Theorem nlelshi 31002

Description: The null space of a linear functional is a subspace. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)

Hypothesis

Ref	Expression
nlelsh.1	⊢ 𝑇 ∈ LinFn

Assertion

Ref	Expression
nlelshi	⊢ (null‘𝑇) ∈ Sℋ

Proof of Theorem nlelshi

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-hv0cl 29945	. . 3 ⊢ 0ℎ ∈ ℋ
2		nlelsh.1	. . . 4 ⊢ 𝑇 ∈ LinFn
3	2	lnfn0i 30984	. . 3 ⊢ (𝑇‘0ℎ) = 0
4	2	lnfnfi 30983	. . . 4 ⊢ 𝑇: ℋ⟶ℂ
5		elnlfn 30870	. . . 4 ⊢ (𝑇: ℋ⟶ℂ → (0ℎ ∈ (null‘𝑇) ↔ (0ℎ ∈ ℋ ∧ (𝑇‘0ℎ) = 0)))
6	4, 5	ax-mp 5	. . 3 ⊢ (0ℎ ∈ (null‘𝑇) ↔ (0ℎ ∈ ℋ ∧ (𝑇‘0ℎ) = 0))
7	1, 3, 6	mpbir2an 709	. 2 ⊢ 0ℎ ∈ (null‘𝑇)
8		nlfnval 30823	. . . . . . . . . 10 ⊢ (𝑇: ℋ⟶ℂ → (null‘𝑇) = (◡𝑇 “ {0}))
9	4, 8	ax-mp 5	. . . . . . . . 9 ⊢ (null‘𝑇) = (◡𝑇 “ {0})
10		cnvimass 6033	. . . . . . . . 9 ⊢ (◡𝑇 “ {0}) ⊆ dom 𝑇
11	9, 10	eqsstri 3978	. . . . . . . 8 ⊢ (null‘𝑇) ⊆ dom 𝑇
12	4	fdmi 6680	. . . . . . . 8 ⊢ dom 𝑇 = ℋ
13	11, 12	sseqtri 3980	. . . . . . 7 ⊢ (null‘𝑇) ⊆ ℋ
14	13	sseli 3940	. . . . . 6 ⊢ (𝑥 ∈ (null‘𝑇) → 𝑥 ∈ ℋ)
15	13	sseli 3940	. . . . . 6 ⊢ (𝑦 ∈ (null‘𝑇) → 𝑦 ∈ ℋ)
16		hvaddcl 29954	. . . . . 6 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 +ℎ 𝑦) ∈ ℋ)
17	14, 15, 16	syl2an 596	. . . . 5 ⊢ ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 +ℎ 𝑦) ∈ ℋ)
18	2	lnfnaddi 30985	. . . . . . . 8 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑥 +ℎ 𝑦)) = ((𝑇‘𝑥) + (𝑇‘𝑦)))
19	14, 15, 18	syl2an 596	. . . . . . 7 ⊢ ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 +ℎ 𝑦)) = ((𝑇‘𝑥) + (𝑇‘𝑦)))
20		elnlfn 30870	. . . . . . . . . 10 ⊢ (𝑇: ℋ⟶ℂ → (𝑥 ∈ (null‘𝑇) ↔ (𝑥 ∈ ℋ ∧ (𝑇‘𝑥) = 0)))
21	4, 20	ax-mp 5	. . . . . . . . 9 ⊢ (𝑥 ∈ (null‘𝑇) ↔ (𝑥 ∈ ℋ ∧ (𝑇‘𝑥) = 0))
22	21	simprbi 497	. . . . . . . 8 ⊢ (𝑥 ∈ (null‘𝑇) → (𝑇‘𝑥) = 0)
23		elnlfn 30870	. . . . . . . . . 10 ⊢ (𝑇: ℋ⟶ℂ → (𝑦 ∈ (null‘𝑇) ↔ (𝑦 ∈ ℋ ∧ (𝑇‘𝑦) = 0)))
24	4, 23	ax-mp 5	. . . . . . . . 9 ⊢ (𝑦 ∈ (null‘𝑇) ↔ (𝑦 ∈ ℋ ∧ (𝑇‘𝑦) = 0))
25	24	simprbi 497	. . . . . . . 8 ⊢ (𝑦 ∈ (null‘𝑇) → (𝑇‘𝑦) = 0)
26	22, 25	oveqan12d 7376	. . . . . . 7 ⊢ ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → ((𝑇‘𝑥) + (𝑇‘𝑦)) = (0 + 0))
27	19, 26	eqtrd 2776	. . . . . 6 ⊢ ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 +ℎ 𝑦)) = (0 + 0))
28		00id 11330	. . . . . 6 ⊢ (0 + 0) = 0
29	27, 28	eqtrdi 2792	. . . . 5 ⊢ ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 +ℎ 𝑦)) = 0)
30		elnlfn 30870	. . . . . 6 ⊢ (𝑇: ℋ⟶ℂ → ((𝑥 +ℎ 𝑦) ∈ (null‘𝑇) ↔ ((𝑥 +ℎ 𝑦) ∈ ℋ ∧ (𝑇‘(𝑥 +ℎ 𝑦)) = 0)))
31	4, 30	ax-mp 5	. . . . 5 ⊢ ((𝑥 +ℎ 𝑦) ∈ (null‘𝑇) ↔ ((𝑥 +ℎ 𝑦) ∈ ℋ ∧ (𝑇‘(𝑥 +ℎ 𝑦)) = 0))
32	17, 29, 31	sylanbrc 583	. . . 4 ⊢ ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 +ℎ 𝑦) ∈ (null‘𝑇))
