ipasslem10 - Metamath Proof Explorer

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Theorem ipasslem10 30079

Description: Lemma for ipassi 30081. Show the inner product associative law for the imaginary number i. (Contributed by NM, 24-Aug-2007.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
ip1i.1	⊢ 𝑋 = (BaseSet‘𝑈)
ip1i.2	⊢ 𝐺 = ( +_𝑣 ‘𝑈)
ip1i.4	⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
ip1i.7	⊢ 𝑃 = (·_𝑖OLD‘𝑈)
ip1i.9	⊢ 𝑈 ∈ CPreHil_OLD
ipasslem10.a	⊢ 𝐴 ∈ 𝑋
ipasslem10.b	⊢ 𝐵 ∈ 𝑋
ipasslem10.6	⊢ 𝑁 = (norm_CV‘𝑈)

**Assertion**
Ref	Expression
ipasslem10	⊢ ((i𝑆𝐴)𝑃𝐵) = (i · (𝐴𝑃𝐵))

**Proof of Theorem ipasslem10**
Step	Hyp	Ref	Expression
1		ip1i.9	. . . . . . 7 ⊢ 𝑈 ∈ CPreHil_OLD
2	1	phnvi 30056	. . . . . 6 ⊢ 𝑈 ∈ NrmCVec
3		ipasslem10.b	. . . . . 6 ⊢ 𝐵 ∈ 𝑋
4		ax-icn 11165	. . . . . . 7 ⊢ i ∈ ℂ
5		ipasslem10.a	. . . . . . 7 ⊢ 𝐴 ∈ 𝑋
6		ip1i.1	. . . . . . . 8 ⊢ 𝑋 = (BaseSet‘𝑈)
7		ip1i.4	. . . . . . . 8 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
8	6, 7	nvscl 29866	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ i ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (i𝑆𝐴) ∈ 𝑋)
9	2, 4, 5, 8	mp3an 1461	. . . . . 6 ⊢ (i𝑆𝐴) ∈ 𝑋
10		ip1i.2	. . . . . . 7 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
11		ipasslem10.6	. . . . . . 7 ⊢ 𝑁 = (norm_CV‘𝑈)
12		ip1i.7	. . . . . . 7 ⊢ 𝑃 = (·_𝑖OLD‘𝑈)
13	6, 10, 7, 11, 12	4ipval2 29948	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ (i𝑆𝐴) ∈ 𝑋) → (4 · (𝐵𝑃(i𝑆𝐴))) = ((((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺(-1𝑆(i𝑆𝐴))))↑2)) + (i · (((𝑁‘(𝐵𝐺(i𝑆(i𝑆𝐴))))↑2) − ((𝑁‘(𝐵𝐺(-i𝑆(i𝑆𝐴))))↑2)))))
14	2, 3, 9, 13	mp3an 1461	. . . . 5 ⊢ (4 · (𝐵𝑃(i𝑆𝐴))) = ((((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺(-1𝑆(i𝑆𝐴))))↑2)) + (i · (((𝑁‘(𝐵𝐺(i𝑆(i𝑆𝐴))))↑2) − ((𝑁‘(𝐵𝐺(-i𝑆(i𝑆𝐴))))↑2))))
15		4cn 12293	. . . . . . 7 ⊢ 4 ∈ ℂ
16		negicn 11457	. . . . . . 7 ⊢ -i ∈ ℂ
17	6, 12	dipcl 29952	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐵𝑃𝐴) ∈ ℂ)
18	2, 3, 5, 17	mp3an 1461	. . . . . . 7 ⊢ (𝐵𝑃𝐴) ∈ ℂ
19	15, 16, 18	mul12i 11405	. . . . . 6 ⊢ (4 · (-i · (𝐵𝑃𝐴))) = (-i · (4 · (𝐵𝑃𝐴)))
20	6, 10	nvgcl 29860	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ (i𝑆𝐴) ∈ 𝑋) → (𝐵𝐺(i𝑆𝐴)) ∈ 𝑋)
