Theorem h1datom 5483

Description: A 1-dimensional subspace is an atom.

Hypotheses

Ref	Expression
h1datom.1	|- A e. CH
h1datom.2	|- B e. H~

Assertion

Ref	Expression
h1datom	|- (A (_ (_|_` (_|_` {B})) -> (A = (_|_` (_|_` {B})) \/ A = 0H))

Proof of Theorem h1datom

Step	Hyp	Ref	Expression
1		ssel 1502	. . . . . . . . 9 |- (A (_ (_|_` (_|_` {B})) -> (x e. A -> x e. (_|_` (_|_` {B}))))
2		cleq1 1107	. . . . . . . . . . . . . . . . . 18 |- (x = (y .s B) -> (x = 0v <-> (y .s B) = 0v))
3		opreq1 3006	. . . . . . . . . . . . . . . . . . 19 |- (y = 0 -> (y .s B) = (0 .s B))
4		h1datom.2	. . . . . . . . . . . . . . . . . . . 20 |- B e. H~
5		ax-hvmulzer 4995	. . . . . . . . . . . . . . . . . . . 20 |- (B e. H~ -> (0 .s B) = 0v)
6	4, 5	ax-mp 6	. . . . . . . . . . . . . . . . . . 19 |- (0 .s B) = 0v
7	3, 6	syl6eq 1140	. . . . . . . . . . . . . . . . . 18 |- (y = 0 -> (y .s B) = 0v)
8	2, 7	syl5bir 184	. . . . . . . . . . . . . . . . 17 |- (x = (y .s B) -> (y = 0 -> x = 0v))
9	8	con3d 87	. . . . . . . . . . . . . . . 16 |- (x = (y .s B) -> (-. x = 0v -> -. y = 0))
10		df-ne 1192	. . . . . . . . . . . . . . . 16 |- (y =/= 0 <-> -. y = 0)
11	9, 10	syl6ibr 186	. . . . . . . . . . . . . . 15 |- (x = (y .s B) -> (-. x = 0v -> y =/= 0))
12	11	adantl 305	. . . . . . . . . . . . . 14 |- ((y e. CC /\ x = (y .s B)) -> (-. x = 0v -> y =/= 0))
13		1cn 4101	. . . . . . . . . . . . . . . . . . . . 21 |- 1 e. CC
14		divclt 4223	. . . . . . . . . . . . . . . . . . . . 21 |- (((1 e. CC /\ y e. CC) /\ y =/= 0) -> (1 / y) e. CC)
15	13, 14	mpan11 529	. . . . . . . . . . . . . . . . . . . 20 |- ((y e. CC /\ y =/= 0) -> (1 / y) e. CC)
16		h1datom.1	. . . . . . . . . . . . . . . . . . . . . . 23 |- A e. CH
17	16	chshi 5132	. . . . . . . . . . . . . . . . . . . . . 22 |- A e. SH
18		shmulclt 5124	. . . . . . . . . . . . . . . . . . . . . 22 |- (A e. SH -> (((1 / y) e. CC /\ x e. A) -> ((1 / y) .s x) e. A))
19	17, 18	ax-mp 6	. . . . . . . . . . . . . . . . . . . . 21 |- (((1 / y) e. CC /\ x e. A) -> ((1 / y) .s x) e. A)
20	19	exp 291	. . . . . . . . . . . . . . . . . . . 20 |- ((1 / y) e. CC -> (x e. A -> ((1 / y) .s x) e. A))
21	15, 20	syl 12	. . . . . . . . . . . . . . . . . . 19 |- ((y e. CC /\ y =/= 0) -> (x e. A -> ((1 / y) .s x) e. A))
22	21	adantr 306	. . . . . . . . . . . . . . . . . 18 |- (((y e. CC /\ y =/= 0) /\ x = (y .s B)) -> (x e. A -> ((1 / y) .s x) e. A))
23		opreq2 3007	. . . . . . . . . . . . . . . . . . . 20 |- (x = (y .s B) -> ((1 / y) .s x) = ((1 / y) .s (y .s B)))
