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Theorem ipidsq 30729

Description: The inner product of a vector with itself is the square of the vector's norm. Equation I4 of [Ponnusamy] p. 362. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
ipid.1	⊢ 𝑋 = (BaseSet‘𝑈)
ipid.6	⊢ 𝑁 = (norm_CV‘𝑈)
ipid.7	⊢ 𝑃 = (·_𝑖OLD‘𝑈)

**Assertion**
Ref	Expression
ipidsq	⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝑃𝐴) = ((𝑁‘𝐴)↑2))

**Proof of Theorem ipidsq**
Step	Hyp	Ref	Expression
1		ipid.1	. . . 4 ⊢ 𝑋 = (BaseSet‘𝑈)
2		eqid 2737	. . . 4 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
3		eqid 2737	. . . 4 ⊢ ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘𝑈)
4		ipid.6	. . . 4 ⊢ 𝑁 = (norm_CV‘𝑈)
5		ipid.7	. . . 4 ⊢ 𝑃 = (·_𝑖OLD‘𝑈)
6	1, 2, 3, 4, 5	ipval2 30726	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴𝑃𝐴) = (((((𝑁‘(𝐴( +_𝑣 ‘𝑈)𝐴))↑2) − ((𝑁‘(𝐴( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝐴)))↑2)) + (i · (((𝑁‘(𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)))↑2) − ((𝑁‘(𝐴( +_𝑣 ‘𝑈)(-i( ·_𝑠OLD ‘𝑈)𝐴)))↑2)))) / 4))
7	6	3anidm23 1423	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝑃𝐴) = (((((𝑁‘(𝐴( +_𝑣 ‘𝑈)𝐴))↑2) − ((𝑁‘(𝐴( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝐴)))↑2)) + (i · (((𝑁‘(𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)))↑2) − ((𝑁‘(𝐴( +_𝑣 ‘𝑈)(-i( ·_𝑠OLD ‘𝑈)𝐴)))↑2)))) / 4))
8	1, 2, 3	nv2 30651	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴( +_𝑣 ‘𝑈)𝐴) = (2( ·_𝑠OLD ‘𝑈)𝐴))
9	8	fveq2d 6910	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝐴( +_𝑣 ‘𝑈)𝐴)) = (𝑁‘(2( ·_𝑠OLD ‘𝑈)𝐴)))
10		2re 12340	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
11		0le2 12368	. . . . . . . . . . . 12 ⊢ 0 ≤ 2
12	10, 11	pm3.2i 470	. . . . . . . . . . 11 ⊢ (2 ∈ ℝ ∧ 0 ≤ 2)
13	1, 3, 4	nvsge0 30683	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ (2 ∈ ℝ ∧ 0 ≤ 2) ∧ 𝐴 ∈ 𝑋) → (𝑁‘(2( ·_𝑠OLD ‘𝑈)𝐴)) = (2 · (𝑁‘𝐴)))
14	12, 13	mp3an2 1451	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(2( ·_𝑠OLD ‘𝑈)𝐴)) = (2 · (𝑁‘𝐴)))
15	9, 14	eqtrd 2777	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝐴( +_𝑣 ‘𝑈)𝐴)) = (2 · (𝑁‘𝐴)))
16	15	oveq1d 7446	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑁‘(𝐴( +_𝑣 ‘𝑈)𝐴))↑2) = ((2 · (𝑁‘𝐴))↑2))
17	1, 4	nvcl 30680	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℝ)
18	17	recnd 11289	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℂ)
19		2cn 12341	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
20		2nn0 12543	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
21		mulexp 14142	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ (𝑁‘𝐴) ∈ ℂ ∧ 2 ∈ ℕ₀) → ((2 · (𝑁‘𝐴))↑2) = ((2↑2) · ((𝑁‘𝐴)↑2)))
22	19, 20, 21	mp3an13 1454	. . . . . . . . . 10 ⊢ ((𝑁‘𝐴) ∈ ℂ → ((2 · (𝑁‘𝐴))↑2) = ((2↑2) · ((𝑁‘𝐴)↑2)))
23	18, 22	syl 17	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((2 · (𝑁‘𝐴))↑2) = ((2↑2) · ((𝑁‘𝐴)↑2)))
