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Theorem nlelshi 29047
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) ∈ S

Proof of Theorem nlelshi
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-hv0cl 27988 . . 3 0 ∈ ℋ
2 nlelsh.1 . . . 4 𝑇 ∈ LinFn
32lnfn0i 29029 . . 3 (𝑇‘0) = 0
42lnfnfi 29028 . . . 4 𝑇: ℋ⟶ℂ
5 elnlfn 28915 . . . 4 (𝑇: ℋ⟶ℂ → (0 ∈ (null‘𝑇) ↔ (0 ∈ ℋ ∧ (𝑇‘0) = 0)))
64, 5ax-mp 5 . . 3 (0 ∈ (null‘𝑇) ↔ (0 ∈ ℋ ∧ (𝑇‘0) = 0))
71, 3, 6mpbir2an 975 . 2 0 ∈ (null‘𝑇)
8 nlfnval 28868 . . . . . . . . . 10 (𝑇: ℋ⟶ℂ → (null‘𝑇) = (𝑇 “ {0}))
94, 8ax-mp 5 . . . . . . . . 9 (null‘𝑇) = (𝑇 “ {0})
10 cnvimass 5520 . . . . . . . . 9 (𝑇 “ {0}) ⊆ dom 𝑇
119, 10eqsstri 3668 . . . . . . . 8 (null‘𝑇) ⊆ dom 𝑇
124fdmi 6090 . . . . . . . 8 dom 𝑇 = ℋ
1311, 12sseqtri 3670 . . . . . . 7 (null‘𝑇) ⊆ ℋ
1413sseli 3632 . . . . . 6 (𝑥 ∈ (null‘𝑇) → 𝑥 ∈ ℋ)
1513sseli 3632 . . . . . 6 (𝑦 ∈ (null‘𝑇) → 𝑦 ∈ ℋ)
16 hvaddcl 27997 . . . . . 6 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 + 𝑦) ∈ ℋ)
1714, 15, 16syl2an 493 . . . . 5 ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 + 𝑦) ∈ ℋ)
182lnfnaddi 29030 . . . . . . . 8 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑥 + 𝑦)) = ((𝑇𝑥) + (𝑇𝑦)))
1914, 15, 18syl2an 493 . . . . . . 7 ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 + 𝑦)) = ((𝑇𝑥) + (𝑇𝑦)))
20 elnlfn 28915 . . . . . . . . . 10 (𝑇: ℋ⟶ℂ → (𝑥 ∈ (null‘𝑇) ↔ (𝑥 ∈ ℋ ∧ (𝑇𝑥) = 0)))
214, 20ax-mp 5 . . . . . . . . 9 (𝑥 ∈ (null‘𝑇) ↔ (𝑥 ∈ ℋ ∧ (𝑇𝑥) = 0))
2221simprbi 479 . . . . . . . 8 (𝑥 ∈ (null‘𝑇) → (𝑇𝑥) = 0)
23 elnlfn 28915 . . . . . . . . . 10 (𝑇: ℋ⟶ℂ → (𝑦 ∈ (null‘𝑇) ↔ (𝑦 ∈ ℋ ∧ (𝑇𝑦) = 0)))
244, 23ax-mp 5 . . . . . . . . 9 (𝑦 ∈ (null‘𝑇) ↔ (𝑦 ∈ ℋ ∧ (𝑇𝑦) = 0))
2524simprbi 479 . . . . . . . 8 (𝑦 ∈ (null‘𝑇) → (𝑇𝑦) = 0)
2622, 25oveqan12d 6709 . . . . . . 7 ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → ((𝑇𝑥) + (𝑇𝑦)) = (0 + 0))
2719, 26eqtrd 2685 . . . . . 6 ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 + 𝑦)) = (0 + 0))
28 00id 10249 . . . . . 6 (0 + 0) = 0
2927, 28syl6eq 2701 . . . . 5 ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 + 𝑦)) = 0)
30 elnlfn 28915 . . . . . 6 (𝑇: ℋ⟶ℂ → ((𝑥 + 𝑦) ∈ (null‘𝑇) ↔ ((𝑥 + 𝑦) ∈ ℋ ∧ (𝑇‘(𝑥 + 𝑦)) = 0)))
314, 30ax-mp 5 . . . . 5 ((𝑥 + 𝑦) ∈ (null‘𝑇) ↔ ((𝑥 + 𝑦) ∈ ℋ ∧ (𝑇‘(𝑥 + 𝑦)) = 0))
3217, 29, 31sylanbrc 699 . . . 4 ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 + 𝑦) ∈ (null‘𝑇))