21	2, 3, 9, 20	mp3an 1461	. . . . . . . . . . . . 13 ⊢ (𝐵𝐺(i𝑆𝐴)) ∈ 𝑋
22	6, 11, 2, 21	nvcli 29902	. . . . . . . . . . . 12 ⊢ (𝑁‘(𝐵𝐺(i𝑆𝐴))) ∈ ℝ
23	22	recni 11224	. . . . . . . . . . 11 ⊢ (𝑁‘(𝐵𝐺(i𝑆𝐴))) ∈ ℂ
24	23	sqcli 14141	. . . . . . . . . 10 ⊢ ((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) ∈ ℂ
25		neg1cn 12322	. . . . . . . . . . . . . . 15 ⊢ -1 ∈ ℂ
26	6, 7	nvscl 29866	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ (i𝑆𝐴) ∈ 𝑋) → (-1𝑆(i𝑆𝐴)) ∈ 𝑋)
27	2, 25, 9, 26	mp3an 1461	. . . . . . . . . . . . . 14 ⊢ (-1𝑆(i𝑆𝐴)) ∈ 𝑋
28	6, 10	nvgcl 29860	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ (-1𝑆(i𝑆𝐴)) ∈ 𝑋) → (𝐵𝐺(-1𝑆(i𝑆𝐴))) ∈ 𝑋)
29	2, 3, 27, 28	mp3an 1461	. . . . . . . . . . . . 13 ⊢ (𝐵𝐺(-1𝑆(i𝑆𝐴))) ∈ 𝑋
30	6, 11, 2, 29	nvcli 29902	. . . . . . . . . . . 12 ⊢ (𝑁‘(𝐵𝐺(-1𝑆(i𝑆𝐴)))) ∈ ℝ
31	30	recni 11224	. . . . . . . . . . 11 ⊢ (𝑁‘(𝐵𝐺(-1𝑆(i𝑆𝐴)))) ∈ ℂ
32	31	sqcli 14141	. . . . . . . . . 10 ⊢ ((𝑁‘(𝐵𝐺(-1𝑆(i𝑆𝐴))))↑2) ∈ ℂ
33	24, 32	subcli 11532	. . . . . . . . 9 ⊢ (((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺(-1𝑆(i𝑆𝐴))))↑2)) ∈ ℂ
34	6, 7	nvscl 29866	. . . . . . . . . . . . . . . 16 ⊢ ((𝑈 ∈ NrmCVec ∧ i ∈ ℂ ∧ (i𝑆𝐴) ∈ 𝑋) → (i𝑆(i𝑆𝐴)) ∈ 𝑋)
35	2, 4, 9, 34	mp3an 1461	. . . . . . . . . . . . . . 15 ⊢ (i𝑆(i𝑆𝐴)) ∈ 𝑋
36	6, 10	nvgcl 29860	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ (i𝑆(i𝑆𝐴)) ∈ 𝑋) → (𝐵𝐺(i𝑆(i𝑆𝐴))) ∈ 𝑋)
37	2, 3, 35, 36	mp3an 1461	. . . . . . . . . . . . . 14 ⊢ (𝐵𝐺(i𝑆(i𝑆𝐴))) ∈ 𝑋
38	6, 11, 2, 37	nvcli 29902	. . . . . . . . . . . . 13 ⊢ (𝑁‘(𝐵𝐺(i𝑆(i𝑆𝐴)))) ∈ ℝ
39	38	recni 11224	. . . . . . . . . . . 12 ⊢ (𝑁‘(𝐵𝐺(i𝑆(i𝑆𝐴)))) ∈ ℂ
40	39	sqcli 14141	. . . . . . . . . . 11 ⊢ ((𝑁‘(𝐵𝐺(i𝑆(i𝑆𝐴))))↑2) ∈ ℂ
41	6, 7	nvscl 29866	. . . . . . . . . . . . . . . 16 ⊢ ((𝑈 ∈ NrmCVec ∧ -i ∈ ℂ ∧ (i𝑆𝐴) ∈ 𝑋) → (-i𝑆(i𝑆𝐴)) ∈ 𝑋)
42	2, 16, 9, 41	mp3an 1461	. . . . . . . . . . . . . . 15 ⊢ (-i𝑆(i𝑆𝐴)) ∈ 𝑋
43	6, 10	nvgcl 29860	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ (-i𝑆(i𝑆𝐴)) ∈ 𝑋) → (𝐵𝐺(-i𝑆(i𝑆𝐴))) ∈ 𝑋)
44	2, 3, 42, 43	mp3an 1461	. . . . . . . . . . . . . 14 ⊢ (𝐵𝐺(-i𝑆(i𝑆𝐴))) ∈ 𝑋
45	6, 11, 2, 44	nvcli 29902	. . . . . . . . . . . . 13 ⊢ (𝑁‘(𝐵𝐺(-i𝑆(i𝑆𝐴)))) ∈ ℝ
46	45	recni 11224	. . . . . . . . . . . 12 ⊢ (𝑁‘(𝐵𝐺(-i𝑆(i𝑆𝐴)))) ∈ ℂ