24		sq2 14236	. . . . . . . . . 10 ⊢ (2↑2) = 4
25	24	oveq1i 7441	. . . . . . . . 9 ⊢ ((2↑2) · ((𝑁‘𝐴)↑2)) = (4 · ((𝑁‘𝐴)↑2))
26	23, 25	eqtrdi 2793	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((2 · (𝑁‘𝐴))↑2) = (4 · ((𝑁‘𝐴)↑2)))
27	16, 26	eqtrd 2777	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑁‘(𝐴( +_𝑣 ‘𝑈)𝐴))↑2) = (4 · ((𝑁‘𝐴)↑2)))
28		eqid 2737	. . . . . . . . . . 11 ⊢ (0_vec‘𝑈) = (0_vec‘𝑈)
29	1, 2, 3, 28	nvrinv 30670	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝐴)) = (0_vec‘𝑈))
30	29	fveq2d 6910	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝐴( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝐴))) = (𝑁‘(0_vec‘𝑈)))
31	28, 4	nvz0 30687	. . . . . . . . . 10 ⊢ (𝑈 ∈ NrmCVec → (𝑁‘(0_vec‘𝑈)) = 0)
32	31	adantr 480	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(0_vec‘𝑈)) = 0)
33	30, 32	eqtrd 2777	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝐴( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝐴))) = 0)
34	33	sq0id 14233	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑁‘(𝐴( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝐴)))↑2) = 0)
35	27, 34	oveq12d 7449	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (((𝑁‘(𝐴( +_𝑣 ‘𝑈)𝐴))↑2) − ((𝑁‘(𝐴( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝐴)))↑2)) = ((4 · ((𝑁‘𝐴)↑2)) − 0))
36		4cn 12351	. . . . . . . 8 ⊢ 4 ∈ ℂ
37	18	sqcld 14184	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)↑2) ∈ ℂ)
38		mulcl 11239	. . . . . . . 8 ⊢ ((4 ∈ ℂ ∧ ((𝑁‘𝐴)↑2) ∈ ℂ) → (4 · ((𝑁‘𝐴)↑2)) ∈ ℂ)
39	36, 37, 38	sylancr 587	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (4 · ((𝑁‘𝐴)↑2)) ∈ ℂ)
40	39	subid1d 11609	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((4 · ((𝑁‘𝐴)↑2)) − 0) = (4 · ((𝑁‘𝐴)↑2)))
41	35, 40	eqtrd 2777	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (((𝑁‘(𝐴( +_𝑣 ‘𝑈)𝐴))↑2) − ((𝑁‘(𝐴( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝐴)))↑2)) = (4 · ((𝑁‘𝐴)↑2)))
42		1re 11261	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ
43		neg1rr 12381	. . . . . . . . . . . . . . . 16 ⊢ -1 ∈ ℝ
44		absreim 15332	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℝ ∧ -1 ∈ ℝ) → (abs‘(1 + (i · -1))) = (√‘((1↑2) + (-1↑2))))
45	42, 43, 44	mp2an 692	. . . . . . . . . . . . . . 15 ⊢ (abs‘(1 + (i · -1))) = (√‘((1↑2) + (-1↑2)))
46		ax-icn 11214	. . . . . . . . . . . . . . . . . . 19 ⊢ i ∈ ℂ
47		ax-1cn 11213	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℂ
48	46, 47	mulneg2i 11710	. . . . . . . . . . . . . . . . . 18 ⊢ (i · -1) = -(i · 1)
49	46	mulridi 11265	. . . . . . . . . . . . . . . . . . 19 ⊢ (i · 1) = i
50	49	negeqi 11501	. . . . . . . . . . . . . . . . . 18 ⊢ -(i · 1) = -i
51	48, 50	eqtri 2765	. . . . . . . . . . . . . . . . 17 ⊢ (i · -1) = -i
52	51	oveq2i 7442	. . . . . . . . . . . . . . . 16 ⊢ (1 + (i · -1)) = (1 + -i)
