Theorem h1de2b 9432

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

Hypotheses

Ref	Expression
h1de2.1	|- A e. H~
h1de2.2	|- B e. H~

Assertion

Ref	Expression
h1de2b	|- (B =/= 0h -> (A e. (_|_` (_|_` {B})) <-> A = (((A .ih B) / (B .ih B)) .h B)))

Proof of Theorem h1de2b

Step	Hyp	Ref	Expression
1		h1de2.2	. . . 4 |- B e. H~
2		his6t 8920	. . . 4 |- (B e. H~ -> ((B .ih B) = 0 <-> B = 0h))
3	1, 2	ax-mp 7	. . 3 |- ((B .ih B) = 0 <-> B = 0h)
4	3	necon3bii 1596	. 2 |- ((B .ih B) =/= 0 <-> B =/= 0h)
5		h1de2.1	. . . . . . . 8 |- A e. H~
6	5, 1	h1de2 9431	. . . . . . 7 |- (A e. (_|_` (_|_` {B})) -> ((B .ih B) .h A) = ((A .ih B) .h B))
7	6	adantl 388	. . . . . 6 |- (((B .ih B) =/= 0 /\ A e. (_|_` (_|_` {B}))) -> ((B .ih B) .h A) = ((A .ih B) .h B))
8	7	opreq2d 3971	. . . . 5 |- (((B .ih B) =/= 0 /\ A e. (_|_` (_|_` {B}))) -> ((1 / (B .ih B)) .h ((B .ih B) .h A)) = ((1 / (B .ih B)) .h ((A .ih B) .h B)))
9	1, 1	hicl 8903	. . . . . . . . . 10 |- (B .ih B) e. CC
10	9	recclz 5693	. . . . . . . . 9 |- ((B .ih B) =/= 0 -> (1 / (B .ih B)) e. CC)
11		ax-hvmulass 8832	. . . . . . . . . 10 |- (((1 / (B .ih B)) e. CC /\ (B .ih B) e. CC /\ A e. H~) -> (((1 / (B .ih B)) x. (B .ih B)) .h A) = ((1 / (B .ih B)) .h ((B .ih B) .h A)))
12	9, 5, 11	mp3an23 907	. . . . . . . . 9 |- ((1 / (B .ih B)) e. CC -> (((1 / (B .ih B)) x. (B .ih B)) .h A) = ((1 / (B .ih B)) .h ((B .ih B) .h A)))
13	10, 12	syl 10	. . . . . . . 8 |- ((B .ih B) =/= 0 -> (((1 / (B .ih B)) x. (B .ih B)) .h A) = ((1 / (B .ih B)) .h ((B .ih B) .h A)))
14		ax1cn 5252	. . . . . . . . . 10 |- 1 e. CC
15	9, 14	divcan1z 5697	. . . . . . . . 9 |- ((B .ih B) =/= 0 -> ((1 / (B .ih B)) x. (B .ih B)) = 1)
16	15	opreq1d 3970	. . . . . . . 8 |- ((B .ih B) =/= 0 -> (((1 / (B .ih B)) x. (B .ih B)) .h A) = (1 .h A))
17	13, 16	eqtr3d 1507	. . . . . . 7 |- ((B .ih B) =/= 0 -> ((1 / (B .ih B)) .h ((B .ih B) .h A)) = (1 .h A))
18		ax-hvmulid 8831	. . . . . . . 8 |- (A e. H~ -> (1 .h A) = A)
19	5, 18	ax-mp 7	. . . . . . 7 |- (1 .h A) = A
20	17, 19	syl6eq 1521	. . . . . 6 |- ((B .ih B) =/= 0 -> ((1 / (B .ih B)) .h ((B .ih B) .h A)) = A)
21	20	adantr 389	. . . . 5 |- (((B .ih B) =/= 0 /\ A e. (_|_` (_|_` {B}))) -> ((1 / (B .ih B)) .h ((B .ih B) .h A)) = A)
22	5, 1	hicl 8903	. . . . . . . . 9 |- (A .ih B) e. CC
23		ax-hvmulass 8832	. . . . . . . . 9 |- (((1 / (B .ih B)) e. CC /\ (A .ih B) e. CC /\ B e. H~) -> (((1 / (B .ih B)) x. (A .ih B)) .h B) = ((1 / (B .ih B)) .h ((A .ih B) .h B)))
24	22, 1, 23	mp3an23 907	. . . . . . . 8 |- ((1 / (B .ih B)) e. CC -> (((1 / (B .ih B)) x. (A .ih B)) .h B) = ((1 / (B .ih B)) .h ((A .ih B) .h B)))
25	10, 24	syl 10	. . . . . . 7 |- ((B .ih B) =/= 0 -> (((1 / (B .ih B)) x. (A .ih B)) .h B) = ((1 / (B .ih B)) .h ((A .ih B) .h B)))
26		axmulcom 5259	. . . . . . . . . . 11 |- (((1 / (B .ih B)) e. CC /\ (A .ih B) e. CC) -> ((1 / (B .ih B)) x. (A .ih B)) = ((A .ih B) x. (1 / (B .ih B))))
27	22, 26	mpan2 695	. . . . . . . . . 10 |- ((1 / (B .ih B)) e. CC -> ((1 / (B .ih B)) x. (A .ih B)) = ((A .ih B) x. (1 / (B .ih B))))