47	46	sqcli 14141	. . . . . . . . . . 11 ⊢ ((𝑁‘(𝐵𝐺(-i𝑆(i𝑆𝐴))))↑2) ∈ ℂ
48	40, 47	subcli 11532	. . . . . . . . . 10 ⊢ (((𝑁‘(𝐵𝐺(i𝑆(i𝑆𝐴))))↑2) − ((𝑁‘(𝐵𝐺(-i𝑆(i𝑆𝐴))))↑2)) ∈ ℂ
49	4, 48	mulcli 11217	. . . . . . . . 9 ⊢ (i · (((𝑁‘(𝐵𝐺(i𝑆(i𝑆𝐴))))↑2) − ((𝑁‘(𝐵𝐺(-i𝑆(i𝑆𝐴))))↑2))) ∈ ℂ
50	33, 49	addcomi 11401	. . . . . . . 8 ⊢ ((((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺(-1𝑆(i𝑆𝐴))))↑2)) + (i · (((𝑁‘(𝐵𝐺(i𝑆(i𝑆𝐴))))↑2) − ((𝑁‘(𝐵𝐺(-i𝑆(i𝑆𝐴))))↑2)))) = ((i · (((𝑁‘(𝐵𝐺(i𝑆(i𝑆𝐴))))↑2) − ((𝑁‘(𝐵𝐺(-i𝑆(i𝑆𝐴))))↑2))) + (((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺(-1𝑆(i𝑆𝐴))))↑2)))
51	6, 10	nvgcl 29860	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐵𝐺𝐴) ∈ 𝑋)
52	2, 3, 5, 51	mp3an 1461	. . . . . . . . . . . . . 14 ⊢ (𝐵𝐺𝐴) ∈ 𝑋
53	6, 11, 2, 52	nvcli 29902	. . . . . . . . . . . . 13 ⊢ (𝑁‘(𝐵𝐺𝐴)) ∈ ℝ
54	53	recni 11224	. . . . . . . . . . . 12 ⊢ (𝑁‘(𝐵𝐺𝐴)) ∈ ℂ
55	54	sqcli 14141	. . . . . . . . . . 11 ⊢ ((𝑁‘(𝐵𝐺𝐴))↑2) ∈ ℂ
56	6, 7	nvscl 29866	. . . . . . . . . . . . . . . 16 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) ∈ 𝑋)
57	2, 25, 5, 56	mp3an 1461	. . . . . . . . . . . . . . 15 ⊢ (-1𝑆𝐴) ∈ 𝑋
58	6, 10	nvgcl 29860	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ (-1𝑆𝐴) ∈ 𝑋) → (𝐵𝐺(-1𝑆𝐴)) ∈ 𝑋)
59	2, 3, 57, 58	mp3an 1461	. . . . . . . . . . . . . 14 ⊢ (𝐵𝐺(-1𝑆𝐴)) ∈ 𝑋
60	6, 11, 2, 59	nvcli 29902	. . . . . . . . . . . . 13 ⊢ (𝑁‘(𝐵𝐺(-1𝑆𝐴))) ∈ ℝ
61	60	recni 11224	. . . . . . . . . . . 12 ⊢ (𝑁‘(𝐵𝐺(-1𝑆𝐴))) ∈ ℂ
62	61	sqcli 14141	. . . . . . . . . . 11 ⊢ ((𝑁‘(𝐵𝐺(-1𝑆𝐴)))↑2) ∈ ℂ
63	55, 62	subcli 11532	. . . . . . . . . 10 ⊢ (((𝑁‘(𝐵𝐺𝐴))↑2) − ((𝑁‘(𝐵𝐺(-1𝑆𝐴)))↑2)) ∈ ℂ
64	6, 7	nvscl 29866	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈 ∈ NrmCVec ∧ -i ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (-i𝑆𝐴) ∈ 𝑋)
65	2, 16, 5, 64	mp3an 1461	. . . . . . . . . . . . . . . 16 ⊢ (-i𝑆𝐴) ∈ 𝑋
66	6, 10	nvgcl 29860	. . . . . . . . . . . . . . . 16 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ (-i𝑆𝐴) ∈ 𝑋) → (𝐵𝐺(-i𝑆𝐴)) ∈ 𝑋)
67	2, 3, 65, 66	mp3an 1461	. . . . . . . . . . . . . . 15 ⊢ (𝐵𝐺(-i𝑆𝐴)) ∈ 𝑋
68	6, 11, 2, 67	nvcli 29902	. . . . . . . . . . . . . 14 ⊢ (𝑁‘(𝐵𝐺(-i𝑆𝐴))) ∈ ℝ
69	68	recni 11224	. . . . . . . . . . . . 13 ⊢ (𝑁‘(𝐵𝐺(-i𝑆𝐴))) ∈ ℂ
