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Theorem cphipval2 24687

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 1133	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝑊 ∈ ℂPreHil)
3		cphngp 24619	. . . . . . . . . . 11 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp)
4	3	adantr 481	. . . . . . . . . 10 ⊢ ((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) → 𝑊 ∈ NrmGrp)
5		ngpgrp 24037	. . . . . . . . . 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 18802	. . . . . . . . 9 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 + 𝐵) ∈ 𝑋)
10	6, 9	syl3an1 1163	. . . . . . . 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 24640	. . . . . . . 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 1137	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ 𝑋)
16		simp3 1138	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐵 ∈ 𝑋)
17	11, 7, 8, 2, 15, 16, 15, 16	cph2di 24653	. . . . . . 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 2771	. . . . . 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 18877	. . . . . . . . 9 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 − 𝐵) ∈ 𝑋)
21	6, 20	syl3an1 1163	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 − 𝐵) ∈ 𝑋)
22	7, 11, 12	nmsq 24640	. . . . . . . 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 24656	. . . . . . 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 2771	. . . . . 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 7411	. . . . 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 24643	. . . . . . . . . 10 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑋) → (𝐴 , 𝐴) ∈ ℝ)
28	27	adantlr 713	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋) → (𝐴 , 𝐴) ∈ ℝ)
29	28	recnd 11224	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋) → (𝐴 , 𝐴) ∈ ℂ)
30	29	3adant3 1132	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 , 𝐴) ∈ ℂ)
31	7, 11	reipcl 24643	. . . . . . . . . 10 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐵 ∈ 𝑋) → (𝐵 , 𝐵) ∈ ℝ)
32	31	adantlr 713	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐵 ∈ 𝑋) → (𝐵 , 𝐵) ∈ ℝ)
33	32	recnd 11224	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐵 ∈ 𝑋) → (𝐵 , 𝐵) ∈ ℂ)
34	33	3adant2 1131	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵 , 𝐵) ∈ ℂ)
35	30, 34	addcld 11215	. . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 , 𝐴) + (𝐵 , 𝐵)) ∈ ℂ)
36	7, 11	cphipcl 24637	. . . . . . . 8 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 , 𝐵) ∈ ℂ)
37	1, 36	syl3an1 1163	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 , 𝐵) ∈ ℂ)
38	7, 11	cphipcl 24637	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐵 , 𝐴) ∈ ℂ)
39	1, 38	syl3an1 1163	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐵 , 𝐴) ∈ ℂ)
40	39	3com23 1126	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵 , 𝐴) ∈ ℂ)
41	37, 40	addcld 11215	. . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 , 𝐵) + (𝐵 , 𝐴)) ∈ ℂ)
42	35, 41, 41	pnncand 11592	. . . . 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 2771	. . . 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 1133	. . . . . . . . . 10 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝑊 ∈ Grp)
45		cphlmod 24620	. . . . . . . . . . . . . 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 767	. . . . . . . . . . . 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 20438	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ i ∈ 𝐾 ∧ 𝐵 ∈ 𝑋) → (i · 𝐵) ∈ 𝑋)
54	47, 48, 49, 53	syl3anc 1371	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐵 ∈ 𝑋) → (i · 𝐵) ∈ 𝑋)
55	54	3adant2 1131	. . . . . . . . . 10 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (i · 𝐵) ∈ 𝑋)
56	7, 8	grpcl 18802	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (i · 𝐵) ∈ 𝑋) → (𝐴 + (i · 𝐵)) ∈ 𝑋)
57	44, 15, 55, 56	syl3anc 1371	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 + (i · 𝐵)) ∈ 𝑋)
58	7, 11, 12	nmsq 24640	. . . . . . . . 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 24653	. . . . . . . 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 2771	. . . . . . 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 18877	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (i · 𝐵) ∈ 𝑋) → (𝐴 − (i · 𝐵)) ∈ 𝑋)
63	44, 15, 55, 62	syl3anc 1371	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 − (i · 𝐵)) ∈ 𝑋)
64	7, 11, 12	nmsq 24640	. . . . . . . . 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 24656	. . . . . . . 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 2771	. . . . . . 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 7411	. . . . . 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 7409	. . . . 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 24637	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℂPreHil ∧ (i · 𝐵) ∈ 𝑋 ∧ (i · 𝐵) ∈ 𝑋) → ((i · 𝐵) , (i · 𝐵)) ∈ ℂ)
71	2, 55, 55, 70	syl3anc 1371	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((i · 𝐵) , (i · 𝐵)) ∈ ℂ)
72	30, 71	addcld 11215	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 , 𝐴) + ((i · 𝐵) , (i · 𝐵))) ∈ ℂ)
73	7, 11	cphipcl 24637	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑋 ∧ (i · 𝐵) ∈ 𝑋) → (𝐴 , (i · 𝐵)) ∈ ℂ)
74	2, 15, 55, 73	syl3anc 1371	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 , (i · 𝐵)) ∈ ℂ)
75	7, 11	cphipcl 24637	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℂPreHil ∧ (i · 𝐵) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((i · 𝐵) , 𝐴) ∈ ℂ)
76	2, 55, 15, 75	syl3anc 1371	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((i · 𝐵) , 𝐴) ∈ ℂ)
77	74, 76	addcld 11215	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 , (i · 𝐵)) + ((i · 𝐵) , 𝐴)) ∈ ℂ)
