ipidsq - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ipidsq		Structured version Visualization version GIF version

Theorem ipidsq 28174

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 2797	. . . 4 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
3		eqid 2797	. . . 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 28171	. . 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 1414	. 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 28096	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴( +_𝑣 ‘𝑈)𝐴) = (2( ·_𝑠OLD ‘𝑈)𝐴))
9	8	fveq2d 6549	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝐴( +_𝑣 ‘𝑈)𝐴)) = (𝑁‘(2( ·_𝑠OLD ‘𝑈)𝐴)))
10		2re 11565	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
11		0le2 11593	. . . . . . . . . . . 12 ⊢ 0 ≤ 2
12	10, 11	pm3.2i 471	. . . . . . . . . . 11 ⊢ (2 ∈ ℝ ∧ 0 ≤ 2)
13	1, 3, 4	nvsge0 28128	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ (2 ∈ ℝ ∧ 0 ≤ 2) ∧ 𝐴 ∈ 𝑋) → (𝑁‘(2( ·_𝑠OLD ‘𝑈)𝐴)) = (2 · (𝑁‘𝐴)))
14	12, 13	mp3an2 1441	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(2( ·_𝑠OLD ‘𝑈)𝐴)) = (2 · (𝑁‘𝐴)))
15	9, 14	eqtrd 2833	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝐴( +_𝑣 ‘𝑈)𝐴)) = (2 · (𝑁‘𝐴)))
16	15	oveq1d 7038	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑁‘(𝐴( +_𝑣 ‘𝑈)𝐴))↑2) = ((2 · (𝑁‘𝐴))↑2))
17	1, 4	nvcl 28125	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℝ)
18	17	recnd 10522	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℂ)
19		2cn 11566	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
20		2nn0 11768	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
21		mulexp 13322	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ (𝑁‘𝐴) ∈ ℂ ∧ 2 ∈ ℕ₀) → ((2 · (𝑁‘𝐴))↑2) = ((2↑2) · ((𝑁‘𝐴)↑2)))
22	19, 20, 21	mp3an13 1444	. . . . . . . . . 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 13414	. . . . . . . . . 10 ⊢ (2↑2) = 4
25	24	oveq1i 7033	. . . . . . . . 9 ⊢ ((2↑2) · ((𝑁‘𝐴)↑2)) = (4 · ((𝑁‘𝐴)↑2))
26	23, 25	syl6eq 2849	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((2 · (𝑁‘𝐴))↑2) = (4 · ((𝑁‘𝐴)↑2)))
27	16, 26	eqtrd 2833	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑁‘(𝐴( +_𝑣 ‘𝑈)𝐴))↑2) = (4 · ((𝑁‘𝐴)↑2)))
28		eqid 2797	. . . . . . . . . . 11 ⊢ (0_vec‘𝑈) = (0_vec‘𝑈)
29	1, 2, 3, 28	nvrinv 28115	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝐴)) = (0_vec‘𝑈))
30	29	fveq2d 6549	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝐴( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝐴))) = (𝑁‘(0_vec‘𝑈)))
31	28, 4	nvz0 28132	. . . . . . . . . 10 ⊢ (𝑈 ∈ NrmCVec → (𝑁‘(0_vec‘𝑈)) = 0)
32	31	adantr 481	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(0_vec‘𝑈)) = 0)
33	30, 32	eqtrd 2833	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝐴( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝐴))) = 0)
34	33	sq0id 13411	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑁‘(𝐴( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝐴)))↑2) = 0)
35	27, 34	oveq12d 7041	. . . . . 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 11576	. . . . . . . 8 ⊢ 4 ∈ ℂ
37	18	sqcld 13362	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)↑2) ∈ ℂ)
38		mulcl 10474	. . . . . . . 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 10840	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((4 · ((𝑁‘𝐴)↑2)) − 0) = (4 · ((𝑁‘𝐴)↑2)))
41	35, 40	eqtrd 2833	. . . . 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 10494	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ
43		neg1rr 11606	. . . . . . . . . . . . . . . 16 ⊢ -1 ∈ ℝ
44		absreim 14491	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℝ ∧ -1 ∈ ℝ) → (abs‘(1 + (i · -1))) = (√‘((1↑2) + (-1↑2))))
45	42, 43, 44	mp2an 688	. . . . . . . . . . . . . . 15 ⊢ (abs‘(1 + (i · -1))) = (√‘((1↑2) + (-1↑2)))