28	10, 27	syl 10	. . . . . . . . 9 |- ((B .ih B) =/= 0 -> ((1 / (B .ih B)) x. (A .ih B)) = ((A .ih B) x. (1 / (B .ih B))))
29	22, 9	divrecz 5711	. . . . . . . . 9 |- ((B .ih B) =/= 0 -> ((A .ih B) / (B .ih B)) = ((A .ih B) x. (1 / (B .ih B))))
30	28, 29	eqtr4d 1508	. . . . . . . 8 |- ((B .ih B) =/= 0 -> ((1 / (B .ih B)) x. (A .ih B)) = ((A .ih B) / (B .ih B)))
31	30	opreq1d 3970	. . . . . . 7 |- ((B .ih B) =/= 0 -> (((1 / (B .ih B)) x. (A .ih B)) .h B) = (((A .ih B) / (B .ih B)) .h B))
32	25, 31	eqtr3d 1507	. . . . . 6 |- ((B .ih B) =/= 0 -> ((1 / (B .ih B)) .h ((A .ih B) .h B)) = (((A .ih B) / (B .ih B)) .h B))
33	32	adantr 389	. . . . 5 |- (((B .ih B) =/= 0 /\ A e. (_|_` (_|_` {B}))) -> ((1 / (B .ih B)) .h ((A .ih B) .h B)) = (((A .ih B) / (B .ih B)) .h B))
34	8, 21, 33	3eqtr3d 1513	. . . 4 |- (((B .ih B) =/= 0 /\ A e. (_|_` (_|_` {B}))) -> A = (((A .ih B) / (B .ih B)) .h B))
35	34	ex 373	. . 3 |- ((B .ih B) =/= 0 -> (A e. (_|_` (_|_` {B})) -> A = (((A .ih B) / (B .ih B)) .h B)))
36		eleq1 1532	. . . 4 |- (A = (((A .ih B) / (B .ih B)) .h B) -> (A e. (_|_` (_|_` {B})) <-> (((A .ih B) / (B .ih B)) .h B) e. (_|_` (_|_` {B}))))
37	22, 9	divclz 5690	. . . . 5 |- ((B .ih B) =/= 0 -> ((A .ih B) / (B .ih B)) e. CC)
38		h1did 9429	. . . . . . 7 |- (B e. H~ -> B e. (_|_` (_|_` {B})))
39	1, 38	ax-mp 7	. . . . . 6 |- B e. (_|_` (_|_` {B}))
40		snssi 2463	. . . . . . . . . . 11 |- (B e. H~ -> {B} (_ H~)
41	1, 40	ax-mp 7	. . . . . . . . . 10 |- {B} (_ H~
42	41	occl 9136	. . . . . . . . 9 |- (_|_` {B}) e. CH
43	42	choccl 9140	. . . . . . . 8 |- (_|_` (_|_` {B})) e. CH
44	43	chshi 9052	. . . . . . 7 |- (_|_` (_|_` {B})) e. SH
45		shmulcltOLD 9043	. . . . . . 7 |- ((_|_` (_|_` {B})) e. SH -> ((((A .ih B) / (B .ih B)) e. CC /\ B e. (_|_` (_|_` {B}))) -> (((A .ih B) / (B .ih B)) .h B) e. (_|_` (_|_` {B}))))
46	44, 45	ax-mp 7	. . . . . 6 |- ((((A .ih B) / (B .ih B)) e. CC /\ B e. (_|_` (_|_` {B}))) -> (((A .ih B) / (B .ih B)) .h B) e. (_|_` (_|_` {B})))
47	39, 46	mpan2 695	. . . . 5 |- (((A .ih B) / (B .ih B)) e. CC -> (((A .ih B) / (B .ih B)) .h B) e. (_|_` (_|_` {B})))
48	37, 47	syl 10	. . . 4 |- ((B .ih B) =/= 0 -> (((A .ih B) / (B .ih B)) .h B) e. (_|_` (_|_` {B})))
49	36, 48	syl5cbir 211	. . 3 |- ((B .ih B) =/= 0 -> (A = (((A .ih B) / (B .ih B)) .h B) -> A e. (_|_` (_|_` {B}))))
50	35, 49	impbid 515	. 2 |- ((B .ih B) =/= 0 -> (A e. (_|_` (_|_` {B})) <-> A = (((A .ih B) / (B .ih B)) .h B)))
51	4, 50	sylbir 201	1 |- (B =/= 0h -> (A e. (_|_` (_|_` {B})) <-> A = (((A .ih B) / (B .ih B)) .h B)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 955   e. wcel 957   =/= wne 1583   (_ wss 2044  {csn 2406  ` cfv 3178  (class class class)co 3958  CCcc 5215  0cc0 5217  1c1 5218   x. cmul 5222   / cdiv 5277  H~chil 8743   .h csm 8745  0hc0v 8746   .ih csp 8748  SHcsh 8752  _|_cort 8754

This theorem is referenced by:  elspansn2t 9446

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-reg 4576  ax-inf2 4608  ax-ac 4727  ax-hilex 8824  ax-hfvadd 8825  ax-hvcom 8826  ax-hvass 8827  ax-hv0cl 8828  ax-hvaddid 8829  ax-hfv