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Theorem ipidsq 30691

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 2735	. . . 4 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
3		eqid 2735	. . . 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 30688	. . 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 30613	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴( +_𝑣 ‘𝑈)𝐴) = (2( ·_𝑠OLD ‘𝑈)𝐴))
9	8	fveq2d 6880	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝐴( +_𝑣 ‘𝑈)𝐴)) = (𝑁‘(2( ·_𝑠OLD ‘𝑈)𝐴)))
10		2re 12314	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
11		0le2 12342	. . . . . . . . . . . 12 ⊢ 0 ≤ 2
12	10, 11	pm3.2i 470	. . . . . . . . . . 11 ⊢ (2 ∈ ℝ ∧ 0 ≤ 2)
13	1, 3, 4	nvsge0 30645	. . . . . . . . . . 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 2770	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝐴( +_𝑣 ‘𝑈)𝐴)) = (2 · (𝑁‘𝐴)))
16	15	oveq1d 7420	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑁‘(𝐴( +_𝑣 ‘𝑈)𝐴))↑2) = ((2 · (𝑁‘𝐴))↑2))
17	1, 4	nvcl 30642	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℝ)
18	17	recnd 11263	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℂ)
19		2cn 12315	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
20		2nn0 12518	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
21		mulexp 14119	. . . . . . . . . . 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 14215	. . . . . . . . . 10 ⊢ (2↑2) = 4
25	24	oveq1i 7415	. . . . . . . . 9 ⊢ ((2↑2) · ((𝑁‘𝐴)↑2)) = (4 · ((𝑁‘𝐴)↑2))
26	23, 25	eqtrdi 2786	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((2 · (𝑁‘𝐴))↑2) = (4 · ((𝑁‘𝐴)↑2)))
27	16, 26	eqtrd 2770	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑁‘(𝐴( +_𝑣 ‘𝑈)𝐴))↑2) = (4 · ((𝑁‘𝐴)↑2)))
28		eqid 2735	. . . . . . . . . . 11 ⊢ (0_vec‘𝑈) = (0_vec‘𝑈)
29	1, 2, 3, 28	nvrinv 30632	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝐴)) = (0_vec‘𝑈))
30	29	fveq2d 6880	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝐴( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝐴))) = (𝑁‘(0_vec‘𝑈)))
31	28, 4	nvz0 30649	. . . . . . . . . 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 2770	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝐴( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝐴))) = 0)
34	33	sq0id 14212	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑁‘(𝐴( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝐴)))↑2) = 0)
35	27, 34	oveq12d 7423	. . . . . 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 12325	. . . . . . . 8 ⊢ 4 ∈ ℂ
37	18	sqcld 14162	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)↑2) ∈ ℂ)
38		mulcl 11213	. . . . . . . 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 11583	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((4 · ((𝑁‘𝐴)↑2)) − 0) = (4 · ((𝑁‘𝐴)↑2)))
41	35, 40	eqtrd 2770	. . . . 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 11235	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ
43		neg1rr 12355	. . . . . . . . . . . . . . . 16 ⊢ -1 ∈ ℝ
44		absreim 15312	. . . . . . . . . . . . . . . 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 11188	. . . . . . . . . . . . . . . . . . 19 ⊢ i ∈ ℂ
47		ax-1cn 11187	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℂ
48	46, 47	mulneg2i 11684	. . . . . . . . . . . . . . . . . 18 ⊢ (i · -1) = -(i · 1)
49	46	mulridi 11239	. . . . . . . . . . . . . . . . . . 19 ⊢ (i · 1) = i
50	49	negeqi 11475	. . . . . . . . . . . . . . . . . 18 ⊢ -(i · 1) = -i
51	48, 50	eqtri 2758	. . . . . . . . . . . . . . . . 17 ⊢ (i · -1) = -i
52	51	oveq2i 7416	. . . . . . . . . . . . . . . 16 ⊢ (1 + (i · -1)) = (1 + -i)
53	52	fveq2i 6879	. . . . . . . . . . . . . . 15 ⊢ (abs‘(1 + (i · -1))) = (abs‘(1 + -i))
54		sqneg 14133	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ ℂ → (-1↑2) = (1↑2))
55	47, 54	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (-1↑2) = (1↑2)
56	55	oveq2i 7416	. . . . . . . . . . . . . . . 16 ⊢ ((1↑2) + (-1↑2)) = ((1↑2) + (1↑2))
57	56	fveq2i 6879	. . . . . . . . . . . . . . 15 ⊢ (√‘((1↑2) + (-1↑2))) = (√‘((1↑2) + (1↑2)))
58	45, 53, 57	3eqtr3i 2766	. . . . . . . . . . . . . 14 ⊢ (abs‘(1 + -i)) = (√‘((1↑2) + (1↑2)))
59		absreim 15312	. . . . . . . . . . . . . . 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 7416	. . . . . . . . . . . . . . 15 ⊢ (1 + (i · 1)) = (1 + i)