53	52	fveq2i 6909	. . . . . . . . . . . . . . 15 ⊢ (abs‘(1 + (i · -1))) = (abs‘(1 + -i))
54		sqneg 14156	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ ℂ → (-1↑2) = (1↑2))
55	47, 54	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (-1↑2) = (1↑2)
56	55	oveq2i 7442	. . . . . . . . . . . . . . . 16 ⊢ ((1↑2) + (-1↑2)) = ((1↑2) + (1↑2))
57	56	fveq2i 6909	. . . . . . . . . . . . . . 15 ⊢ (√‘((1↑2) + (-1↑2))) = (√‘((1↑2) + (1↑2)))
58	45, 53, 57	3eqtr3i 2773	. . . . . . . . . . . . . 14 ⊢ (abs‘(1 + -i)) = (√‘((1↑2) + (1↑2)))
59		absreim 15332	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℝ ∧ 1 ∈ ℝ) → (abs‘(1 + (i · 1))) = (√‘((1↑2) + (1↑2))))
60	42, 42, 59	mp2an 692	. . . . . . . . . . . . . 14 ⊢ (abs‘(1 + (i · 1))) = (√‘((1↑2) + (1↑2)))
61	49	oveq2i 7442	. . . . . . . . . . . . . . 15 ⊢ (1 + (i · 1)) = (1 + i)
62	61	fveq2i 6909	. . . . . . . . . . . . . 14 ⊢ (abs‘(1 + (i · 1))) = (abs‘(1 + i))
63	58, 60, 62	3eqtr2i 2771	. . . . . . . . . . . . 13 ⊢ (abs‘(1 + -i)) = (abs‘(1 + i))
64	63	oveq1i 7441	. . . . . . . . . . . 12 ⊢ ((abs‘(1 + -i)) · (𝑁‘𝐴)) = ((abs‘(1 + i)) · (𝑁‘𝐴))
65		negicn 11509	. . . . . . . . . . . . . 14 ⊢ -i ∈ ℂ
66	47, 65	addcli 11267	. . . . . . . . . . . . 13 ⊢ (1 + -i) ∈ ℂ
67	1, 3, 4	nvs 30682	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ (1 + -i) ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (𝑁‘((1 + -i)( ·_𝑠OLD ‘𝑈)𝐴)) = ((abs‘(1 + -i)) · (𝑁‘𝐴)))
68	66, 67	mp3an2 1451	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘((1 + -i)( ·_𝑠OLD ‘𝑈)𝐴)) = ((abs‘(1 + -i)) · (𝑁‘𝐴)))
69	47, 46	addcli 11267	. . . . . . . . . . . . 13 ⊢ (1 + i) ∈ ℂ
70	1, 3, 4	nvs 30682	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ (1 + i) ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (𝑁‘((1 + i)( ·_𝑠OLD ‘𝑈)𝐴)) = ((abs‘(1 + i)) · (𝑁‘𝐴)))
71	69, 70	mp3an2 1451	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘((1 + i)( ·_𝑠OLD ‘𝑈)𝐴)) = ((abs‘(1 + i)) · (𝑁‘𝐴)))
72	64, 68, 71	3eqtr4a 2803	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘((1 + -i)( ·_𝑠OLD ‘𝑈)𝐴)) = (𝑁‘((1 + i)( ·_𝑠OLD ‘𝑈)𝐴)))
73	1, 2, 3	nvdir 30650	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ NrmCVec ∧ (1 ∈ ℂ ∧ -i ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → ((1 + -i)( ·_𝑠OLD ‘𝑈)𝐴) = ((1( ·_𝑠OLD ‘𝑈)𝐴)( +_𝑣 ‘𝑈)(-i( ·_𝑠OLD ‘𝑈)𝐴)))
74	47, 73	mp3anr1 1460	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ NrmCVec ∧ (-i ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → ((1 + -i)( ·_𝑠OLD ‘𝑈)𝐴) = ((1( ·_𝑠OLD ‘𝑈)𝐴)( +_𝑣 ‘𝑈)(-i( ·_𝑠OLD ‘𝑈)𝐴)))
75	65, 74	mpanr1 703	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((1 + -i)( ·_𝑠OLD ‘𝑈)𝐴) = ((1( ·_𝑠OLD ‘𝑈)𝐴)( +_𝑣 ‘𝑈)(-i( ·_𝑠OLD ‘𝑈)𝐴)))
76	1, 3	nvsid 30646	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (1( ·_𝑠OLD ‘𝑈)𝐴) = 𝐴)
77	76	oveq1d 7446	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((1( ·_𝑠OLD ‘𝑈)𝐴)( +_𝑣 ‘𝑈)(-i( ·_𝑠OLD ‘𝑈)𝐴)) = (𝐴( +_𝑣 ‘𝑈)(-i( ·_𝑠OLD ‘𝑈)𝐴)))
