Theorem nlelsh 9988

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 9965	. . . . 5 |- T:H~-->CC
3		nlfnvalt 9803	. . . . 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 2134	. . . 4 |- {x e. H~ | (T` x) = 0} (_ H~
6	4, 5	eqsstr 2094	. . 3 |- (null` T) (_ H~
7		sh2 9086	. . 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 9966	. . 3 |- (T` 0h) = 0
10		ax-hv0cl 8868	. . . 4 |- 0h e. H~
11		elnlfnt 9847	. . . 4 |- ((T:H~-->CC /\ 0h e. H~) -> (0h e. (null` T) <-> (T` 0h) = 0))
12	2, 10, 11	mp2an 699	. . 3 |- (0h e. (null` T) <-> (T` 0h) = 0)
13	9, 12	mpbir 190	. 2 |- 0h e. (null` T)
14	1	lnfnadd 9967	. . . . . . 7 |- ((x e. H~ /\ y e. H~) -> (T` (x +h y)) = ((T` x) + (T` y)))
15	6	sseli 2068	. . . . . . 7 |- (x e. (null` T) -> x e. H~)
16	6	sseli 2068	. . . . . . 7 |- (y e. (null` T) -> y e. H~)
17	14, 15, 16	syl2an 456	. . . . . 6 |- ((x e. (null` T) /\ y e. (null` T)) -> (T` (x +h y)) = ((T` x) + (T` y)))
18		elnlfn2t 9848	. . . . . . . . 9 |- ((T:H~-->CC /\ x e. (null` T)) -> (T` x) = 0)
19	2, 18	mpan 697	. . . . . . . 8 |- (x e. (null` T) -> (T` x) = 0)
20		elnlfn2t 9848	. . . . . . . . 9 |- ((T:H~-->CC /\ y e. (null` T)) -> (T` y) = 0)
21	2, 20	mpan 697	. . . . . . . 8 |- (y e. (null` T) -> (T` y) = 0)
22	19, 21	opreqan12d 3985	. . . . . . 7 |- ((x e. (null` T) /\ y e. (null` T)) -> ((T` x) + (T` y)) = (0 + 0))
23		0cn 5340	. . . . . . . 8 |- 0 e. CC
24	23	addid1 5342	. . . . . . 7 |- (0 + 0) = 0
25	22, 24	syl6eq 1526	. . . . . 6 |- ((x e. (null` T) /\ y e. (null` T)) -> ((T` x) + (T` y)) = 0)
26	17, 25	eqtrd 1510	. . . . 5 |- ((x e. (null` T) /\ y e. (null` T)) -> (T` (x +h y)) = 0)
27		hvaddclt 8877	. . . . . . 7 |- ((x e. H~ /\ y e. H~) -> (x +h y) e. H~)
28	27, 15, 16	syl2an 456	. . . . . 6 |- ((x e. (null` T) /\ y e. (null` T)) -> (x +h y) e. H~)
29		elnlfnt 9847	. . . . . . 7 |- ((T:H~-->CC /\ (x +h y) e. H~) -> ((x +h y) e. (null` T) <-> (T` (x +h y)) = 0))
30	2, 29	mpan 697	. . . . . 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 1726	. . 3 |- A.x e. (null` T)A.y e. (null` T)(x +h y) e. (null` T)
34	1	lnfnmul 9968	. . . . . . 7 |- ((x e. CC /\ y e. H~) -> (T` (x .h y)) = (x x. (T` y)))
35	34, 16	sylan2 453	. . . . . 6 |- ((x e. CC /\ y e. (null` T)) -> (T` (x .h y)) = (x x. (T` y)))
36	21	opreq2d 3982	. . . . . . 7 |- (y e. (null` T) -> (x x. (T` y)) = (x x. 0))
37	36	adantl 390	. . . . . 6 |- ((x e. CC /\ y e. (null` T)) -> (x x. (T` y)) = (x x. 0))
38		mul01t 5455	. . . . . . 7 |- (x e. CC -> (x x. 0) = 0)
39	38	adantr 391	. . . . . 6 |- ((x e. CC /\ y e. (null` T)) -> (x x. 0) = 0)
40	35, 37, 39	3eqtrd 1514	. . . . 5 |- ((x e. CC /\ y e. (null` T)) -> (T` (x .h y)) = 0)
41		hvmulclt 8878	. . . . . . 7 |- ((x e. CC /\ y e. H~) -> (x .h y) e. H~)
42	41, 16	sylan2 453	. . . . . 6 |- ((x e. CC /\ y e. (null` T)) -> (x .h y) e. H~)
43		elnlfnt 9847	. . . . . . 7 |- ((T:H~-->CC /\ (x .h y) e. H~) -> ((x .h y) e. (null` T) <-> (T` (x .h y)) = 0))
44	2, 43	mpan 697	. . . . . 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 1726	. . 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 732	1 |- (null` T) e. SH

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  A.wral 1648  {crab 1651   (_ wss 2050  -->wf 3184  ` cfv 3188  (class class class)co 3969  CCcc 5244  0cc0 5246   + caddc 5249   x. cmul 5251  H~chil 8783   +h cva 8784   .h csm 8785  0hc0v 8786  SHcsh 8792  nullcnl 8816  LinFnclf 8818

This theorem is referenced by:  nlelch 9989

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634  ax-hilex 8864  ax-hfvadd 8865  ax-hv0cl 8868  ax-hvaddid 8869  ax-hfvmul 8870  ax-hvmulid 8871

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-map 4330  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-sub 5368  df-neg 5370  df-sh 9071  df-nlfn 9767  df-lnfn 9769

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