Hilbert Space Explorer

Hilbert Space Explorer

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GIF version

Theorem h1de2 5458

Description: Membership in 1-dimensional subspace. All members are collinear with the generating vector.

**Hypotheses**
Ref	Expression
h1de2.1	⊢ A ∈ ℋ
h1de2.2	⊢ B ∈ ℋ

**Assertion**
Ref	Expression
h1de2	⊢ (A ∈ (⊥ ‘(⊥ ‘{B})) → ((B ·_i B) ·_s A) = ((A ·_i B) ·_s B))

**Proof of Theorem h1de2**
Step	Hyp	Ref	Expression
1		h1de2.2	. . . . . . . . . 10 ⊢ B ∈ ℋ
2	1, 1	hicl 5044	. . . . . . . . 9 ⊢ (B ·_i B) ∈ ℂ
3		h1de2.1	. . . . . . . . 9 ⊢ A ∈ ℋ
4	2, 3	hvmulcl 4990	. . . . . . . 8 ⊢ ((B ·_i B) ·_s A) ∈ ℋ
5	3, 1	hicl 5044	. . . . . . . . 9 ⊢ (A ·_i B) ∈ ℂ
6	5, 1	hvmulcl 4990	. . . . . . . 8 ⊢ ((A ·_i B) ·_s B) ∈ ℋ
7		his2subt 5052	. . . . . . . 8 ⊢ ((((B ·_i B) ·_s A) ∈ ℋ ∧ ((A ·_i B) ·_s B) ∈ ℋ ∧ B ∈ ℋ ) → ((((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B)) ·_i B) = ((((B ·_i B) ·_s A) ·_i B) − (((A ·_i B) ·_s B) ·_i B)))
8	4, 6, 1, 7	mp3an 642	. . . . . . 7 ⊢ ((((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B)) ·_i B) = ((((B ·_i B) ·_s A) ·_i B) − (((A ·_i B) ·_s B) ·_i B))
9	2, 5	mulcom 4107	. . . . . . . . 9 ⊢ ((B ·_i B) · (A ·_i B)) = ((A ·_i B) · (B ·_i B))
10		ax-his3 5047	. . . . . . . . . 10 ⊢ (((B ·_i B) ∈ ℂ ∧ A ∈ ℋ ∧ B ∈ ℋ ) → (((B ·_i B) ·_s A) ·_i B) = ((B ·_i B) · (A ·_i B)))
11	2, 3, 1, 10	mp3an 642	. . . . . . . . 9 ⊢ (((B ·_i B) ·_s A) ·_i B) = ((B ·_i B) · (A ·_i B))
12		ax-his3 5047	. . . . . . . . . 10 ⊢ (((A ·_i B) ∈ ℂ ∧ B ∈ ℋ ∧ B ∈ ℋ ) → (((A ·_i B) ·_s B) ·_i B) = ((A ·_i B) · (B ·_i B)))
13	5, 1, 1, 12	mp3an 642	. . . . . . . . 9 ⊢ (((A ·_i B) ·_s B) ·_i B) = ((A ·_i B) · (B ·_i B))
14	9, 11, 13	3eqtr4 1126	. . . . . . . 8 ⊢ (((B ·_i B) ·_s A) ·_i B) = (((A ·_i B) ·_s B) ·_i B)
15	4, 1	hicl 5044	. . . . . . . . 9 ⊢ (((B ·_i B) ·_s A) ·_i B) ∈ ℂ
16	6, 1	hicl 5044	. . . . . . . . 9 ⊢ (((A ·_i B) ·_s B) ·_i B) ∈ ℂ
17	15, 16	subeq0 4163	. . . . . . . 8 ⊢ (((((B ·_i B) ·_s A) ·_i B) − (((A ·_i B) ·_s B) ·_i B)) = 0 ↔ (((B ·_i B) ·_s A) ·_i B) = (((A ·_i B) ·_s B) ·_i B))
18	14, 17	mpbir 165	. . . . . . 7 ⊢ ((((B ·_i B) ·_s A) ·_i B) − (((A ·_i B) ·_s B) ·_i B)) = 0
19	8, 18	eqtr 1119	. . . . . 6 ⊢ ((((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B)) ·_i B) = 0
20	1	h1det 5455	. . . . . . . . 9 ⊢ (A ∈ (⊥ ‘(⊥ ‘{B})) ↔ (A ∈ ℋ ∧ ∀x ∈ ℋ ((B ·_i x) = 0 → (A ·_i x) = 0)))
21	20, 3	mpbiran 547	. . . . . . . 8 ⊢ (A ∈ (⊥ ‘(⊥ ‘{B})) ↔ ∀x ∈ ℋ ((B ·_i x) = 0 → (A ·_i x) = 0))
22	4, 6	hvsubcl 5002	. . . . . . . . 9 ⊢ (((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B)) ∈ ℋ