24		ax-hvmulass 4992	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (((1 / y) e. CC /\ y e. CC /\ B e. H~) -> (((1 / y) x. y) .s B) = ((1 / y) .s (y .s B)))
25	4, 24	mp3an3 641	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((1 / y) e. CC /\ y e. CC) -> (((1 / y) x. y) .s B) = ((1 / y) .s (y .s B)))
26		pm3.26 256	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((y e. CC /\ y =/= 0) -> y e. CC)
27	25, 15, 26	sylanc 361	. . . . . . . . . . . . . . . . . . . . . 22 |- ((y e. CC /\ y =/= 0) -> (((1 / y) x. y) .s B) = ((1 / y) .s (y .s B)))
28		axmulcom 4071	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((y e. CC /\ (1 / y) e. CC) -> (y x. (1 / y)) = ((1 / y) x. y))
29	28, 26, 15	sylanc 361	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((y e. CC /\ y =/= 0) -> (y x. (1 / y)) = ((1 / y) x. y))
30		recidt 4235	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((y e. CC /\ y =/= 0) -> (y x. (1 / y)) = 1)
31	29, 30	eqtr3d 1130	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((y e. CC /\ y =/= 0) -> ((1 / y) x. y) = 1)
32	31	opreq1d 3012	. . . . . . . . . . . . . . . . . . . . . 22 |- ((y e. CC /\ y =/= 0) -> (((1 / y) x. y) .s B) = (1 .s B))
33	27, 32	eqtr3d 1130	. . . . . . . . . . . . . . . . . . . . 21 |- ((y e. CC /\ y =/= 0) -> ((1 / y) .s (y .s B)) = (1 .s B))
34		ax-hvmulid 4991	. . . . . . . . . . . . . . . . . . . . . 22 |- (B e. H~ -> (1 .s B) = B)
35	4, 34	ax-mp 6	. . . . . . . . . . . . . . . . . . . . 21 |- (1 .s B) = B
36	33, 35	syl6eq 1140	. . . . . . . . . . . . . . . . . . . 20 |- ((y e. CC /\ y =/= 0) -> ((1 / y) .s (y .s B)) = B)
37	23, 36	sylan9eqr 1145	. . . . . . . . . . . . . . . . . . 19 |- (((y e. CC /\ y =/= 0) /\ x = (y .s B)) -> ((1 / y) .s x) = B)
38	37	eleq1d 1155	. . . . . . . . . . . . . . . . . 18 |- (((y e. CC /\ y =/= 0) /\ x = (y .s B)) -> (((1 / y) .s x) e. A <-> B e. A))
39	22, 38	sylibd 177	. . . . . . . . . . . . . . . . 17 |- (((y e. CC /\ y =/= 0) /\ x = (y .s B)) -> (x e. A -> B e. A))
40	39	exp31 293	. . . . . . . . . . . . . . . 16 |- (y e. CC -> (y =/= 0 -> (x = (y .s B) -> (x e. A -> B e. A))))
41	40	com23 32	. . . . . . . . . . . . . . 15 |- (y e. CC -> (x = (y .s B) -> (y =/= 0 -> (x e. A -> B e. A))))
42	41	imp 277	. . . . . . . . . . . . . 14 |- ((y e. CC /\ x = (y .s B)) -> (y =/= 0 -> (x e. A -> B e. A)))
43	12, 42	syld 27	. . . . . . . . . . . . 13 |- ((y e. CC /\ x = (y .s B)) -> (-. x = 0v -> (x e. A -> B e. A)))
44	43	com3r 35	. . . . . . . . . . . 12 |- (x e. A -> ((y e. CC /\ x = (y .s B)) -> (-. x = 0v -> B e. A)))