46		ax-icn 10449	. . . . . . . . . . . . . . . . . . 19 ⊢ i ∈ ℂ
47		ax-1cn 10448	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℂ
48	46, 47	mulneg2i 10941	. . . . . . . . . . . . . . . . . 18 ⊢ (i · -1) = -(i · 1)
49	46	mulid1i 10498	. . . . . . . . . . . . . . . . . . 19 ⊢ (i · 1) = i
50	49	negeqi 10732	. . . . . . . . . . . . . . . . . 18 ⊢ -(i · 1) = -i
51	48, 50	eqtri 2821	. . . . . . . . . . . . . . . . 17 ⊢ (i · -1) = -i
52	51	oveq2i 7034	. . . . . . . . . . . . . . . 16 ⊢ (1 + (i · -1)) = (1 + -i)
53	52	fveq2i 6548	. . . . . . . . . . . . . . 15 ⊢ (abs‘(1 + (i · -1))) = (abs‘(1 + -i))
54		sqneg 13336	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ ℂ → (-1↑2) = (1↑2))
55	47, 54	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (-1↑2) = (1↑2)
56	55	oveq2i 7034	. . . . . . . . . . . . . . . 16 ⊢ ((1↑2) + (-1↑2)) = ((1↑2) + (1↑2))
57	56	fveq2i 6548	. . . . . . . . . . . . . . 15 ⊢ (√‘((1↑2) + (-1↑2))) = (√‘((1↑2) + (1↑2)))
58	45, 53, 57	3eqtr3i 2829	. . . . . . . . . . . . . 14 ⊢ (abs‘(1 + -i)) = (√‘((1↑2) + (1↑2)))
59		absreim 14491	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℝ ∧ 1 ∈ ℝ) → (abs‘(1 + (i · 1))) = (√‘((1↑2) + (1↑2))))
60	42, 42, 59	mp2an 688	. . . . . . . . . . . . . 14 ⊢ (abs‘(1 + (i · 1))) = (√‘((1↑2) + (1↑2)))
61	49	oveq2i 7034	. . . . . . . . . . . . . . 15 ⊢ (1 + (i · 1)) = (1 + i)
62	61	fveq2i 6548	. . . . . . . . . . . . . 14 ⊢ (abs‘(1 + (i · 1))) = (abs‘(1 + i))
63	58, 60, 62	3eqtr2i 2827	. . . . . . . . . . . . 13 ⊢ (abs‘(1 + -i)) = (abs‘(1 + i))
64	63	oveq1i 7033	. . . . . . . . . . . 12 ⊢ ((abs‘(1 + -i)) · (𝑁‘𝐴)) = ((abs‘(1 + i)) · (𝑁‘𝐴))
65		negicn 10740	. . . . . . . . . . . . . 14 ⊢ -i ∈ ℂ
66	47, 65	addcli 10500	. . . . . . . . . . . . 13 ⊢ (1 + -i) ∈ ℂ
67	1, 3, 4	nvs 28127	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ (1 + -i) ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (𝑁‘((1 + -i)( ·_𝑠OLD ‘𝑈)𝐴)) = ((abs‘(1 + -i)) · (𝑁‘𝐴)))
68	66, 67	mp3an2 1441	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘((1 + -i)( ·_𝑠OLD ‘𝑈)𝐴)) = ((abs‘(1 + -i)) · (𝑁‘𝐴)))
69	47, 46	addcli 10500	. . . . . . . . . . . . 13 ⊢ (1 + i) ∈ ℂ
70	1, 3, 4	nvs 28127	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ (1 + i) ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (𝑁‘((1 + i)( ·_𝑠OLD ‘𝑈)𝐴)) = ((abs‘(1 + i)) · (𝑁‘𝐴)))
71	69, 70	mp3an2 1441	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘((1 + i)( ·_𝑠OLD ‘𝑈)𝐴)) = ((abs‘(1 + i)) · (𝑁‘𝐴)))
72	64, 68, 71	3eqtr4a 2859	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘((1 + -i)( ·_𝑠OLD ‘𝑈)𝐴)) = (𝑁‘((1 + i)( ·_𝑠OLD ‘𝑈)𝐴)))
73	1, 2, 3	nvdir 28095	. . . . . . . . . . . . . . 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 1450	. . . . . . . . . . . . . 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 699	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((1 + -i)( ·_𝑠OLD ‘𝑈)𝐴) = ((1( ·_𝑠OLD ‘𝑈)𝐴)( +_𝑣 ‘𝑈)(-i( ·_𝑠OLD ‘𝑈)𝐴)))
76	1, 3	nvsid 28091	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (1( ·_𝑠OLD ‘𝑈)𝐴) = 𝐴)
77	76	oveq1d 7038	. . . . . . . . . . . . 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 2833	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((1 + -i)( ·_𝑠OLD ‘𝑈)𝐴) = (𝐴( +_𝑣 ‘𝑈)(-i( ·_𝑠OLD ‘𝑈)𝐴)))
79	78	fveq2d 6549	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘((1 + -i)( ·_𝑠OLD ‘𝑈)𝐴)) = (𝑁‘(𝐴( +_𝑣 ‘𝑈)(-i( ·_𝑠OLD ‘𝑈)𝐴))))
80	1, 2, 3	nvdir 28095	. . . . . . . . . . . . . . 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 1450	. . . . . . . . . . . . . 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 699	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((1 + i)( ·_𝑠OLD ‘𝑈)𝐴) = ((1( ·_𝑠OLD ‘𝑈)𝐴)( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)))
