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Theorem ipidsq 31003

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 2769	. . . 4 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
3		eqid 2769	. . . 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 31000	. . 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 1446	. 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 30925	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴( +_𝑣 ‘𝑈)𝐴) = (2( ·_𝑠OLD ‘𝑈)𝐴))
9	8	fveq2d 6886	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝐴( +_𝑣 ‘𝑈)𝐴)) = (𝑁‘(2( ·_𝑠OLD ‘𝑈)𝐴)))
10		2re 12315	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
11		0le2 12343	. . . . . . . . . . . 12 ⊢ 0 ≤ 2
12	10, 11	pm3.2i 475	. . . . . . . . . . 11 ⊢ (2 ∈ ℝ ∧ 0 ≤ 2)
13	1, 3, 4	nvsge0 30957	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ (2 ∈ ℝ ∧ 0 ≤ 2) ∧ 𝐴 ∈ 𝑋) → (𝑁‘(2( ·_𝑠OLD ‘𝑈)𝐴)) = (2 · (𝑁‘𝐴)))
14	12, 13	mp3an2 1475	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(2( ·_𝑠OLD ‘𝑈)𝐴)) = (2 · (𝑁‘𝐴)))
15	9, 14	eqtrd 2804	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝐴( +_𝑣 ‘𝑈)𝐴)) = (2 · (𝑁‘𝐴)))
16	15	oveq1d 7426	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑁‘(𝐴( +_𝑣 ‘𝑈)𝐴))↑2) = ((2 · (𝑁‘𝐴))↑2))
17	1, 4	nvcl 30954	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℝ)
18	17	recnd 11237	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℂ)
19		2cn 12316	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
20		2nn0 12521	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
21		mulexp 14137	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ (𝑁‘𝐴) ∈ ℂ ∧ 2 ∈ ℕ₀) → ((2 · (𝑁‘𝐴))↑2) = ((2↑2) · ((𝑁‘𝐴)↑2)))
22	19, 20, 21	mp3an13 1478	. . . . . . . . . 10 ⊢ ((𝑁‘𝐴) ∈ ℂ → ((2 · (𝑁‘𝐴))↑2) = ((2↑2) · ((𝑁‘𝐴)↑2)))
23	18, 22	syl 18	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((2 · (𝑁‘𝐴))↑2) = ((2↑2) · ((𝑁‘𝐴)↑2)))
24		sq2 14233	. . . . . . . . . 10 ⊢ (2↑2) = 4
25	24	oveq1i 7421	. . . . . . . . 9 ⊢ ((2↑2) · ((𝑁‘𝐴)↑2)) = (4 · ((𝑁‘𝐴)↑2))
26	23, 25	eqtrdi 2820	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((2 · (𝑁‘𝐴))↑2) = (4 · ((𝑁‘𝐴)↑2)))
27	16, 26	eqtrd 2804	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑁‘(𝐴( +_𝑣 ‘𝑈)𝐴))↑2) = (4 · ((𝑁‘𝐴)↑2)))
28		eqid 2769	. . . . . . . . . . 11 ⊢ (0_vec‘𝑈) = (0_vec‘𝑈)
29	1, 2, 3, 28	nvrinv 30944	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝐴)) = (0_vec‘𝑈))
30	29	fveq2d 6886	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝐴( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝐴))) = (𝑁‘(0_vec‘𝑈)))
31	28, 4	nvz0 30961	. . . . . . . . . 10 ⊢ (𝑈 ∈ NrmCVec → (𝑁‘(0_vec‘𝑈)) = 0)
32	31	adantr 485	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(0_vec‘𝑈)) = 0)
33	30, 32	eqtrd 2804	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝐴( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝐴))) = 0)
34	33	sq0id 14230	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑁‘(𝐴( +_𝑣 ‘𝑈)(-1( ·_𝑠OLD ‘𝑈)𝐴)))↑2) = 0)
35	27, 34	oveq12d 7429	. . . . . 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 12326	. . . . . . . 8 ⊢ 4 ∈ ℂ
37	18	sqcld 14180	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)↑2) ∈ ℂ)
38		mulcl 11184	. . . . . . . 8 ⊢ ((4 ∈ ℂ ∧ ((𝑁‘𝐴)↑2) ∈ ℂ) → (4 · ((𝑁‘𝐴)↑2)) ∈ ℂ)
39	36, 37, 38	sylancr 598	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (4 · ((𝑁‘𝐴)↑2)) ∈ ℂ)
40	39	subid1d 11558	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((4 · ((𝑁‘𝐴)↑2)) − 0) = (4 · ((𝑁‘𝐴)↑2)))
41	35, 40	eqtrd 2804	. . . . 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 11208	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ
43		neg1rr 12204	. . . . . . . . . . . . . . . 16 ⊢ -1 ∈ ℝ
