HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  nlelshi Structured version   Visualization version   GIF version

Theorem nlelshi 28807
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 27748 . . 3 0 ∈ ℋ
2 nlelsh.1 . . . 4 𝑇 ∈ LinFn
32lnfn0i 28789 . . 3 (𝑇‘0) = 0
42lnfnfi 28788 . . . 4 𝑇: ℋ⟶ℂ
5 elnlfn 28675 . . . 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 954 . 2 0 ∈ (null‘𝑇)
8 nlfnval 28628 . . . . . . . . . 10 (𝑇: ℋ⟶ℂ → (null‘𝑇) = (𝑇 “ {0}))
94, 8ax-mp 5 . . . . . . . . 9 (null‘𝑇) = (𝑇 “ {0})
10 cnvimass 5454 . . . . . . . . 9 (𝑇 “ {0}) ⊆ dom 𝑇
119, 10eqsstri 3620 . . . . . . . 8 (null‘𝑇) ⊆ dom 𝑇
124fdmi 6019 . . . . . . . 8 dom 𝑇 = ℋ
1311, 12sseqtri 3622 . . . . . . 7 (null‘𝑇) ⊆ ℋ
1413sseli 3584 . . . . . 6 (𝑥 ∈ (null‘𝑇) → 𝑥 ∈ ℋ)
1513sseli 3584 . . . . . 6 (𝑦 ∈ (null‘𝑇) → 𝑦 ∈ ℋ)
16 hvaddcl 27757 . . . . . 6 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 + 𝑦) ∈ ℋ)
1714, 15, 16syl2an 494 . . . . 5 ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 + 𝑦) ∈ ℋ)
182lnfnaddi 28790 . . . . . . . 8 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑥 + 𝑦)) = ((𝑇𝑥) + (𝑇𝑦)))
1914, 15, 18syl2an 494 . . . . . . 7 ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 + 𝑦)) = ((𝑇𝑥) + (𝑇𝑦)))
20 elnlfn 28675 . . . . . . . . . 10 (𝑇: ℋ⟶ℂ → (𝑥 ∈ (null‘𝑇) ↔ (𝑥 ∈ ℋ ∧ (𝑇𝑥) = 0)))
214, 20ax-mp 5 . . . . . . . . 9 (𝑥 ∈ (null‘𝑇) ↔ (𝑥 ∈ ℋ ∧ (𝑇𝑥) = 0))
2221simprbi 480 . . . . . . . 8 (𝑥 ∈ (null‘𝑇) → (𝑇𝑥) = 0)
23 elnlfn 28675 . . . . . . . . . 10 (𝑇: ℋ⟶ℂ → (𝑦 ∈ (null‘𝑇) ↔ (𝑦 ∈ ℋ ∧ (𝑇𝑦) = 0)))
244, 23ax-mp 5 . . . . . . . . 9 (𝑦 ∈ (null‘𝑇) ↔ (𝑦 ∈ ℋ ∧ (𝑇𝑦) = 0))
2524simprbi 480 . . . . . . . 8 (𝑦 ∈ (null‘𝑇) → (𝑇𝑦) = 0)
2622, 25oveqan12d 6634 . . . . . . 7 ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → ((𝑇𝑥) + (𝑇𝑦)) = (0 + 0))
2719, 26eqtrd 2655 . . . . . 6 ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 + 𝑦)) = (0 + 0))
28 00id 10171 . . . . . 6 (0 + 0) = 0
2927, 28syl6eq 2671 . . . . 5 ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 + 𝑦)) = 0)
30 elnlfn 28675 . . . . . 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 697 . . . 4 ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 + 𝑦) ∈ (null‘𝑇))
3332rgen2 2971 . . 3 𝑥 ∈ (null‘𝑇)∀𝑦 ∈ (null‘𝑇)(𝑥 + 𝑦) ∈ (null‘𝑇)
34 hvmulcl 27758 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 · 𝑦) ∈ ℋ)
3515, 34sylan2 491 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 · 𝑦) ∈ ℋ)
362lnfnmuli 28791 . . . . . . 7 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑥 · 𝑦)) = (𝑥 · (𝑇𝑦)))
3715, 36sylan2 491 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 · 𝑦)) = (𝑥 · (𝑇𝑦)))
3825oveq2d 6631 . . . . . . 7 (𝑦 ∈ (null‘𝑇) → (𝑥 · (𝑇𝑦)) = (𝑥 · 0))
39 mul01 10175 . . . . . . 7 (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
4038, 39sylan9eqr 2677 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 · (𝑇𝑦)) = 0)
4137, 40eqtrd 2655 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 · 𝑦)) = 0)
42 elnlfn 28675 . . . . . 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 697 . . . 4 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 · 𝑦) ∈ (null‘𝑇))
4544rgen2 2971 . . 3 𝑥 ∈ ℂ ∀𝑦 ∈ (null‘𝑇)(𝑥 · 𝑦) ∈ (null‘𝑇)
4633, 45pm3.2i 471 . 2 (∀𝑥 ∈ (null‘𝑇)∀𝑦 ∈ (null‘𝑇)(𝑥 + 𝑦) ∈ (null‘𝑇) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (null‘𝑇)(𝑥 · 𝑦) ∈ (null‘𝑇))
47 issh3 27964 . . 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 954 1 (null‘𝑇) ∈ S
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2908  wss 3560  {csn 4155  ccnv 5083  dom cdm 5084  cima 5087  wf 5853  cfv 5857  (class class class)co 6615  cc 9894  0cc0 9896   + caddc 9899   · cmul 9901  chil 27664   + cva 27665   · csm 27666  0c0v 27669   S csh 27673  nullcnl 27697  LinFnclf 27699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-hilex 27744  ax-hfvadd 27745  ax-hv0cl 27748  ax-hvaddid 27749  ax-hfvmul 27750  ax-hvmulid 27751
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-po 5005  df-so 5006  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-sdom 7918  df-pnf 10036  df-mnf 10037  df-ltxr 10039  df-sub 10228  df-sh 27952  df-nlfn 28593  df-lnfn 28595
This theorem is referenced by:  nlelchi  28808
  Copyright terms: Public domain W3C validator