70	69	sqcli 14141	. . . . . . . . . . . 12 ⊢ ((𝑁‘(𝐵𝐺(-i𝑆𝐴)))↑2) ∈ ℂ
71	24, 70	subcli 11532	. . . . . . . . . . 11 ⊢ (((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺(-i𝑆𝐴)))↑2)) ∈ ℂ
72	4, 71	mulcli 11217	. . . . . . . . . 10 ⊢ (i · (((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺(-i𝑆𝐴)))↑2))) ∈ ℂ
73	16, 63, 72	adddii 11222	. . . . . . . . 9 ⊢ (-i · ((((𝑁‘(𝐵𝐺𝐴))↑2) − ((𝑁‘(𝐵𝐺(-1𝑆𝐴)))↑2)) + (i · (((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺(-i𝑆𝐴)))↑2))))) = ((-i · (((𝑁‘(𝐵𝐺𝐴))↑2) − ((𝑁‘(𝐵𝐺(-1𝑆𝐴)))↑2))) + (-i · (i · (((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺(-i𝑆𝐴)))↑2)))))
74	4, 4, 5	3pm3.2i 1339	. . . . . . . . . . . . . . . . . 18 ⊢ (i ∈ ℂ ∧ i ∈ ℂ ∧ 𝐴 ∈ 𝑋)
75	6, 7	nvsass 29868	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑈 ∈ NrmCVec ∧ (i ∈ ℂ ∧ i ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → ((i · i)𝑆𝐴) = (i𝑆(i𝑆𝐴)))
76	2, 74, 75	mp2an 690	. . . . . . . . . . . . . . . . 17 ⊢ ((i · i)𝑆𝐴) = (i𝑆(i𝑆𝐴))
77		ixi 11839	. . . . . . . . . . . . . . . . . 18 ⊢ (i · i) = -1
78	77	oveq1i 7415	. . . . . . . . . . . . . . . . 17 ⊢ ((i · i)𝑆𝐴) = (-1𝑆𝐴)
79	76, 78	eqtr3i 2762	. . . . . . . . . . . . . . . 16 ⊢ (i𝑆(i𝑆𝐴)) = (-1𝑆𝐴)
80	79	oveq2i 7416	. . . . . . . . . . . . . . 15 ⊢ (𝐵𝐺(i𝑆(i𝑆𝐴))) = (𝐵𝐺(-1𝑆𝐴))
81	80	fveq2i 6891	. . . . . . . . . . . . . 14 ⊢ (𝑁‘(𝐵𝐺(i𝑆(i𝑆𝐴)))) = (𝑁‘(𝐵𝐺(-1𝑆𝐴)))
82	81	oveq1i 7415	. . . . . . . . . . . . 13 ⊢ ((𝑁‘(𝐵𝐺(i𝑆(i𝑆𝐴))))↑2) = ((𝑁‘(𝐵𝐺(-1𝑆𝐴)))↑2)
83	4, 4	mulneg1i 11656	. . . . . . . . . . . . . . . . . . 19 ⊢ (-i · i) = -(i · i)
84	77	negeqi 11449	. . . . . . . . . . . . . . . . . . 19 ⊢ -(i · i) = --1
85		negneg1e1 12326	. . . . . . . . . . . . . . . . . . 19 ⊢ --1 = 1
86	83, 84, 85	3eqtri 2764	. . . . . . . . . . . . . . . . . 18 ⊢ (-i · i) = 1
87	86	oveq1i 7415	. . . . . . . . . . . . . . . . 17 ⊢ ((-i · i)𝑆𝐴) = (1𝑆𝐴)
88	16, 4, 5	3pm3.2i 1339	. . . . . . . . . . . . . . . . . 18 ⊢ (-i ∈ ℂ ∧ i ∈ ℂ ∧ 𝐴 ∈ 𝑋)
89	6, 7	nvsass 29868	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑈 ∈ NrmCVec ∧ (-i ∈ ℂ ∧ i ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → ((-i · i)𝑆𝐴) = (-i𝑆(i𝑆𝐴)))
90	2, 88, 89	mp2an 690	. . . . . . . . . . . . . . . . 17 ⊢ ((-i · i)𝑆𝐴) = (-i𝑆(i𝑆𝐴))
