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 30785

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 2736	. . . 4 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
3		eqid 2736	. . . 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 30782	. . 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 30707	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴( +_𝑣 ‘𝑈)𝐴) = (2( ·_𝑠OLD ‘𝑈)𝐴))
9	8	fveq2d 6838	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝐴( +_𝑣 ‘𝑈)𝐴)) = (𝑁‘(2( ·_𝑠OLD ‘𝑈)𝐴)))
10		2re 12219	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
11		0le2 12247	. . . . . . . . . . . 12 ⊢ 0 ≤ 2
12	10, 11	pm3.2i 470	. . . . . . . . . . 11 ⊢ (2 ∈ ℝ ∧ 0 ≤ 2)
13	1, 3, 4	nvsge0 30739	. . . . . . . . . . 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 2771	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝐴( +_𝑣 ‘𝑈)𝐴)) = (2 · (𝑁‘𝐴)))
16	15	oveq1d 7373	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑁‘(𝐴( +_𝑣 ‘𝑈)𝐴))↑2) = ((2 · (𝑁‘𝐴))↑2))
17	1, 4	nvcl 30736	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℝ)
18	17	recnd 11160	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℂ)
19		2cn 12220	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
20		2nn0 12418	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
21		mulexp 14024	. . . . . . . . . . 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 14120	. . . . . . . . . 10 ⊢ (2↑2) = 4
25	24	oveq1i 7368	. . . . . . . . 9 ⊢ ((2↑2) · ((𝑁‘𝐴)↑2)) = (4 · ((𝑁‘𝐴)↑2))
26	23, 25	eqtrdi 2787	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((2 · (𝑁‘𝐴))↑2) = (4 · ((𝑁‘𝐴)↑2)))
27	16, 26	eqtrd 2771	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑁‘(𝐴( +_𝑣 ‘𝑈)𝐴))↑2) = (4 · ((𝑁‘𝐴)↑2)))
28		eqid 2736	. . . . . . . . . . 11 ⊢ (0_vec‘𝑈) = (0_vec‘𝑈)
29	1, 2, 3, 28	nvrinv 30726	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝐴)) = (0_vec‘𝑈))
30	29	fveq2d 6838	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝐴( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝐴))) = (𝑁‘(0_vec‘𝑈)))
31	28, 4	nvz0 30743	. . . . . . . . . 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 2771	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝐴( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝐴))) = 0)
34	33	sq0id 14117	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑁‘(𝐴( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝐴)))↑2) = 0)
35	27, 34	oveq12d 7376	. . . . . 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 12230	. . . . . . . 8 ⊢ 4 ∈ ℂ
37	18	sqcld 14067	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)↑2) ∈ ℂ)
38		mulcl 11110	. . . . . . . 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 11481	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((4 · ((𝑁‘𝐴)↑2)) − 0) = (4 · ((𝑁‘𝐴)↑2)))
41	35, 40	eqtrd 2771	. . . . 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 11132	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ
43		neg1rr 12131	. . . . . . . . . . . . . . . 16 ⊢ -1 ∈ ℝ
44		absreim 15216	. . . . . . . . . . . . . . . 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 11085	. . . . . . . . . . . . . . . . . . 19 ⊢ i ∈ ℂ
47		ax-1cn 11084	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℂ
48	46, 47	mulneg2i 11584	. . . . . . . . . . . . . . . . . 18 ⊢ (i · -1) = -(i · 1)
49	46	mulridi 11136	. . . . . . . . . . . . . . . . . . 19 ⊢ (i · 1) = i
50	49	negeqi 11373	. . . . . . . . . . . . . . . . . 18 ⊢ -(i · 1) = -i
51	48, 50	eqtri 2759	. . . . . . . . . . . . . . . . 17 ⊢ (i · -1) = -i
52	51	oveq2i 7369	. . . . . . . . . . . . . . . 16 ⊢ (1 + (i · -1)) = (1 + -i)
53	52	fveq2i 6837	. . . . . . . . . . . . . . 15 ⊢ (abs‘(1 + (i · -1))) = (abs‘(1 + -i))
54		sqneg 14038	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ ℂ → (-1↑2) = (1↑2))
55	47, 54	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (-1↑2) = (1↑2)
56	55	oveq2i 7369	. . . . . . . . . . . . . . . 16 ⊢ ((1↑2) + (-1↑2)) = ((1↑2) + (1↑2))
57	56	fveq2i 6837	. . . . . . . . . . . . . . 15 ⊢ (√‘((1↑2) + (-1↑2))) = (√‘((1↑2) + (1↑2)))
58	45, 53, 57	3eqtr3i 2767	. . . . . . . . . . . . . 14 ⊢ (abs‘(1 + -i)) = (√‘((1↑2) + (1↑2)))