78	72, 77, 77	pnncand 11592	. . . . . 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 7409	. . . . 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 24661	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 , (i · 𝐵)) = (-i · (𝐴 , 𝐵)))
81	7, 51, 11, 50, 52	cphassi 24660	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((i · 𝐵) , 𝐴) = (i · (𝐵 , 𝐴)))
82	80, 81	oveq12d 7411	. . . . . . . 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 7411	. . . . . . 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 7409	. . . . . 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 11151	. . . . . . . 8 ⊢ i ∈ ℂ
86	85	a1i 11	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → i ∈ ℂ)
87		negicn 11443	. . . . . . . . . 10 ⊢ -i ∈ ℂ
88	87	a1i 11	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → -i ∈ ℂ)
89	88, 37	mulcld 11216	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (-i · (𝐴 , 𝐵)) ∈ ℂ)
90	86, 40	mulcld 11216	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (i · (𝐵 , 𝐴)) ∈ ℂ)
91	89, 90	addcld 11215	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((-i · (𝐴 , 𝐵)) + (i · (𝐵 , 𝐴))) ∈ ℂ)
92	86, 91, 91	adddid 11220	. . . . . 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 11220	. . . . . . . . 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 11219	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((i · -i) · (𝐴 , 𝐵)) = (i · (-i · (𝐴 , 𝐵))))
95	85, 85	mulneg2i 11643	. . . . . . . . . . . . 13 ⊢ (i · -i) = -(i · i)
96		ixi 11825	. . . . . . . . . . . . . 14 ⊢ (i · i) = -1
97	96	negeqi 11435	. . . . . . . . . . . . 13 ⊢ -(i · i) = --1
98		negneg1e1 12312	. . . . . . . . . . . . 13 ⊢ --1 = 1
99	95, 97, 98	3eqtri 2763	. . . . . . . . . . . 12 ⊢ (i · -i) = 1
100	99	oveq1i 7403	. . . . . . . . . . 11 ⊢ ((i · -i) · (𝐴 , 𝐵)) = (1 · (𝐴 , 𝐵))
101	94, 100	eqtr3di 2786	. . . . . . . . . 10 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (i · (-i · (𝐴 , 𝐵))) = (1 · (𝐴 , 𝐵)))
102	86, 86, 40	mulassd 11219	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((i · i) · (𝐵 , 𝐴)) = (i · (i · (𝐵 , 𝐴))))
103	96	oveq1i 7403	. . . . . . . . . . 11 ⊢ ((i · i) · (𝐵 , 𝐴)) = (-1 · (𝐵 , 𝐴))
104	102, 103	eqtr3di 2786	. . . . . . . . . 10 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (i · (i · (𝐵 , 𝐴))) = (-1 · (𝐵 , 𝐴)))
105	101, 104	oveq12d 7411	. . . . . . . . 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 2771	. . . . . . . 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 7411	. . . . . . 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 11214	. . . . . . . . . 10 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (1 · (𝐴 , 𝐵)) = (𝐴 , 𝐵))
109	108	oveq1d 7408	. . . . . . . . 9 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((1 · (𝐴 , 𝐵)) + (-1 · (𝐵 , 𝐴))) = ((𝐴 , 𝐵) + (-1 · (𝐵 , 𝐴))))
110		addneg1mul 11638	. . . . . . . . . 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 2771	. . . . . . . 8 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((1 · (𝐴 , 𝐵)) + (-1 · (𝐵 , 𝐴))) = ((𝐴 , 𝐵) − (𝐵 , 𝐴)))
113	112, 112	oveq12d 7411	. . . . . . 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 2771	. . . . . 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 2775	. . . . 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 2775	. . . 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 7411	. . 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 7408	. 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 11553	. . . . 5 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 , 𝐵) − (𝐵 , 𝐴)) ∈ ℂ)
120	41, 41, 119, 119	add4d 11424	. . . 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 11593	. . . . 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 7411	. . . 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 2771	. . 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 7408	. 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 12437	. . . . . . 7 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (2 · (𝐴 , 𝐵)) = ((𝐴 , 𝐵) + (𝐴 , 𝐵)))
126	125	eqcomd 2737	. . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 , 𝐵) + (𝐴 , 𝐵)) = (2 · (𝐴 , 𝐵)))
127	126, 126	oveq12d 7411	. . . . 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 12272	. . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 2 ∈ ℂ)
129	128, 128, 37	adddird 11221	. . . . 5 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((2 + 2) · (𝐴 , 𝐵)) = ((2 · (𝐴 , 𝐵)) + (2 · (𝐴 , 𝐵))))
130		2p2e4 12329	. . . . . . 7 ⊢ (2 + 2) = 4
131	130	a1i 11	. . . . . 6 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (2 + 2) = 4)
132	131	oveq1d 7408	. . . . 5 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((2 + 2) · (𝐴 , 𝐵)) = (4 · (𝐴 , 𝐵)))
133	127, 129, 132	3eqtr2d 2777	. . . 4 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝐴 , 𝐵) + (𝐴 , 𝐵)) + ((𝐴 , 𝐵) + (𝐴 , 𝐵))) = (4 · (𝐴 , 𝐵)))
134	133	oveq1d 7408	. . 3 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((((𝐴 , 𝐵) + (𝐴 , 𝐵)) + ((𝐴 , 𝐵) + (𝐴 , 𝐵))) / 4) = ((4 · (𝐴 , 𝐵)) / 4))
135		4cn 12279	. . . . 5 ⊢ 4 ∈ ℂ
136	135	a1i 11	. . . 4 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 4 ∈ ℂ)
137		4ne0 12302	. . . . 5 ⊢ 4 ≠ 0
138	137	a1i 11	. . . 4 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 4 ≠ 0)
139	37, 136, 138	divcan3d 11977	. . 3 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((4 · (𝐴 , 𝐵)) / 4) = (𝐴 , 𝐵))
140	134, 139	eqtrd 2771	. 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 2776	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 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 ‘cfv 6532 (class class class)co 7393 ℂcc 11090 ℝcr 11091 0cc0 11092 1c1 11093 ici 11094 + caddc 11095 · cmul 11097 − cmin 11426 -cneg 11427 / cdiv 11853 2c2 12249 4c4 12251 ↑cexp 14009 Basecbs 17126 +_gcplusg 17179 Scalarcsca 17182 ·_𝑠 cvsca 17183 ·_𝑖cip 17184 Grpcgrp 18794 -_gcsg 18796 LModclmod 20420 normcnm 24014 NrmGrpcngp 24015 ℂPreHilccph 24612