33	32	rgen2 3194	. . 3 ⊢ ∀𝑥 ∈ (null‘𝑇)∀𝑦 ∈ (null‘𝑇)(𝑥 +ℎ 𝑦) ∈ (null‘𝑇)
34		hvmulcl 29955	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ 𝑦) ∈ ℋ)
35	15, 34	sylan2 593	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 ·ℎ 𝑦) ∈ ℋ)
36	2	lnfnmuli 30986	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑥 ·ℎ 𝑦)) = (𝑥 · (𝑇‘𝑦)))
37	15, 36	sylan2 593	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 ·ℎ 𝑦)) = (𝑥 · (𝑇‘𝑦)))
38	25	oveq2d 7373	. . . . . . 7 ⊢ (𝑦 ∈ (null‘𝑇) → (𝑥 · (𝑇‘𝑦)) = (𝑥 · 0))
39		mul01 11334	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
40	38, 39	sylan9eqr 2798	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 · (𝑇‘𝑦)) = 0)
41	37, 40	eqtrd 2776	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 ·ℎ 𝑦)) = 0)
42		elnlfn 30870	. . . . . 6 ⊢ (𝑇: ℋ⟶ℂ → ((𝑥 ·ℎ 𝑦) ∈ (null‘𝑇) ↔ ((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ (𝑇‘(𝑥 ·ℎ 𝑦)) = 0)))
43	4, 42	ax-mp 5	. . . . 5 ⊢ ((𝑥 ·ℎ 𝑦) ∈ (null‘𝑇) ↔ ((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ (𝑇‘(𝑥 ·ℎ 𝑦)) = 0))
44	35, 41, 43	sylanbrc 583	. . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 ·ℎ 𝑦) ∈ (null‘𝑇))
45	44	rgen2 3194	. . 3 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (null‘𝑇)(𝑥 ·ℎ 𝑦) ∈ (null‘𝑇)
46	33, 45	pm3.2i 471	. 2 ⊢ (∀𝑥 ∈ (null‘𝑇)∀𝑦 ∈ (null‘𝑇)(𝑥 +ℎ 𝑦) ∈ (null‘𝑇) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (null‘𝑇)(𝑥 ·ℎ 𝑦) ∈ (null‘𝑇))
47		issh3 30161	. . 3 ⊢ ((null‘𝑇) ⊆ ℋ → ((null‘𝑇) ∈ Sℋ ↔ (0ℎ ∈ (null‘𝑇) ∧ (∀𝑥 ∈ (null‘𝑇)∀𝑦 ∈ (null‘𝑇)(𝑥 +ℎ 𝑦) ∈ (null‘𝑇) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (null‘𝑇)(𝑥 ·ℎ 𝑦) ∈ (null‘𝑇)))))
48	13, 47	ax-mp 5	. 2 ⊢ ((null‘𝑇) ∈ Sℋ ↔ (0ℎ ∈ (null‘𝑇) ∧ (∀𝑥 ∈ (null‘𝑇)∀𝑦 ∈ (null‘𝑇)(𝑥 +ℎ 𝑦) ∈ (null‘𝑇) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (null‘𝑇)(𝑥 ·ℎ 𝑦) ∈ (null‘𝑇))))
49	7, 46, 48	mpbir2an 709	1 ⊢ (null‘𝑇) ∈ Sℋ

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3064   ⊆ wss 3910  {csn 4586  ^◡ccnv 5632  dom cdm 5633   “ cima 5636  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  ℂcc 11049  0cc0 11051   + caddc 11054   · cmul 11056   ℋchba 29861   +_ℎ cva 29862   ·_ℎ csm 29863  0_ℎc0v 29866   S_ℋ csh 29870  nullcnl 29894  LinFnclf 29896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-hilex 29941  ax-hfvadd 29942  ax-hv0cl 29945  ax-hvaddid 29946  ax-hfvmul 29947  ax-hvmulid 29948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-po 5545  df-so 5546  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-ltxr 11194  df-sub 11387  df-sh 30149  df-nlfn 30788  df-lnfn 30790

This theorem is referenced by:  nlelchi  31003