91	6, 7	nvsid 29867	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴)
92	2, 5, 91	mp2an 690	. . . . . . . . . . . . . . . . 17 ⊢ (1𝑆𝐴) = 𝐴
93	87, 90, 92	3eqtr3i 2768	. . . . . . . . . . . . . . . 16 ⊢ (-i𝑆(i𝑆𝐴)) = 𝐴
94	93	oveq2i 7416	. . . . . . . . . . . . . . 15 ⊢ (𝐵𝐺(-i𝑆(i𝑆𝐴))) = (𝐵𝐺𝐴)
95	94	fveq2i 6891	. . . . . . . . . . . . . 14 ⊢ (𝑁‘(𝐵𝐺(-i𝑆(i𝑆𝐴)))) = (𝑁‘(𝐵𝐺𝐴))
96	95	oveq1i 7415	. . . . . . . . . . . . 13 ⊢ ((𝑁‘(𝐵𝐺(-i𝑆(i𝑆𝐴))))↑2) = ((𝑁‘(𝐵𝐺𝐴))↑2)
97	82, 96	oveq12i 7417	. . . . . . . . . . . 12 ⊢ (((𝑁‘(𝐵𝐺(i𝑆(i𝑆𝐴))))↑2) − ((𝑁‘(𝐵𝐺(-i𝑆(i𝑆𝐴))))↑2)) = (((𝑁‘(𝐵𝐺(-1𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺𝐴))↑2))
98	97	oveq2i 7416	. . . . . . . . . . 11 ⊢ (i · (((𝑁‘(𝐵𝐺(i𝑆(i𝑆𝐴))))↑2) − ((𝑁‘(𝐵𝐺(-i𝑆(i𝑆𝐴))))↑2))) = (i · (((𝑁‘(𝐵𝐺(-1𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺𝐴))↑2)))
99	63	mulm1i 11655	. . . . . . . . . . . . . 14 ⊢ (-1 · (((𝑁‘(𝐵𝐺𝐴))↑2) − ((𝑁‘(𝐵𝐺(-1𝑆𝐴)))↑2))) = -(((𝑁‘(𝐵𝐺𝐴))↑2) − ((𝑁‘(𝐵𝐺(-1𝑆𝐴)))↑2))
100	55, 62	negsubdi2i 11542	. . . . . . . . . . . . . 14 ⊢ -(((𝑁‘(𝐵𝐺𝐴))↑2) − ((𝑁‘(𝐵𝐺(-1𝑆𝐴)))↑2)) = (((𝑁‘(𝐵𝐺(-1𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺𝐴))↑2))
101	99, 100	eqtr2i 2761	. . . . . . . . . . . . 13 ⊢ (((𝑁‘(𝐵𝐺(-1𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺𝐴))↑2)) = (-1 · (((𝑁‘(𝐵𝐺𝐴))↑2) − ((𝑁‘(𝐵𝐺(-1𝑆𝐴)))↑2)))
102	101	oveq2i 7416	. . . . . . . . . . . 12 ⊢ (i · (((𝑁‘(𝐵𝐺(-1𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺𝐴))↑2))) = (i · (-1 · (((𝑁‘(𝐵𝐺𝐴))↑2) − ((𝑁‘(𝐵𝐺(-1𝑆𝐴)))↑2))))
103	4, 25, 63	mulassi 11221	. . . . . . . . . . . 12 ⊢ ((i · -1) · (((𝑁‘(𝐵𝐺𝐴))↑2) − ((𝑁‘(𝐵𝐺(-1𝑆𝐴)))↑2))) = (i · (-1 · (((𝑁‘(𝐵𝐺𝐴))↑2) − ((𝑁‘(𝐵𝐺(-1𝑆𝐴)))↑2))))
104	102, 103	eqtr4i 2763	. . . . . . . . . . 11 ⊢ (i · (((𝑁‘(𝐵𝐺(-1𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺𝐴))↑2))) = ((i · -1) · (((𝑁‘(𝐵𝐺𝐴))↑2) − ((𝑁‘(𝐵𝐺(-1𝑆𝐴)))↑2)))
105	4	mulm1i 11655	. . . . . . . . . . . . 13 ⊢ (-1 · i) = -i
106	25, 4, 105	mulcomli 11219	. . . . . . . . . . . 12 ⊢ (i · -1) = -i
107	106	oveq1i 7415	. . . . . . . . . . 11 ⊢ ((i · -1) · (((𝑁‘(𝐵𝐺𝐴))↑2) − ((𝑁‘(𝐵𝐺(-1𝑆𝐴)))↑2))) = (-i · (((𝑁‘(𝐵𝐺𝐴))↑2) − ((𝑁‘(𝐵𝐺(-1𝑆𝐴)))↑2)))
108	98, 104, 107	3eqtri 2764	. . . . . . . . . 10 ⊢ (i · (((𝑁‘(𝐵𝐺(i𝑆(i𝑆𝐴))))↑2) − ((𝑁‘(𝐵𝐺(-i𝑆(i𝑆𝐴))))↑2))) = (-i · (((𝑁‘(𝐵𝐺𝐴))↑2) − ((𝑁‘(𝐵𝐺(-1𝑆𝐴)))↑2)))
