) = 4
131	130	a1i 11	. . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (2 + 2) = 4)
132	131	oveq1d 7290	. . . . 5 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((2 + 2) · (𝐴 , 𝐵)) = (4 · (𝐴 , 𝐵)))
133	127, 129, 132	3eqtr2d 2784	. . . 4 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝐴 , 𝐵) + (𝐴 , 𝐵)) + ((𝐴 , 𝐵) + (𝐴 , 𝐵))) = (4 · (𝐴 , 𝐵)))
134	133	oveq1d 7290	. . 3 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((((𝐴 , 𝐵) + (𝐴 , 𝐵)) + ((𝐴 , 𝐵) + (𝐴 , 𝐵))) / 4) = ((4 · (𝐴 , 𝐵)) / 4))
135		4cn 12058	. . . . 5 ⊢ 4 ∈ ℂ
136	135	a1i 11	. . . 4 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 4 ∈ ℂ)
137		4ne0 12081	. . . . 5 ⊢ 4 ≠ 0
138	137	a1i 11	. . . 4 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 4 ≠ 0)
139	37, 136, 138	divcan3d 11756	. . 3 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((4 · (𝐴 , 𝐵)) / 4) = (𝐴 , 𝐵))
140	134, 139	eqtrd 2778	. 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 2783	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 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 ici 10873 + caddc 10874 · cmul 10876 − cmin 11205 -cneg 11206 / cdiv 11632 2c2 12028 4c4 12030 ↑cexp 13782 Basecbs 16912 +_gcplusg 16962 Scalarcsca 16965 ·_𝑠 cvsca 16966 ·_𝑖cip 16967 Grpcgrp 18577 -_gcsg 18579 LModclmod 20123 normcnm 23732 NrmGrpcngp 23733 ℂPreHilccph 24330

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8042 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-inf 9202 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-fz 13240 df-seq 13722 df-exp 13783 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-0g 17152 df-topgen 17154 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-grp 18580 df-minusg 18581 df-sbg 18582 df-subg 18752 df-ghm 18832 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-cring 19786 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-rnghom 19959 df-drng 19993 df-subrg 20022 df-staf 20105 df-srng 20106 df-lmod 20125 df-lmhm 20284 df-lvec 20365 df-sra 20434 df-rgmod 20435 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-cnfld 20598 df-phl 20831 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-xms 23473 df-ms 23474 df-nm 23738 df-ngp 23739 df-nlm 23742 df-clm 24226 df-cph 24332

This theorem is referenced by: 4cphipval2 24406 cphipval 24407

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