78	75, 77	eqtrd 2777	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((1 + -i)( ·_𝑠OLD ‘𝑈)𝐴) = (𝐴( +_𝑣 ‘𝑈)(-i( ·_𝑠OLD ‘𝑈)𝐴)))
79	78	fveq2d 6910	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘((1 + -i)( ·_𝑠OLD ‘𝑈)𝐴)) = (𝑁‘(𝐴( +_𝑣 ‘𝑈)(-i( ·_𝑠OLD ‘𝑈)𝐴))))
80	1, 2, 3	nvdir 30650	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ NrmCVec ∧ (1 ∈ ℂ ∧ i ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → ((1 + i)( ·_𝑠OLD ‘𝑈)𝐴) = ((1( ·_𝑠OLD ‘𝑈)𝐴)( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)))
81	47, 80	mp3anr1 1460	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ NrmCVec ∧ (i ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → ((1 + i)( ·_𝑠OLD ‘𝑈)𝐴) = ((1( ·_𝑠OLD ‘𝑈)𝐴)( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)))
82	46, 81	mpanr1 703	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((1 + i)( ·_𝑠OLD ‘𝑈)𝐴) = ((1( ·_𝑠OLD ‘𝑈)𝐴)( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)))
83	76	oveq1d 7446	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((1( ·_𝑠OLD ‘𝑈)𝐴)( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)) = (𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)))
84	82, 83	eqtrd 2777	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((1 + i)( ·_𝑠OLD ‘𝑈)𝐴) = (𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)))
85	84	fveq2d 6910	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘((1 + i)( ·_𝑠OLD ‘𝑈)𝐴)) = (𝑁‘(𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴))))
86	72, 79, 85	3eqtr3d 2785	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝐴( +_𝑣 ‘𝑈)(-i( ·_𝑠OLD ‘𝑈)𝐴))) = (𝑁‘(𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴))))
87	86	oveq1d 7446	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑁‘(𝐴( +_𝑣 ‘𝑈)(-i( ·_𝑠OLD ‘𝑈)𝐴)))↑2) = ((𝑁‘(𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)))↑2))
88	87	oveq2d 7447	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (((𝑁‘(𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)))↑2) − ((𝑁‘(𝐴( +_𝑣 ‘𝑈)(-i( ·_𝑠OLD ‘𝑈)𝐴)))↑2)) = (((𝑁‘(𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)))↑2) − ((𝑁‘(𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)))↑2)))
89	1, 2, 3, 4, 5	ipval2lem4 30725	. . . . . . . . . . 11 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ i ∈ ℂ) → ((𝑁‘(𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)))↑2) ∈ ℂ)
90	46, 89	mpan2 691	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝑁‘(𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)))↑2) ∈ ℂ)
91	90	3anidm23 1423	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑁‘(𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)))↑2) ∈ ℂ)
92	91	subidd 11608	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (((𝑁‘(𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)))↑2) − ((𝑁‘(𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)))↑2)) = 0)