23		opreq2 3007	. . . . . . . . . . . 12 ⊢ (x = (((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B)) → (B ·_i x) = (B ·_i (((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B))))
24	23	cleq1d 1109	. . . . . . . . . . 11 ⊢ (x = (((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B)) → ((B ·_i x) = 0 ↔ (B ·_i (((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B))) = 0))
25		opreq2 3007	. . . . . . . . . . . 12 ⊢ (x = (((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B)) → (A ·_i x) = (A ·_i (((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B))))
26	25	cleq1d 1109	. . . . . . . . . . 11 ⊢ (x = (((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B)) → ((A ·_i x) = 0 ↔ (A ·_i (((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B))) = 0))
27	24, 26	imbi12d 474	. . . . . . . . . 10 ⊢ (x = (((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B)) → (((B ·_i x) = 0 → (A ·_i x) = 0) ↔ ((B ·_i (((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B))) = 0 → (A ·_i (((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B))) = 0)))
28	27	rcla4v 1402	. . . . . . . . 9 ⊢ (∀x ∈ ℋ ((B ·_i x) = 0 → (A ·_i x) = 0) → ((((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B)) ∈ ℋ → ((B ·_i (((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B))) = 0 → (A ·_i (((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B))) = 0)))
29	22, 28	mpi 44	. . . . . . . 8 ⊢ (∀x ∈ ℋ ((B ·_i x) = 0 → (A ·_i x) = 0) → ((B ·_i (((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B))) = 0 → (A ·_i (((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B))) = 0))
30	21, 29	sylbi 174	. . . . . . 7 ⊢ (A ∈ (⊥ ‘(⊥ ‘{B})) → ((B ·_i (((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B))) = 0 → (A ·_i (((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B))) = 0))
31		orthcom 5061	. . . . . . . 8 ⊢ (((((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B)) ∈ ℋ ∧ B ∈ ℋ ) → (((((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B)) ·_i B) = 0 ↔ (B ·_i (((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B))) = 0))
32	22, 1, 31	mp2an 520	. . . . . . 7 ⊢ (((((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B)) ·_i B) = 0 ↔ (B ·_i (((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B))) = 0)
33		orthcom 5061	. . . . . . . 8 ⊢ (((((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B)) ∈ ℋ ∧ A ∈ ℋ ) → (((((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B)) ·_i A) = 0 ↔ (A ·_i (((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B))) = 0))