45	44	exp3a 292	. . . . . . . . . . 11 |- (x e. A -> (y e. CC -> (x = (y .s B) -> (-. x = 0v -> B e. A))))
46	45	r19.23adv 1286	. . . . . . . . . 10 |- (x e. A -> (E.y e. CC x = (y .s B) -> (-. x = 0v -> B e. A)))
47	4	h1de2ct 5461	. . . . . . . . . 10 |- (x e. (_|_` (_|_` {B})) <-> E.y e. CC x = (y .s B))
48	46, 47	syl5ib 181	. . . . . . . . 9 |- (x e. A -> (x e. (_|_` (_|_` {B})) -> (-. x = 0v -> B e. A)))
49	1, 48	sylcom 51	. . . . . . . 8 |- (A (_ (_|_` (_|_` {B})) -> (x e. A -> (-. x = 0v -> B e. A)))
50	49	r19.23adv 1286	. . . . . . 7 |- (A (_ (_|_` (_|_` {B})) -> (E.x e. A -. x = 0v -> B e. A))
51	16	chne0 5375	. . . . . . 7 |- (-. A = 0H <-> E.x e. A -. x = 0v)
52	50, 51	syl5ib 181	. . . . . 6 |- (A (_ (_|_` (_|_` {B})) -> (-. A = 0H -> B e. A))
53		snssi 1851	. . . . . . . 8 |- (B e. A -> {B} (_ A)
54		snssi 1851	. . . . . . . . . 10 |- (B e. H~ -> {B} (_ H~)
55	4, 54	ax-mp 6	. . . . . . . . 9 |- {B} (_ H~
56	16	chssi 5136	. . . . . . . . 9 |- A (_ H~
57	55, 56	occon2 5170	. . . . . . . 8 |- ({B} (_ A -> (_|_` (_|_` {B})) (_ (_|_` (_|_` A)))
58	53, 57	syl 12	. . . . . . 7 |- (B e. A -> (_|_` (_|_` {B})) (_ (_|_` (_|_` A)))
59	16	ococ 5252	. . . . . . 7 |- (_|_` (_|_` A)) = A
60	58, 59	syl6ss 1546	. . . . . 6 |- (B e. A -> (_|_` (_|_` {B})) (_ A)
61	52, 60	syl6 23	. . . . 5 |- (A (_ (_|_` (_|_` {B})) -> (-. A = 0H -> (_|_` (_|_` {B})) (_ A))
62	61	anc2li 250	. . . 4 |- (A (_ (_|_` (_|_` {B})) -> (-. A = 0H -> (A (_ (_|_` (_|_` {B})) /\ (_|_` (_|_` {B})) (_ A)))
63		eqss 1516	. . . 4 |- (A = (_|_` (_|_` {B})) <-> (A (_ (_|_` (_|_` {B})) /\ (_|_` (_|_` {B})) (_ A))
64	62, 63	syl6ibr 186	. . 3 |- (A (_ (_|_` (_|_` {B})) -> (-. A = 0H -> A = (_|_` (_|_` {B}))))
65	64	con1d 85	. 2 |- (A (_ (_|_` (_|_` {B})) -> (-. A = (_|_` (_|_` {B})) -> A = 0H))
66	65	orrd 203	1 |- (A (_ (_|_` (_|_` {B})) -> (A = (_|_` (_|_` {B})) \/ A = 0H))

Syntax hints:  -. wn 1   -> wi 2   \/ wo 195   /\ wa 196   = wceq 1091   e. wcel 1092   =/= wne 1190  E.wrex 1202   (_ wss 1487  {csn 1808  ` cfv 2422  (class class class)co 3001  CCcc 4026  0cc0 4028  1c1 4029   x. cmulc 4032   / cdiv 4091  H~chil 4958   .s csm 4960  0vc0v 4961  SHcsh 4967  CHcch 4968  _|_cort 4969  0Hc0h 4974

This theorem is referenced by:  h1datomt 5484

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-ac 1080  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-hvdistr1 4993  ax-hvdistr2 4994  ax-hvmulzer 4995  ax-hicl 5043  ax-his1 5045  ax-his2 5046  ax-his3 5047  ax-his4 5048  ax-hcompl 5113

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