109	25, 4, 5	3pm3.2i 1339	. . . . . . . . . . . . . . . . . . 19 ⊢ (-1 ∈ ℂ ∧ i ∈ ℂ ∧ 𝐴 ∈ 𝑋)
110	6, 7	nvsass 29868	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑈 ∈ NrmCVec ∧ (-1 ∈ ℂ ∧ i ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → ((-1 · i)𝑆𝐴) = (-1𝑆(i𝑆𝐴)))
111	2, 109, 110	mp2an 690	. . . . . . . . . . . . . . . . . 18 ⊢ ((-1 · i)𝑆𝐴) = (-1𝑆(i𝑆𝐴))
112	105	oveq1i 7415	. . . . . . . . . . . . . . . . . 18 ⊢ ((-1 · i)𝑆𝐴) = (-i𝑆𝐴)
113	111, 112	eqtr3i 2762	. . . . . . . . . . . . . . . . 17 ⊢ (-1𝑆(i𝑆𝐴)) = (-i𝑆𝐴)
114	113	oveq2i 7416	. . . . . . . . . . . . . . . 16 ⊢ (𝐵𝐺(-1𝑆(i𝑆𝐴))) = (𝐵𝐺(-i𝑆𝐴))
115	114	fveq2i 6891	. . . . . . . . . . . . . . 15 ⊢ (𝑁‘(𝐵𝐺(-1𝑆(i𝑆𝐴)))) = (𝑁‘(𝐵𝐺(-i𝑆𝐴)))
116	115	oveq1i 7415	. . . . . . . . . . . . . 14 ⊢ ((𝑁‘(𝐵𝐺(-1𝑆(i𝑆𝐴))))↑2) = ((𝑁‘(𝐵𝐺(-i𝑆𝐴)))↑2)
117	116	oveq2i 7416	. . . . . . . . . . . . 13 ⊢ (((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺(-1𝑆(i𝑆𝐴))))↑2)) = (((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺(-i𝑆𝐴)))↑2))
118	71	mullidi 11215	. . . . . . . . . . . . 13 ⊢ (1 · (((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺(-i𝑆𝐴)))↑2))) = (((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺(-i𝑆𝐴)))↑2))
119	117, 118	eqtr4i 2763	. . . . . . . . . . . 12 ⊢ (((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺(-1𝑆(i𝑆𝐴))))↑2)) = (1 · (((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺(-i𝑆𝐴)))↑2)))
120	86	oveq1i 7415	. . . . . . . . . . . 12 ⊢ ((-i · i) · (((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺(-i𝑆𝐴)))↑2))) = (1 · (((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺(-i𝑆𝐴)))↑2)))
121	119, 120	eqtr4i 2763	. . . . . . . . . . 11 ⊢ (((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺(-1𝑆(i𝑆𝐴))))↑2)) = ((-i · i) · (((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺(-i𝑆𝐴)))↑2)))
122	16, 4, 71	mulassi 11221	. . . . . . . . . . 11 ⊢ ((-i · i) · (((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺(-i𝑆𝐴)))↑2))) = (-i · (i · (((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺(-i𝑆𝐴)))↑2))))
123	121, 122	eqtri 2760	. . . . . . . . . 10 ⊢ (((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺(-1𝑆(i𝑆𝐴))))↑2)) = (-i · (i · (((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺(-i𝑆𝐴)))↑2))))