34	22, 3, 33	mp2an 520	. . . . . . 7 ⊢ (((((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B)) ·_i A) = 0 ↔ (A ·_i (((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B))) = 0)
35	30, 32, 34	3imtr4g 426	. . . . . 6 ⊢ (A ∈ (⊥ ‘(⊥ ‘{B})) → (((((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B)) ·_i B) = 0 → ((((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B)) ·_i A) = 0))
36	19, 35	mpi 44	. . . . 5 ⊢ (A ∈ (⊥ ‘(⊥ ‘{B})) → ((((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B)) ·_i A) = 0)
37		his2subt 5052	. . . . . . 7 ⊢ ((((B ·_i B) ·_s A) ∈ ℋ ∧ ((A ·_i B) ·_s B) ∈ ℋ ∧ A ∈ ℋ ) → ((((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B)) ·_i A) = ((((B ·_i B) ·_s A) ·_i A) − (((A ·_i B) ·_s B) ·_i A)))
38	4, 6, 3, 37	mp3an 642	. . . . . 6 ⊢ ((((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B)) ·_i A) = ((((B ·_i B) ·_s A) ·_i A) − (((A ·_i B) ·_s B) ·_i A))
39		ax-his3 5047	. . . . . . . . 9 ⊢ (((B ·_i B) ∈ ℂ ∧ A ∈ ℋ ∧ A ∈ ℋ ) → (((B ·_i B) ·_s A) ·_i A) = ((B ·_i B) · (A ·_i A)))
40	2, 3, 3, 39	mp3an 642	. . . . . . . 8 ⊢ (((B ·_i B) ·_s A) ·_i A) = ((B ·_i B) · (A ·_i A))
41	3, 3	hicl 5044	. . . . . . . . 9 ⊢ (A ·_i A) ∈ ℂ
42	2, 41	mulcom 4107	. . . . . . . 8 ⊢ ((B ·_i B) · (A ·_i A)) = ((A ·_i A) · (B ·_i B))
43	40, 42	eqtr 1119	. . . . . . 7 ⊢ (((B ·_i B) ·_s A) ·_i A) = ((A ·_i A) · (B ·_i B))
44		ax-his3 5047	. . . . . . . 8 ⊢ (((A ·_i B) ∈ ℂ ∧ B ∈ ℋ ∧ A ∈ ℋ ) → (((A ·_i B) ·_s B) ·_i A) = ((A ·_i B) · (B ·_i A)))
45	5, 1, 3, 44	mp3an 642	. . . . . . 7 ⊢ (((A ·_i B) ·_s B) ·_i A) = ((A ·_i B) · (B ·_i A))
46	43, 45	opreq12i 3011	. . . . . 6 ⊢ ((((B ·_i B) ·_s A) ·_i A) − (((A ·_i B) ·_s B) ·_i A)) = (((A ·_i A) · (B ·_i B)) − ((A ·_i B) · (B ·_i A)))
47	38, 46	eqtr2 1120	. . . . 5 ⊢ (((A ·_i A) · (B ·_i B)) − ((A ·_i B) · (B ·_i A))) = ((((B ·_i B) ·_s A) −_v ((A ·_i B) ·_s B)) ·_i A)
48	36, 47	syl5eq 1136	. . . 4 ⊢ (A ∈ (⊥ ‘(⊥ ‘{B})) → (((A ·_i A) · (B ·_i B)) − ((A ·_i B) · (B ·_i A))) = 0)
49	41, 2	mulcl 4105	. . . . 5 ⊢ ((A ·_i A) · (B ·_i B)) ∈ ℂ
50	1, 3	hicl 5044	. . . . . 6 ⊢ (B ·_i A) ∈ ℂ
51	5, 50	mulcl 4105	. . . . 5 ⊢ ((A ·_i B) · (B ·_i A)) ∈ ℂ
52	49, 51	subeq0 4163	. . . 4 ⊢ ((((A ·_i A) · (B ·_i B)) − ((A ·_i B) · (B ·_i A))) = 0 ↔ ((A ·_i A) · (B ·_i B)) = ((A ·_i B) · (B ·_i A)))
53	48, 52	sylib 173	. . 3 ⊢ (A ∈ (⊥ ‘(⊥ ‘{B})) → ((A ·_i A) · (B ·_i B)) = ((A ·_i B) · (B ·_i A)))
54	53	cleqcomd 1106	. 2 ⊢ (A ∈ (⊥ ‘(⊥ ‘{B})) → ((A ·_i B) · (B ·_i A)) = ((A ·_i A) · (B ·_i B)))
55	3, 1	bcseq 5073	. 2 ⊢ (((A ·_i B) · (B ·_i A)) = ((A ·_i A) · (B ·_i B)) ↔ ((B ·_i B) ·_s A) = ((A ·_i B) ·_s B))
56	54, 55	sylib 173	1 ⊢ (A ∈ (⊥ ‘(⊥ ‘{B})) → ((B ·_i B) ·_s A) = ((A ·_i B) ·_s B))

