Theorem nlelsh 9908

Description: The null space of a linear functional is a subspace.

Hypothesis

Ref	Expression
nlelsh.1	|- T e. LinFn

Assertion

Ref	Expression
nlelsh	|- (null` T) e. SH

Proof of Theorem nlelsh

Step	Hyp	Ref	Expression
1		nlelsh.1	. . . . . 6 |- T e. LinFn
2	1	lnfnf 9885	. . . . 5 |- T:H~-->CC
3		nlfnvalt 9725	. . . . 5 |- (T:H~-->CC -> (null` T) = {x e. H~ | (T` x) = 0})
4	2, 3	ax-mp 7	. . . 4 |- (null` T) = {x e. H~ | (T` x) = 0}
5		ssrab2 2121	. . . 4 |- {x e. H~ | (T` x) = 0} (_ H~
6	4, 5	eqsstr 2081	. . 3 |- (null` T) (_ H~
7		sh2 9012	. . 3 |- ((null` T) (_ H~ -> ((null` T) e. SH <-> (0h e. (null` T) /\ (A.x e. (null` T)A.y e. (null` T)(x +h y) e. (null` T) /\ A.x e. CC A.y e. (null` T)(x .h y) e. (null` T)))))
8	6, 7	ax-mp 7	. 2 |- ((null` T) e. SH <-> (0h e. (null` T) /\ (A.x e. (null` T)A.y e. (null` T)(x +h y) e. (null` T) /\ A.x e. CC A.y e. (null` T)(x .h y) e. (null` T))))
9	1	lnfn0 9886	. . 3 |- (T` 0h) = 0
10		ax-hv0cl 8794	. . . 4 |- 0h e. H~
11		elnlfnt 9768	. . . 4 |- ((T:H~-->CC /\ 0h e. H~) -> (0h e. (null` T) <-> (T` 0h) = 0))
12	2, 10, 11	mp2an 695	. . 3 |- (0h e. (null` T) <-> (T` 0h) = 0)
13	9, 12	mpbir 190	. 2 |- 0h e. (null` T)
14	1	lnfnadd 9887	. . . . . . 7 |- ((x e. H~ /\ y e. H~) -> (T` (x +h y)) = ((T` x) + (T` y)))
15	6	sseli 2055	. . . . . . 7 |- (x e. (null` T) -> x e. H~)
16	6	sseli 2055	. . . . . . 7 |- (y e. (null` T) -> y e. H~)
17	14, 15, 16	syl2an 454	. . . . . 6 |- ((x e. (null` T) /\ y e. (null` T)) -> (T` (x +h y)) = ((T` x) + (T` y)))
18		elnlfn2t 9769	. . . . . . . . 9 |- ((T:H~-->CC /\ x e. (null` T)) -> (T` x) = 0)
19	2, 18	mpan 693	. . . . . . . 8 |- (x e. (null` T) -> (T` x) = 0)
20		elnlfn2t 9769	. . . . . . . . 9 |- ((T:H~-->CC /\ y e. (null` T)) -> (T` y) = 0)
21	2, 20	mpan 693	. . . . . . . 8 |- (y e. (null` T) -> (T` y) = 0)
22	19, 21	opreqan12d 3964	. . . . . . 7 |- ((x e. (null` T) /\ y e. (null` T)) -> ((T` x) + (T` y)) = (0 + 0))
23		0cn 5300	. . . . . . . 8 |- 0 e. CC
24	23	addid1 5302	. . . . . . 7 |- (0 + 0) = 0
25	22, 24	syl6eq 1515	. . . . . 6 |- ((x e. (null` T) /\ y e. (null` T)) -> ((T` x) + (T` y)) = 0)
26	17, 25	eqtrd 1499	. . . . 5 |- ((x e. (null` T) /\ y e. (null` T)) -> (T` (x +h y)) = 0)
27		hvaddclt 8803	. . . . . . 7 |- ((x e. H~ /\ y e. H~) -> (x +h y) e. H~)
28	27, 15, 16	syl2an 454	. . . . . 6 |- ((x e. (null` T) /\ y e. (null` T)) -> (x +h y) e. H~)
29		elnlfnt 9768	. . . . . . 7 |- ((T:H~-->CC /\ (x +h y) e. H~) -> ((x +h y) e. (null` T) <-> (T` (x +h y)) = 0))
