Theorem h1de2b 9477

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 8965	. . . 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 1598	. 2 |- ((B .ih B) =/= 0 <-> B =/= 0h)
5		h1de2.1	. . . . . . . 8 |- A e. H~
6	5, 1	h1de2 9476	. . . . . . 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 3976	. . . . 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 8948	. . . . . . . . . 10 |- (B .ih B) e. CC
10	9	recclz 5714	. . . . . . . . 9 |- ((B .ih B) =/= 0 -> (1 / (B .ih B)) e. CC)
11		ax-hvmulass 8877	. . . . . . . . . 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 908	. . . . . . . . 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 5269	. . . . . . . . . 10 |- 1 e. CC
15	14, 9	divcan1z 5718	. . . . . . . . 9 |- ((B .ih B) =/= 0 -> ((1 / (B .ih B)) x. (B .ih B)) = 1)
16	15	opreq1d 3975	. . . . . . . 8 |- ((B .ih B) =/= 0 -> (((1 / (B .ih B)) x. (B .ih B)) .h A) = (1 .h A))
17	13, 16	eqtr3d 1509	. . . . . . 7 |- ((B .ih B) =/= 0 -> ((1 / (B .ih B)) .h ((B .ih B) .h A)) = (1 .h A))
18		ax-hvmulid 8876	. . . . . . . 8 |- (A e. H~ -> (1 .h A) = A)
19	5, 18	ax-mp 7	. . . . . . 7 |- (1 .h A) = A
20	17, 19	syl6eq 1523	. . . . . 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 8948	. . . . . . . . 9 |- (A .ih B) e. CC
23		ax-hvmulass 8877	. . . . . . . . 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 908	. . . . . . . 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 5276	. . . . . . . . . . 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 696	. . . . . . . . . 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 5738	. . . . . . . . 9 |- ((B .ih B) =/= 0 -> ((A .ih B) / (B .ih B)) = ((A .ih B) x. (1 / (B .ih B))))
30	28, 29	eqtr4d 1510	. . . . . . . 8 |- ((B .ih B) =/= 0 -> ((1 / (B .ih B)) x. (A .ih B)) = ((A .ih B) / (B .ih B)))
31	30	opreq1d 3975	. . . . . . 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 1509	. . . . . 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 1515	. . . 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 1534	. . . 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 5711	. . . . 5 |- ((B .ih B) =/= 0 -> ((A .ih B) / (B .ih B)) e. CC)
38		h1did 9474	. . . . . . 7 |- (B e. H~ -> B e. (_|_` (_|_` {B})))
39	1, 38	ax-mp 7	. . . . . 6 |- B e. (_|_` (_|_` {B}))
40		snssi 2466	. . . . . . . . . . 11 |- (B e. H~ -> {B} (_ H~)
41	1, 40	ax-mp 7	. . . . . . . . . 10 |- {B} (_ H~
42	41	occl 9181	. . . . . . . . 9 |- (_|_` {B}) e. CH
43	42	choccl 9185	. . . . . . . 8 |- (_|_` (_|_` {B})) e. CH
44	43	chshi 9097	. . . . . . 7 |- (_|_` (_|_` {B})) e. SH
45		shmulcltOLD 9088	. . . . . . 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 696	. . . . 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 516	. 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 956   e. wcel 958   =/= wne 1585   (_ wss 2047  {csn 2409  ` cfv 3182  (class class class)co 3963  CCcc 5232  0cc0 5234  1c1 5235   x. cmul 5239   / cdiv 5294  H~chil 8788   .h csm 8790  0hc0v 8791   .ih csp 8793  SHcsh 8797  _|_cort 8799

This theorem is referenced by:  h1de2ctlem 9478  elspansn2t 9490

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-reg 4593  ax-inf2 4625  ax-ac 4744  ax-hilex 8869  ax-hfvadd 8870  ax-hvcom 8871  ax-hvass 8872  ax-hv0cl