3332rgen2 3004 . . 3 𝑥 ∈ (null‘𝑇)∀𝑦 ∈ (null‘𝑇)(𝑥 + 𝑦) ∈ (null‘𝑇)
34 hvmulcl 27998 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 · 𝑦) ∈ ℋ)
3515, 34sylan2 490 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 · 𝑦) ∈ ℋ)
362lnfnmuli 29031 . . . . . . 7 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑥 · 𝑦)) = (𝑥 · (𝑇𝑦)))
3715, 36sylan2 490 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 · 𝑦)) = (𝑥 · (𝑇𝑦)))
3825oveq2d 6706 . . . . . . 7 (𝑦 ∈ (null‘𝑇) → (𝑥 · (𝑇𝑦)) = (𝑥 · 0))
39 mul01 10253 . . . . . . 7 (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
4038, 39sylan9eqr 2707 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 · (𝑇𝑦)) = 0)
4137, 40eqtrd 2685 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 · 𝑦)) = 0)
42 elnlfn 28915 . . . . . 6 (𝑇: ℋ⟶ℂ → ((𝑥 · 𝑦) ∈ (null‘𝑇) ↔ ((𝑥 · 𝑦) ∈ ℋ ∧ (𝑇‘(𝑥 · 𝑦)) = 0)))
434, 42ax-mp 5 . . . . 5 ((𝑥 · 𝑦) ∈ (null‘𝑇) ↔ ((𝑥 · 𝑦) ∈ ℋ ∧ (𝑇‘(𝑥 · 𝑦)) = 0))
4435, 41, 43sylanbrc 699 . . . 4 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 · 𝑦) ∈ (null‘𝑇))
4544rgen2 3004 . . 3 𝑥 ∈ ℂ ∀𝑦 ∈ (null‘𝑇)(𝑥 · 𝑦) ∈ (null‘𝑇)
4633, 45pm3.2i 470 . 2 (∀𝑥 ∈ (null‘𝑇)∀𝑦 ∈ (null‘𝑇)(𝑥 + 𝑦) ∈ (null‘𝑇) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (null‘𝑇)(𝑥 · 𝑦) ∈ (null‘𝑇))
47 issh3 28204 . . 3 ((null‘𝑇) ⊆ ℋ → ((null‘𝑇) ∈ S ↔ (0 ∈ (null‘𝑇) ∧ (∀𝑥 ∈ (null‘𝑇)∀𝑦 ∈ (null‘𝑇)(𝑥 + 𝑦) ∈ (null‘𝑇) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (null‘𝑇)(𝑥 · 𝑦) ∈ (null‘𝑇)))))
4813, 47ax-mp 5 . 2 ((null‘𝑇) ∈ S ↔ (0 ∈ (null‘𝑇) ∧ (∀𝑥 ∈ (null‘𝑇)∀𝑦 ∈ (null‘𝑇)(𝑥 + 𝑦) ∈ (null‘𝑇) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (null‘𝑇)(𝑥 · 𝑦) ∈ (null‘𝑇))))
497, 46, 48mpbir2an 975 1 (null‘𝑇) ∈ S
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wss 3607  {csn 4210  ccnv 5142  dom cdm 5143  cima 5146  wf 5922  cfv 5926  (class class class)co 6690  cc 9972  0cc0 9974   + caddc 9977   · cmul 9979  chil 27904   + cva 27905   · csm 27906  0c0v 27909   S csh 27913  nullcnl 27937  LinFnclf 27939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-hilex 27984  ax-hfvadd 27985  ax-hv0cl 27988  ax-hvaddid 27989  ax-hfvmul 27990  ax-hvmulid 27991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-ltxr 10117  df-sub 10306  df-sh 28192  df-nlfn 28833  df-lnfn 28835
This theorem is referenced by:  nlelchi  29048
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