30	2, 29	mpan 693	. . . . . 6 |- ((x +h y) e. H~ -> ((x +h y) e. (null` T) <-> (T` (x +h y)) = 0))
31	28, 30	syl 10	. . . . 5 |- ((x e. (null` T) /\ y e. (null` T)) -> ((x +h y) e. (null` T) <-> (T` (x +h y)) = 0))
32	26, 31	mpbird 196	. . . 4 |- ((x e. (null` T) /\ y e. (null` T)) -> (x +h y) e. (null` T))
33	32	rgen2 1715	. . 3 |- A.x e. (null` T)A.y e. (null` T)(x +h y) e. (null` T)
34	1	lnfnmul 9888	. . . . . . 7 |- ((x e. CC /\ y e. H~) -> (T` (x .h y)) = (x x. (T` y)))
35	34, 16	sylan2 451	. . . . . 6 |- ((x e. CC /\ y e. (null` T)) -> (T` (x .h y)) = (x x. (T` y)))
36	21	opreq2d 3961	. . . . . . 7 |- (y e. (null` T) -> (x x. (T` y)) = (x x. 0))
37	36	adantl 388	. . . . . 6 |- ((x e. CC /\ y e. (null` T)) -> (x x. (T` y)) = (x x. 0))
38		mul01t 5415	. . . . . . 7 |- (x e. CC -> (x x. 0) = 0)
39	38	adantr 389	. . . . . 6 |- ((x e. CC /\ y e. (null` T)) -> (x x. 0) = 0)
40	35, 37, 39	3eqtrd 1503	. . . . 5 |- ((x e. CC /\ y e. (null` T)) -> (T` (x .h y)) = 0)
41		hvmulclt 8804	. . . . . . 7 |- ((x e. CC /\ y e. H~) -> (x .h y) e. H~)
42	41, 16	sylan2 451	. . . . . 6 |- ((x e. CC /\ y e. (null` T)) -> (x .h y) e. H~)
43		elnlfnt 9768	. . . . . . 7 |- ((T:H~-->CC /\ (x .h y) e. H~) -> ((x .h y) e. (null` T) <-> (T` (x .h y)) = 0))
44	2, 43	mpan 693	. . . . . 6 |- ((x .h y) e. H~ -> ((x .h y) e. (null` T) <-> (T` (x .h y)) = 0))
45	42, 44	syl 10	. . . . 5 |- ((x e. CC /\ y e. (null` T)) -> ((x .h y) e. (null` T) <-> (T` (x .h y)) = 0))
46	40, 45	mpbird 196	. . . 4 |- ((x e. CC /\ y e. (null` T)) -> (x .h y) e. (null` T))
47	46	rgen2 1715	. . 3 |- A.x e. CC A.y e. (null` T)(x .h y) e. (null` T)
48	33, 47	pm3.2i 285	. 2 |- (A.x e. (null` T)A.y e. (null` T)(x +h y) e. (null` T) /\ A.x e. CC A.y e. (null` T)(x .h y) e. (null` T))
49	8, 13, 48	mpbir2an 728	1 |- (null` T) e. SH

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  A.wral 1637  {crab 1640   (_ wss 2037  -->wf 3168  ` cfv 3172  (class class class)co 3948  CCcc 5204  0cc0 5206   + caddc 5209   x. cmul 5211  H~chil 8727   +h cva 8728   .h csm 8729  0hc0v 8730  SHcsh 8736  nullcnl 8760  LinFnclf 8762

This theorem is referenced by:  nlelch 9909

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597  ax-hilex 8790  ax-hfvadd 8791  ax-hv0cl 8794  ax-hvaddid 8795  ax-hfvmul 8796  ax-hvmulid 8797

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-map 4308  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218  df-sub 5328  df-neg 5330  df-sh 8997  df-nlfn 9689  df-lnfn 9691

Copyright terms: Public domain