Colors of variables: wff set class

Syntax hints: → wi 2 ↔ wb 127 = wceq 1091 ∈ wcel 1092 ∀wral 1201 {csn 1808 ‘cfv 2422 (class class class)co 3001 ℂcc 4026 0cc0 4028 · cmulc 4032 − cmin 4089 ℋ chil 4958 ·_s csm 4960 −_v cmv 4962 ·_i csp 4963 ⊥cort 4969

This theorem is referenced by: h1de2b 5459

This theorem was proved from axioms: ax-1 3 ax-2 4 ax-3 5 ax-mp 6 ax-4 673 ax-5 674 ax-6 675 ax-7 676 ax-gen 677 ax-8 798 ax-9 799 ax-10 800 ax-11 801 ax-12 802 ax-13 804 ax-14 805 ax-16 922 ax-17 925 ax-ext 1074 ax-rep 1075 ax-un 1076 ax-pow 1077 ax-reg 1078 ax-inf 1079 ax-hilex 4983 ax-hvaddcl 4984 ax-hvcom 4985 ax-hvass 4986 ax-hvzercl 4987 ax-hvaddid 4988 ax-hvmulcl 4989 ax-hvmulid 4991 ax-hvmulass 4992 ax-hvdistr2 4994 ax-hvmulzer 4995 ax-hicl 5043 ax-his1 5045 ax-his2 5046 ax-his3 5047 ax-his4 5048

This theorem depends on definitions: df-bi 128 df-or 197 df-an 198 df-3or 582 df-3an 583 df-ex 679 df-sb 853 df-eu 1009 df-mo 1010 df-clab 1093 df-cleq 1097 df-clel 1099 df-ne 1192 df-ral 1205 df-rex 1206 df-reu 1207 df-rab 1208 df-v 1349 df-sbc 1441 df-dif 1489 df-un 1490 df-in 1491 df-ss 1492 df-pss 1494 df-nul 1708 df-if 1777 df-pw 1799 df-sn 1811 df-pr 1812 df-tp 1814 df-op 1815 df-uni 1920 df-int 1966 df-iun 1996 df-tr 2042 df-br 2063 df-opab 2098 df-eprel 2122 df-id 2125 df-po 2128 df-so 2138 df-sup 2154 df-fr 2169 df-we 2186 df-ord 2202 df-on 2203 df-lim 2204 df-suc 2205 df-om 2373 df-xp 2424 df-rel 2425 df-cnv 2426 df-co 2427 df-dm 2428 df-rn 2429 df-res 2430 df-ima 2431 df-fun 2432 df-fn 2433 df-f 2434 df-f1 2435 df-fo 2436 df-f1o 2437 df-fv 2438 df-rdg 2970 df-opr 3003 df-oprab 3004 df-1st 3087 df-2nd 3088 df-1o 3104 df-oadd 3106 df-omul 3107 df-er 3200 df-ec 3202 df-qs 3205 df-ni 3794 df-pli 3795 df-mi 3796 df-lti 3797 df-plpq 3829 df-mpq 3830 df-enq 3831 df-nq 3832 df-plq 3833 df-mq 3834 df-rq 3835 df-ltq 3836 df-1q 3837 df-np 3880 df-1p 3881 df-plp 3882 df-mp 3883 df-ltp 3884 df-plpr 3958 df-mpr 3959 df-enr 3960 df-nr 3961 df-plr 3962 df-mr 3963 df-ltr 3964 df-0r 3965 df-1r 3966 df-m1r 3967 df-c 4034 df-0 4035 df-1 4036 df-i 4037 df-r 4038 df-plus 4039 df-mul 4040 df-lt 4041 df-sub 4133 df-neg 4135 df-div 4216 df-le 4277 df-n 4423 df-2 4462 df-3 4463 df-4 4464 df-n0 4535 df-z 4564 df-seq 4661 df-exp 4676 df-sqr 4728 df-re 4790 df-im 4791 df-cj 4792 df-abs 4793 df-clim 4876 df-hvsub 4996 df-hnorm 5074 df-hlim 5107 df-sh 5114 df-ch 5127 df-oc 5156

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