124	108, 123	oveq12i 7417	. . . . . . . . 9 ⊢ ((i · (((𝑁‘(𝐵𝐺(i𝑆(i𝑆𝐴))))↑2) − ((𝑁‘(𝐵𝐺(-i𝑆(i𝑆𝐴))))↑2))) + (((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺(-1𝑆(i𝑆𝐴))))↑2))) = ((-i · (((𝑁‘(𝐵𝐺𝐴))↑2) − ((𝑁‘(𝐵𝐺(-1𝑆𝐴)))↑2))) + (-i · (i · (((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺(-i𝑆𝐴)))↑2)))))
125	73, 124	eqtr4i 2763	. . . . . . . 8 ⊢ (-i · ((((𝑁‘(𝐵𝐺𝐴))↑2) − ((𝑁‘(𝐵𝐺(-1𝑆𝐴)))↑2)) + (i · (((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺(-i𝑆𝐴)))↑2))))) = ((i · (((𝑁‘(𝐵𝐺(i𝑆(i𝑆𝐴))))↑2) − ((𝑁‘(𝐵𝐺(-i𝑆(i𝑆𝐴))))↑2))) + (((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺(-1𝑆(i𝑆𝐴))))↑2)))
126	50, 125	eqtr4i 2763	. . . . . . 7 ⊢ ((((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺(-1𝑆(i𝑆𝐴))))↑2)) + (i · (((𝑁‘(𝐵𝐺(i𝑆(i𝑆𝐴))))↑2) − ((𝑁‘(𝐵𝐺(-i𝑆(i𝑆𝐴))))↑2)))) = (-i · ((((𝑁‘(𝐵𝐺𝐴))↑2) − ((𝑁‘(𝐵𝐺(-1𝑆𝐴)))↑2)) + (i · (((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺(-i𝑆𝐴)))↑2)))))
127	6, 10, 7, 11, 12	4ipval2 29948	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (4 · (𝐵𝑃𝐴)) = ((((𝑁‘(𝐵𝐺𝐴))↑2) − ((𝑁‘(𝐵𝐺(-1𝑆𝐴)))↑2)) + (i · (((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺(-i𝑆𝐴)))↑2)))))
128	2, 3, 5, 127	mp3an 1461	. . . . . . . 8 ⊢ (4 · (𝐵𝑃𝐴)) = ((((𝑁‘(𝐵𝐺𝐴))↑2) − ((𝑁‘(𝐵𝐺(-1𝑆𝐴)))↑2)) + (i · (((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺(-i𝑆𝐴)))↑2))))
129	128	oveq2i 7416	. . . . . . 7 ⊢ (-i · (4 · (𝐵𝑃𝐴))) = (-i · ((((𝑁‘(𝐵𝐺𝐴))↑2) − ((𝑁‘(𝐵𝐺(-1𝑆𝐴)))↑2)) + (i · (((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺(-i𝑆𝐴)))↑2)))))
130	126, 129	eqtr4i 2763	. . . . . 6 ⊢ ((((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺(-1𝑆(i𝑆𝐴))))↑2)) + (i · (((𝑁‘(𝐵𝐺(i𝑆(i𝑆𝐴))))↑2) − ((𝑁‘(𝐵𝐺(-i𝑆(i𝑆𝐴))))↑2)))) = (-i · (4 · (𝐵𝑃𝐴)))
131	19, 130	eqtr4i 2763	. . . . 5 ⊢ (4 · (-i · (𝐵𝑃𝐴))) = ((((𝑁‘(𝐵𝐺(i𝑆𝐴)))↑2) − ((𝑁‘(𝐵𝐺(-1𝑆(i𝑆𝐴))))↑2)) + (i · (((𝑁‘(𝐵𝐺(i𝑆(i𝑆𝐴))))↑2) − ((𝑁‘(𝐵𝐺(-i𝑆(i𝑆𝐴))))↑2))))
132	14, 131	eqtr4i 2763	. . . 4 ⊢ (4 · (𝐵𝑃(i𝑆𝐴))) = (4 · (-i · (𝐵𝑃𝐴)))
133	6, 12	dipcl 29952	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ (i𝑆𝐴) ∈ 𝑋) → (𝐵𝑃(i𝑆𝐴)) ∈ ℂ)
134	2, 3, 9, 133	mp3an 1461	. . . . 5 ⊢ (𝐵𝑃(i𝑆𝐴)) ∈ ℂ