44		absreim 15344	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℝ ∧ -1 ∈ ℝ) → (abs‘(1 + (i · -1))) = (√‘((1↑2) + (-1↑2))))
45	42, 43, 44	mp2an 704	. . . . . . . . . . . . . . 15 ⊢ (abs‘(1 + (i · -1))) = (√‘((1↑2) + (-1↑2)))
46		ax-icn 11159	. . . . . . . . . . . . . . . . . . 19 ⊢ i ∈ ℂ
47		ax-1cn 11158	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℂ
48	46, 47	mulneg2i 11661	. . . . . . . . . . . . . . . . . 18 ⊢ (i · -1) = -(i · 1)
49	46	mulridi 11213	. . . . . . . . . . . . . . . . . . 19 ⊢ (i · 1) = i
50	49	negeqi 11450	. . . . . . . . . . . . . . . . . 18 ⊢ -(i · 1) = -i
51	48, 50	eqtri 2792	. . . . . . . . . . . . . . . . 17 ⊢ (i · -1) = -i
52	51	oveq2i 7422	. . . . . . . . . . . . . . . 16 ⊢ (1 + (i · -1)) = (1 + -i)
53	52	fveq2i 6885	. . . . . . . . . . . . . . 15 ⊢ (abs‘(1 + (i · -1))) = (abs‘(1 + -i))
54		sqneg 14151	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ ℂ → (-1↑2) = (1↑2))
55	47, 54	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (-1↑2) = (1↑2)
56	55	oveq2i 7422	. . . . . . . . . . . . . . . 16 ⊢ ((1↑2) + (-1↑2)) = ((1↑2) + (1↑2))
57	56	fveq2i 6885	. . . . . . . . . . . . . . 15 ⊢ (√‘((1↑2) + (-1↑2))) = (√‘((1↑2) + (1↑2)))
58	45, 53, 57	3eqtr3i 2800	. . . . . . . . . . . . . 14 ⊢ (abs‘(1 + -i)) = (√‘((1↑2) + (1↑2)))
59		absreim 15344	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℝ ∧ 1 ∈ ℝ) → (abs‘(1 + (i · 1))) = (√‘((1↑2) + (1↑2))))
60	42, 42, 59	mp2an 704	. . . . . . . . . . . . . 14 ⊢ (abs‘(1 + (i · 1))) = (√‘((1↑2) + (1↑2)))
61	49	oveq2i 7422	. . . . . . . . . . . . . . 15 ⊢ (1 + (i · 1)) = (1 + i)
62	61	fveq2i 6885	. . . . . . . . . . . . . 14 ⊢ (abs‘(1 + (i · 1))) = (abs‘(1 + i))
63	58, 60, 62	3eqtr2i 2798	. . . . . . . . . . . . 13 ⊢ (abs‘(1 + -i)) = (abs‘(1 + i))
64	63	oveq1i 7421	. . . . . . . . . . . 12 ⊢ ((abs‘(1 + -i)) · (𝑁‘𝐴)) = ((abs‘(1 + i)) · (𝑁‘𝐴))
65		negicn 11458	. . . . . . . . . . . . . 14 ⊢ -i ∈ ℂ
66	47, 65	addcli 11215	. . . . . . . . . . . . 13 ⊢ (1 + -i) ∈ ℂ
67	1, 3, 4	nvs 30956	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ (1 + -i) ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (𝑁‘((1 + -i)( ·_𝑠OLD ‘𝑈)𝐴)) = ((abs‘(1 + -i)) · (𝑁‘𝐴)))
68	66, 67	mp3an2 1475	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘((1 + -i)( ·_𝑠OLD ‘𝑈)𝐴)) = ((abs‘(1 + -i)) · (𝑁‘𝐴)))
69	47, 46	addcli 11215	. . . . . . . . . . . . 13 ⊢ (1 + i) ∈ ℂ
70	1, 3, 4	nvs 30956	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ (1 + i) ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (𝑁‘((1 + i)( ·_𝑠OLD ‘𝑈)𝐴)) = ((abs‘(1 + i)) · (𝑁‘𝐴)))
71	69, 70	mp3an2 1475	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘((1 + i)( ·_𝑠OLD ‘𝑈)𝐴)) = ((abs‘(1 + i)) · (𝑁‘𝐴)))
72	64, 68, 71	3eqtr4a 2830	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘((1 + -i)( ·_𝑠OLD ‘𝑈)𝐴)) = (𝑁‘((1 + i)( ·_𝑠OLD ‘𝑈)𝐴)))
73	1, 2, 3	nvdir 30924	. . . . . . . . . . . . . . 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 1484	. . . . . . . . . . . . . 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 715	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((1 + -i)( ·_𝑠OLD ‘𝑈)𝐴) = ((1( ·_𝑠OLD ‘𝑈)𝐴)( +_𝑣 ‘𝑈)(-i( ·_𝑠OLD ‘𝑈)𝐴)))
76	1, 3	nvsid 30920	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (1( ·_𝑠OLD ‘𝑈)𝐴) = 𝐴)
77	76	oveq1d 7426	. . . . . . . . . . . . 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 2804	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((1 + -i)( ·_𝑠OLD ‘𝑈)𝐴) = (𝐴( +_𝑣 ‘𝑈)(-i( ·_𝑠OLD ‘𝑈)𝐴)))
79	78	fveq2d 6886	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘((1 + -i)( ·_𝑠OLD ‘𝑈)𝐴)) = (𝑁‘(𝐴( +_𝑣 ‘𝑈)(-i( ·_𝑠OLD ‘𝑈)𝐴))))