83	76	oveq1d 7038	. . . . . . . . . . . . 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 2833	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((1 + i)( ·_𝑠OLD ‘𝑈)𝐴) = (𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)))
85	84	fveq2d 6549	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘((1 + i)( ·_𝑠OLD ‘𝑈)𝐴)) = (𝑁‘(𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴))))
86	72, 79, 85	3eqtr3d 2841	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝐴( +_𝑣 ‘𝑈)(-i( ·_𝑠OLD ‘𝑈)𝐴))) = (𝑁‘(𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴))))
87	86	oveq1d 7038	. . . . . . . . 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 7039	. . . . . . . 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 28170	. . . . . . . . . . 11 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ i ∈ ℂ) → ((𝑁‘(𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)))↑2) ∈ ℂ)
90	46, 89	mpan2 687	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝑁‘(𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)))↑2) ∈ ℂ)
91	90	3anidm23 1414	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑁‘(𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)))↑2) ∈ ℂ)
92	91	subidd 10839	. . . . . . . 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 2833	. . . . . . 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 7039	. . . . . 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 11713	. . . . . 6 ⊢ (i · 0) = 0
96	94, 95	syl6eq 2849	. . . . 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 7041	. . . 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	addid1d 10693	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((4 · ((𝑁‘𝐴)↑2)) + 0) = (4 · ((𝑁‘𝐴)↑2)))
99	97, 98	eqtr2d 2834	. . 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 7038	. 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 11599	. . . 4 ⊢ 4 ≠ 0
102		divcan3 11178	. . . 4 ⊢ ((((𝑁‘𝐴)↑2) ∈ ℂ ∧ 4 ∈ ℂ ∧ 4 ≠ 0) → ((4 · ((𝑁‘𝐴)↑2)) / 4) = ((𝑁‘𝐴)↑2))
103	36, 101, 102	mp3an23 1445	. . 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 2839	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝑃𝐴) = ((𝑁‘𝐴)↑2))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1080 = wceq 1525 ∈ wcel 2083 ≠ wne 2986 class class class wbr 4968 ‘cfv 6232 (class class class)co 7023 ℂcc 10388 ℝcr 10389 0cc0 10390 1c1 10391 ici 10392 + caddc 10393 · cmul 10395 ≤ cle 10529 − cmin 10723 -cneg 10724 / cdiv 11151 2c2 11546 4c4 11548 ℕ₀cn0 11751 ↑cexp 13283 √csqrt 14430 abscabs 14431 NrmCVeccnv 28048 +_𝑣 cpv 28049 BaseSetcba 28050 ·_𝑠OLD cns 28051 0_veccn0v 28052 norm_CVcnmcv 28054 ·_𝑖OLDcdip 28164

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-inf2 8957 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 ax-pre-sup 10468

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-fal 1538 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-se 5410 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-isom 6241 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-1st 7552 df-2nd 7553 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-oadd 7964 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-sup 8759 df-oi 8827 df-card 9221 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-div 11152 df-nn 11493 df-2 11554 df-3 11555 df-4 11556 df-n0 11752 df-z 11836 df-uz 12098 df-rp 12244 df-fz 12747 df-fzo 12888 df-seq 13224 df-exp 13284 df-hash 13545 df-cj 14296 df-re 14297 df-im 14298 df-sqrt 14432 df-abs 14433 df-clim 14683 df-sum 14881 df-grpo 27957 df-gid 27958 df-ginv 27959 df-ablo 28009 df-vc 28023 df-nv 28056 df-va 28059 df-ba 28060 df-sm 28061 df-0v 28062 df-nmcv 28064 df-dip 28165

This theorem is referenced by: ipnm 28175 ipz 28183 pythi 28314 siilem1 28315 hlipgt0 28378 htthlem 28381

W3C validator