62	61	fveq2i 6879	. . . . . . . . . . . . . 14 ⊢ (abs‘(1 + (i · 1))) = (abs‘(1 + i))
63	58, 60, 62	3eqtr2i 2764	. . . . . . . . . . . . 13 ⊢ (abs‘(1 + -i)) = (abs‘(1 + i))
64	63	oveq1i 7415	. . . . . . . . . . . 12 ⊢ ((abs‘(1 + -i)) · (𝑁‘𝐴)) = ((abs‘(1 + i)) · (𝑁‘𝐴))
65		negicn 11483	. . . . . . . . . . . . . 14 ⊢ -i ∈ ℂ
66	47, 65	addcli 11241	. . . . . . . . . . . . 13 ⊢ (1 + -i) ∈ ℂ
67	1, 3, 4	nvs 30644	. . . . . . . . . . . . 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 11241	. . . . . . . . . . . . 13 ⊢ (1 + i) ∈ ℂ
70	1, 3, 4	nvs 30644	. . . . . . . . . . . . 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 2796	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘((1 + -i)( ·_𝑠OLD ‘𝑈)𝐴)) = (𝑁‘((1 + i)( ·_𝑠OLD ‘𝑈)𝐴)))
73	1, 2, 3	nvdir 30612	. . . . . . . . . . . . . . 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 30608	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (1( ·_𝑠OLD ‘𝑈)𝐴) = 𝐴)
77	76	oveq1d 7420	. . . . . . . . . . . . 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 2770	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((1 + -i)( ·_𝑠OLD ‘𝑈)𝐴) = (𝐴( +_𝑣 ‘𝑈)(-i( ·_𝑠OLD ‘𝑈)𝐴)))
79	78	fveq2d 6880	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘((1 + -i)( ·_𝑠OLD ‘𝑈)𝐴)) = (𝑁‘(𝐴( +_𝑣 ‘𝑈)(-i( ·_𝑠OLD ‘𝑈)𝐴))))
80	1, 2, 3	nvdir 30612	. . . . . . . . . . . . . . 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 7420	. . . . . . . . . . . . 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 2770	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((1 + i)( ·_𝑠OLD ‘𝑈)𝐴) = (𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)))
85	84	fveq2d 6880	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘((1 + i)( ·_𝑠OLD ‘𝑈)𝐴)) = (𝑁‘(𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴))))
86	72, 79, 85	3eqtr3d 2778	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝐴( +_𝑣 ‘𝑈)(-i( ·_𝑠OLD ‘𝑈)𝐴))) = (𝑁‘(𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴))))
87	86	oveq1d 7420	. . . . . . . . 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 7421	. . . . . . . 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 30687	. . . . . . . . . . 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 11582	. . . . . . . 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 2770	. . . . . . 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 7421	. . . . . 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 12464	. . . . . 6 ⊢ (i · 0) = 0
96	94, 95	eqtrdi 2786	. . . . 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 7423	. . . 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 11435	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((4 · ((𝑁‘𝐴)↑2)) + 0) = (4 · ((𝑁‘𝐴)↑2)))
99	97, 98	eqtr2d 2771	. . 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 7420	. 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 12348	. . . 4 ⊢ 4 ≠ 0
102		divcan3 11922	. . . 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 2776	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝑃𝐴) = ((𝑁‘𝐴)↑2))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 class class class wbr 5119 ‘cfv 6531 (class class class)co 7405 ℂcc 11127 ℝcr 11128 0cc0 11129 1c1 11130 ici 11131 + caddc 11132 · cmul 11134 ≤ cle 11270 − cmin 11466 -cneg 11467 / cdiv 11894 2c2 12295 4c4 12297 ℕ₀cn0 12501 ↑cexp 14079 √csqrt 15252 abscabs 15253 NrmCVeccnv 30565 +_𝑣 cpv 30566 BaseSetcba 30567 ·_𝑠OLD cns 30568 0_veccn0v 30569 norm_CVcnmcv 30571 ·_𝑖OLDcdip 30681

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 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-inf2 9655 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8719 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-sup 9454 df-oi 9524 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-n0 12502 df-z 12589 df-uz 12853 df-rp 13009 df-fz 13525 df-fzo 13672 df-seq 14020 df-exp 14080 df-hash 14349 df-cj 15118 df-re 15119 df-im 15120 df-sqrt 15254 df-abs 15255 df-clim 15504 df-sum 15703 df-grpo 30474 df-gid 30475 df-ginv 30476 df-ablo 30526 df-vc 30540 df-nv 30573 df-va 30576 df-ba 30577 df-sm 30578 df-0v 30579 df-nmcv 30581 df-dip 30682

This theorem is referenced by: ipnm 30692 ipz 30700 pythi 30831 siilem1 30832 hlipgt0 30895 htthlem 30898

W3C validator