135	16, 18	mulcli 11217	. . . . 5 ⊢ (-i · (𝐵𝑃𝐴)) ∈ ℂ
136		4ne0 12316	. . . . 5 ⊢ 4 ≠ 0
137	134, 135, 15, 136	mulcani 11849	. . . 4 ⊢ ((4 · (𝐵𝑃(i𝑆𝐴))) = (4 · (-i · (𝐵𝑃𝐴))) ↔ (𝐵𝑃(i𝑆𝐴)) = (-i · (𝐵𝑃𝐴)))
138	132, 137	mpbi 229	. . 3 ⊢ (𝐵𝑃(i𝑆𝐴)) = (-i · (𝐵𝑃𝐴))
139	138	fveq2i 6891	. 2 ⊢ (∗‘(𝐵𝑃(i𝑆𝐴))) = (∗‘(-i · (𝐵𝑃𝐴)))
140	6, 12	dipcj 29954	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ (i𝑆𝐴) ∈ 𝑋) → (∗‘(𝐵𝑃(i𝑆𝐴))) = ((i𝑆𝐴)𝑃𝐵))
141	2, 3, 9, 140	mp3an 1461	. 2 ⊢ (∗‘(𝐵𝑃(i𝑆𝐴))) = ((i𝑆𝐴)𝑃𝐵)
142	16, 18	cjmuli 15132	. . 3 ⊢ (∗‘(-i · (𝐵𝑃𝐴))) = ((∗‘-i) · (∗‘(𝐵𝑃𝐴)))
143	25, 4	cjmuli 15132	. . . . 5 ⊢ (∗‘(-1 · i)) = ((∗‘-1) · (∗‘i))
144	105	fveq2i 6891	. . . . 5 ⊢ (∗‘(-1 · i)) = (∗‘-i)
145		neg1rr 12323	. . . . . . . 8 ⊢ -1 ∈ ℝ
146	25	cjrebi 15117	. . . . . . . 8 ⊢ (-1 ∈ ℝ ↔ (∗‘-1) = -1)
147	145, 146	mpbi 229	. . . . . . 7 ⊢ (∗‘-1) = -1
148		cji 15102	. . . . . . 7 ⊢ (∗‘i) = -i
149	147, 148	oveq12i 7417	. . . . . 6 ⊢ ((∗‘-1) · (∗‘i)) = (-1 · -i)
150		ax-1cn 11164	. . . . . . 7 ⊢ 1 ∈ ℂ
151	150, 4	mul2negi 11658	. . . . . 6 ⊢ (-1 · -i) = (1 · i)
152	4	mullidi 11215	. . . . . 6 ⊢ (1 · i) = i
153	149, 151, 152	3eqtri 2764	. . . . 5 ⊢ ((∗‘-1) · (∗‘i)) = i
154	143, 144, 153	3eqtr3i 2768	. . . 4 ⊢ (∗‘-i) = i
155	6, 12	dipcj 29954	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (∗‘(𝐵𝑃𝐴)) = (𝐴𝑃𝐵))
156	2, 3, 5, 155	mp3an 1461	. . . 4 ⊢ (∗‘(𝐵𝑃𝐴)) = (𝐴𝑃𝐵)
157	154, 156	oveq12i 7417	. . 3 ⊢ ((∗‘-i) · (∗‘(𝐵𝑃𝐴))) = (i · (𝐴𝑃𝐵))
158	142, 157	eqtri 2760	. 2 ⊢ (∗‘(-i · (𝐵𝑃𝐴))) = (i · (𝐴𝑃𝐵))
159	139, 141, 158	3eqtr3i 2768	1 ⊢ ((i𝑆𝐴)𝑃𝐵) = (i · (𝐴𝑃𝐵))

Colors of variables: wff setvar class

Syntax hints: ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 ℝcr 11105 1c1 11107 ici 11108 + caddc 11109 · cmul 11111 − cmin 11440 -cneg 11441 2c2 12263 4c4 12265 ↑cexp 14023 ∗ccj 15039 NrmCVeccnv 29824 +_𝑣 cpv 29825 BaseSetcba 29826 ·_𝑠OLD cns 29827 norm_CVcnmcv 29830 ·_𝑖OLDcdip 29940 CPreHil_OLDccphlo 30052

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-sum 15629 df-grpo 29733 df-gid 29734 df-ginv 29735 df-ablo 29785 df-vc 29799 df-nv 29832 df-va 29835 df-ba 29836 df-sm 29837 df-0v 29838 df-nmcv 29840 df-dip 29941 df-ph 30053

This theorem is referenced by: ipasslem11 30080

W3C validator