93	88, 92	eqtrd 2777	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (((𝑁‘(𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)))↑2) − ((𝑁‘(𝐴( +_𝑣 ‘𝑈)(-i( ·_𝑠OLD ‘𝑈)𝐴)))↑2)) = 0)
94	93	oveq2d 7447	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (i · (((𝑁‘(𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)))↑2) − ((𝑁‘(𝐴( +_𝑣 ‘𝑈)(-i( ·_𝑠OLD ‘𝑈)𝐴)))↑2))) = (i · 0))
95		it0e0 12488	. . . . . 6 ⊢ (i · 0) = 0
96	94, 95	eqtrdi 2793	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (i · (((𝑁‘(𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)))↑2) − ((𝑁‘(𝐴( +_𝑣 ‘𝑈)(-i( ·_𝑠OLD ‘𝑈)𝐴)))↑2))) = 0)
97	41, 96	oveq12d 7449	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((((𝑁‘(𝐴( +_𝑣 ‘𝑈)𝐴))↑2) − ((𝑁‘(𝐴( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝐴)))↑2)) + (i · (((𝑁‘(𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)))↑2) − ((𝑁‘(𝐴( +_𝑣 ‘𝑈)(-i( ·_𝑠OLD ‘𝑈)𝐴)))↑2)))) = ((4 · ((𝑁‘𝐴)↑2)) + 0))
98	39	addridd 11461	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((4 · ((𝑁‘𝐴)↑2)) + 0) = (4 · ((𝑁‘𝐴)↑2)))
99	97, 98	eqtr2d 2778	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (4 · ((𝑁‘𝐴)↑2)) = ((((𝑁‘(𝐴( +_𝑣 ‘𝑈)𝐴))↑2) − ((𝑁‘(𝐴( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝐴)))↑2)) + (i · (((𝑁‘(𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)))↑2) − ((𝑁‘(𝐴( +_𝑣 ‘𝑈)(-i( ·_𝑠OLD ‘𝑈)𝐴)))↑2)))))
100	99	oveq1d 7446	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((4 · ((𝑁‘𝐴)↑2)) / 4) = (((((𝑁‘(𝐴( +_𝑣 ‘𝑈)𝐴))↑2) − ((𝑁‘(𝐴( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝐴)))↑2)) + (i · (((𝑁‘(𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)))↑2) − ((𝑁‘(𝐴( +_𝑣 ‘𝑈)(-i( ·_𝑠OLD ‘𝑈)𝐴)))↑2)))) / 4))
101		4ne0 12374	. . . 4 ⊢ 4 ≠ 0
102		divcan3 11948	. . . 4 ⊢ ((((𝑁‘𝐴)↑2) ∈ ℂ ∧ 4 ∈ ℂ ∧ 4 ≠ 0) → ((4 · ((𝑁‘𝐴)↑2)) / 4) = ((𝑁‘𝐴)↑2))
103	36, 101, 102	mp3an23 1455	. . 3 ⊢ (((𝑁‘𝐴)↑2) ∈ ℂ → ((4 · ((𝑁‘𝐴)↑2)) / 4) = ((𝑁‘𝐴)↑2))
104	37, 103	syl 17	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((4 · ((𝑁‘𝐴)↑2)) / 4) = ((𝑁‘𝐴)↑2))
105	7, 100, 104	3eqtr2d 2783	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝑃𝐴) = ((𝑁‘𝐴)↑2))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 ici 11157 + caddc 11158 · cmul 11160 ≤ cle 11296 − cmin 11492 -cneg 11493 / cdiv 11920 2c2 12321 4c4 12323 ℕ₀cn0 12526 ↑cexp 14102 √csqrt 15272 abscabs 15273 NrmCVeccnv 30603 +_𝑣 cpv 30604 BaseSetcba 30605 ·_𝑠OLD cns 30606 0_veccn0v 30607 norm_CVcnmcv 30609 ·_𝑖OLDcdip 30719

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723 df-grpo 30512 df-gid 30513 df-ginv 30514 df-ablo 30564 df-vc 30578 df-nv 30611 df-va 30614 df-ba 30615 df-sm 30616 df-0v 30617 df-nmcv 30619 df-dip 30720

This theorem is referenced by: ipnm 30730 ipz 30738 pythi 30869 siilem1 30870 hlipgt0 30933 htthlem 30936

W3C validator