80	1, 2, 3	nvdir 30924	. . . . . . . . . . . . . . 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 1484	. . . . . . . . . . . . . 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 715	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((1 + i)( ·_𝑠OLD ‘𝑈)𝐴) = ((1( ·_𝑠OLD ‘𝑈)𝐴)( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)))
83	76	oveq1d 7426	. . . . . . . . . . . . 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 2804	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((1 + i)( ·_𝑠OLD ‘𝑈)𝐴) = (𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)))
85	84	fveq2d 6886	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘((1 + i)( ·_𝑠OLD ‘𝑈)𝐴)) = (𝑁‘(𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴))))
86	72, 79, 85	3eqtr3d 2812	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝐴( +_𝑣 ‘𝑈)(-i( ·_𝑠OLD ‘𝑈)𝐴))) = (𝑁‘(𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴))))
87	86	oveq1d 7426	. . . . . . . . 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 7427	. . . . . . . 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 30999	. . . . . . . . . . 11 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ i ∈ ℂ) → ((𝑁‘(𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)))↑2) ∈ ℂ)
90	46, 89	mpan2 703	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝑁‘(𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)))↑2) ∈ ℂ)
91	90	3anidm23 1446	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑁‘(𝐴( +_𝑣 ‘𝑈)(i( ·_𝑠OLD ‘𝑈)𝐴)))↑2) ∈ ℂ)
92	91	subidd 11557	. . . . . . . 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 2804	. . . . . . 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 7427	. . . . . 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 12467	. . . . . 6 ⊢ (i · 0) = 0
96	94, 95	eqtrdi 2820	. . . . 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 7429	. . . 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 11410	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((4 · ((𝑁‘𝐴)↑2)) + 0) = (4 · ((𝑁‘𝐴)↑2)))
99	97, 98	eqtr2d 2805	. . 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 7426	. 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 12352	. . . 4 ⊢ 4 ≠ 0
102		divcan3 11898	. . . 4 ⊢ ((((𝑁‘𝐴)↑2) ∈ ℂ ∧ 4 ∈ ℂ ∧ 4 ≠ 0) → ((4 · ((𝑁‘𝐴)↑2)) / 4) = ((𝑁‘𝐴)↑2))
103	36, 101, 102	mp3an23 1479	. . 3 ⊢ (((𝑁‘𝐴)↑2) ∈ ℂ → ((4 · ((𝑁‘𝐴)↑2)) / 4) = ((𝑁‘𝐴)↑2))
104	37, 103	syl 18	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((4 · ((𝑁‘𝐴)↑2)) / 4) = ((𝑁‘𝐴)↑2))
105	7, 100, 104	3eqtr2d 2810	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝑃𝐴) = ((𝑁‘𝐴)↑2))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 class class class wbr 5113 ‘cfv 6537 (class class class)co 7411 ℂcc 11098 ℝcr 11099 0cc0 11100 1c1 11101 ici 11102 + caddc 11103 · cmul 11105 ≤ cle 11244 − cmin 11441 -cneg 11442 / cdiv 11871 2c2 12295 4c4 12297 ℕ₀cn0 12504 ↑cexp 14097 √csqrt 15284 abscabs 15285 NrmCVeccnv 30877 +_𝑣 cpv 30878 BaseSetcba 30879 ·_𝑠OLD cns 30880 0_veccn0v 30881 norm_CVcnmcv 30883 ·_𝑖OLDcdip 30993

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-inf2 9610 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-pre-sup 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-er 8694 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9402 df-oi 9472 df-card 9925 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-4 12305 df-n0 12505 df-z 12592 df-uz 12863 df-rp 13017 df-fz 13536 df-fzo 13683 df-seq 14038 df-exp 14098 df-hash 14367 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-clim 15539 df-sum 15738 df-grpo 30786 df-gid 30787 df-ginv 30788 df-ablo 30838 df-vc 30852 df-nv 30885 df-va 30888 df-ba 30889 df-sm 30890 df-0v 30891 df-nmcv 30893 df-dip 30994

This theorem is referenced by: ipnm 31004 ipz 31012 pythi 31143 siilem1 31144 hlipgt0 31207 htthlem 31210

W3C validator