59		absreim 15216	. . . . . . . . . . . . . . 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 7369	. . . . . . . . . . . . . . 15 ⊢ (1 + (i · 1)) = (1 + i)
62	61	fveq2i 6837	. . . . . . . . . . . . . 14 ⊢ (abs‘(1 + (i · 1))) = (abs‘(1 + i))
63	58, 60, 62	3eqtr2i 2765	. . . . . . . . . . . . 13 ⊢ (abs‘(1 + -i)) = (abs‘(1 + i))
64	63	oveq1i 7368	. . . . . . . . . . . 12 ⊢ ((abs‘(1 + -i)) · (𝑁‘𝐴)) = ((abs‘(1 + i)) · (𝑁‘𝐴))
65		negicn 11381	. . . . . . . . . . . . . 14 ⊢ -i ∈ ℂ
66	47, 65	addcli 11138	. . . . . . . . . . . . 13 ⊢ (1 + -i) ∈ ℂ
67	1, 3, 4	nvs 30738	. . . . . . . . . . . . 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 11138	. . . . . . . . . . . . 13 ⊢ (1 + i) ∈ ℂ
70	1, 3, 4	nvs 30738	. . . . . . . . . . . . 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 2797	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘((1 + -i)( ·_𝑠OLD ‘𝑈)𝐴)) = (𝑁‘((1 + i)( ·_𝑠OLD ‘𝑈)𝐴)))
73	1, 2, 3	nvdir 30706	. . . . . . . . . . . . . . 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 30702	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (1( ·_𝑠OLD ‘𝑈)𝐴) = 𝐴)
77	76	oveq1d 7373	. . . . . . . . . . . . 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 2771	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((1 + -i)( ·_𝑠OLD ‘𝑈)𝐴) = (𝐴( +_𝑣 ‘𝑈)(-i( ·_𝑠OLD ‘𝑈)𝐴)))
79	78	fveq2d 6838	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘((1 + -i)( ·_𝑠OLD ‘𝑈)𝐴)) = (𝑁‘(𝐴( +_𝑣 ‘𝑈)(-i( ·_𝑠OLD ‘𝑈)𝐴))))
80	1, 2, 3	nvdir 30706	. . . . . . . . . . . . . . 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 7373	. . . . . . . . . . . . 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 2771	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((1 + i)( ·_𝑠OLD ‘𝑈)𝐴) = (𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)))
85	84	fveq2d 6838	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘((1 + i)( ·_𝑠OLD ‘𝑈)𝐴)) = (𝑁‘(𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴))))
86	72, 79, 85	3eqtr3d 2779	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝐴( +_𝑣 ‘𝑈)(-i( ·_𝑠OLD ‘𝑈)𝐴))) = (𝑁‘(𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴))))
87	86	oveq1d 7373	. . . . . . . . 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 7374	. . . . . . . 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 30781	. . . . . . . . . . 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 11480	. . . . . . . 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 2771	. . . . . . 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 7374	. . . . . 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 12364	. . . . . 6 ⊢ (i · 0) = 0
96	94, 95	eqtrdi 2787	. . . . 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 7376	. . . 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 11333	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((4 · ((𝑁‘𝐴)↑2)) + 0) = (4 · ((𝑁‘𝐴)↑2)))
99	97, 98	eqtr2d 2772	. . 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 7373	. 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 12253	. . . 4 ⊢ 4 ≠ 0
102		divcan3 11822	. . . 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 2777	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝑃𝐴) = ((𝑁‘𝐴)↑2))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 class class class wbr 5098 ‘cfv 6492 (class class class)co 7358 ℂcc 11024 ℝcr 11025 0cc0 11026 1c1 11027 ici 11028 + caddc 11029 · cmul 11031 ≤ cle 11167 − cmin 11364 -cneg 11365 / cdiv 11794 2c2 12200 4c4 12202 ℕ₀cn0 12401 ↑cexp 13984 √csqrt 15156 abscabs 15157 NrmCVeccnv 30659 +_𝑣 cpv 30660 BaseSetcba 30661 ·_𝑠OLD cns 30662 0_veccn0v 30663 norm_CVcnmcv 30665 ·_𝑖OLDcdip 30775

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9345 df-oi 9415 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-n0 12402 df-z 12489 df-uz 12752 df-rp 12906 df-fz 13424 df-fzo 13571 df-seq 13925 df-exp 13985 df-hash 14254 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-clim 15411 df-sum 15610 df-grpo 30568 df-gid 30569 df-ginv 30570 df-ablo 30620 df-vc 30634 df-nv 30667 df-va 30670 df-ba 30671 df-sm 30672 df-0v 30673 df-nmcv 30675 df-dip 30776

This theorem is referenced by: ipnm 30786 ipz 30794 pythi 30925 siilem1 30926 hlipgt0 30989 htthlem 30992

W3C validator