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-pre-sup 11170 ax-addf 11171 ax-mulf 11172

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-tpos 8193 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-map 8805 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-sup 9419 df-inf 9420 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-div 11854 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-7 12262 df-8 12263 df-9 12264 df-n0 12455 df-z 12541 df-dec 12660 df-uz 12805 df-q 12915 df-rp 12957 df-xneg 13074 df-xadd 13075 df-xmul 13076 df-fz 13467 df-seq 13949 df-exp 14010 df-cj 15028 df-re 15029 df-im 15030 df-sqrt 15164 df-abs 15165 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-ress 17156 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-0g 17369 df-topgen 17371 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-mhm 18647 df-grp 18797 df-minusg 18798 df-sbg 18799 df-subg 18975 df-ghm 19056 df-cmn 19614 df-abl 19615 df-mgp 19947 df-ur 19964 df-ring 20016 df-cring 20017 df-oppr 20102 df-dvdsr 20123 df-unit 20124 df-rnghom 20201 df-drng 20267 df-subrg 20310 df-staf 20402 df-srng 20403 df-lmod 20422 df-lmhm 20582 df-lvec 20663 df-sra 20734 df-rgmod 20735 df-psmet 20870 df-xmet 20871 df-met 20872 df-bl 20873 df-mopn 20874 df-cnfld 20879 df-phl 21112 df-top 22325 df-topon 22342 df-topsp 22364 df-bases 22378 df-xms 23755 df-ms 23756 df-nm 24020 df-ngp 24021 df-nlm 24024 df-clm 24508 df-cph 24614

This theorem is referenced by: 4cphipval2 24688 cphipval 24689

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