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Theorem nlelshi 30710
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 29653 . . 3 0 ∈ ℋ
2 nlelsh.1 . . . 4 𝑇 ∈ LinFn
32lnfn0i 30692 . . 3 (𝑇‘0) = 0
42lnfnfi 30691 . . . 4 𝑇: ℋ⟶ℂ
5 elnlfn 30578 . . . 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 708 . 2 0 ∈ (null‘𝑇)
8 nlfnval 30531 . . . . . . . . . 10 (𝑇: ℋ⟶ℂ → (null‘𝑇) = (𝑇 “ {0}))
94, 8ax-mp 5 . . . . . . . . 9 (null‘𝑇) = (𝑇 “ {0})
10 cnvimass 6019 . . . . . . . . 9 (𝑇 “ {0}) ⊆ dom 𝑇
119, 10eqsstri 3966 . . . . . . . 8 (null‘𝑇) ⊆ dom 𝑇
124fdmi 6663 . . . . . . . 8 dom 𝑇 = ℋ
1311, 12sseqtri 3968 . . . . . . 7 (null‘𝑇) ⊆ ℋ
1413sseli 3928 . . . . . 6 (𝑥 ∈ (null‘𝑇) → 𝑥 ∈ ℋ)
1513sseli 3928 . . . . . 6 (𝑦 ∈ (null‘𝑇) → 𝑦 ∈ ℋ)
16 hvaddcl 29662 . . . . . 6 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 + 𝑦) ∈ ℋ)
1714, 15, 16syl2an 596 . . . . 5 ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 + 𝑦) ∈ ℋ)
182lnfnaddi 30693 . . . . . . . 8 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑥 + 𝑦)) = ((𝑇𝑥) + (𝑇𝑦)))
1914, 15, 18syl2an 596 . . . . . . 7 ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 + 𝑦)) = ((𝑇𝑥) + (𝑇𝑦)))
20 elnlfn 30578 . . . . . . . . . 10 (𝑇: ℋ⟶ℂ → (𝑥 ∈ (null‘𝑇) ↔ (𝑥 ∈ ℋ ∧ (𝑇𝑥) = 0)))
214, 20ax-mp 5 . . . . . . . . 9 (𝑥 ∈ (null‘𝑇) ↔ (𝑥 ∈ ℋ ∧ (𝑇𝑥) = 0))
2221simprbi 497 . . . . . . . 8 (𝑥 ∈ (null‘𝑇) → (𝑇𝑥) = 0)
23 elnlfn 30578 . . . . . . . . . 10 (𝑇: ℋ⟶ℂ → (𝑦 ∈ (null‘𝑇) ↔ (𝑦 ∈ ℋ ∧ (𝑇𝑦) = 0)))
244, 23ax-mp 5 . . . . . . . . 9 (𝑦 ∈ (null‘𝑇) ↔ (𝑦 ∈ ℋ ∧ (𝑇𝑦) = 0))
2524simprbi 497 . . . . . . . 8 (𝑦 ∈ (null‘𝑇) → (𝑇𝑦) = 0)
2622, 25oveqan12d 7356 . . . . . . 7 ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → ((𝑇𝑥) + (𝑇𝑦)) = (0 + 0))
2719, 26eqtrd 2776 . . . . . 6 ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 + 𝑦)) = (0 + 0))
28 00id 11251 . . . . . 6 (0 + 0) = 0
2927, 28eqtrdi 2792 . . . . 5 ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 + 𝑦)) = 0)
30 elnlfn 30578 . . . . . 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 583 . . . 4 ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 + 𝑦) ∈ (null‘𝑇))
3332rgen2 3190 . . 3 𝑥 ∈ (null‘𝑇)∀𝑦 ∈ (null‘𝑇)(𝑥 + 𝑦) ∈ (null‘𝑇)
34 hvmulcl 29663 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 · 𝑦) ∈ ℋ)
3515, 34sylan2 593 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 · 𝑦) ∈ ℋ)
362lnfnmuli 30694 . . . . . . 7 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑥 · 𝑦)) = (𝑥 · (𝑇𝑦)))
3715, 36sylan2 593 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 · 𝑦)) = (𝑥 · (𝑇𝑦)))
3825oveq2d 7353 . . . . . . 7 (𝑦 ∈ (null‘𝑇) → (𝑥 · (𝑇𝑦)) = (𝑥 · 0))
39 mul01 11255 . . . . . . 7 (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
4038, 39sylan9eqr 2798 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 · (𝑇𝑦)) = 0)
4137, 40eqtrd 2776 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 · 𝑦)) = 0)
42 elnlfn 30578 . . . . . 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 583 . . . 4 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 · 𝑦) ∈ (null‘𝑇))
4544rgen2 3190 . . 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 29869 . . 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 708 1 (null‘𝑇) ∈ S
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1540  wcel 2105  wral 3061  wss 3898  {csn 4573  ccnv 5619  dom cdm 5620  cima 5623  wf 6475  cfv 6479  (class class class)co 7337  cc 10970  0cc0 10972   + caddc 10975   · cmul 10977  chba 29569   + cva 29570   · csm 29571  0c0v 29574   S csh 29578  nullcnl 29602  LinFnclf 29604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-cnex 11028  ax-resscn 11029  ax-1cn 11030  ax-icn 11031  ax-addcl 11032  ax-addrcl 11033  ax-mulcl 11034  ax-mulrcl 11035  ax-mulcom 11036  ax-addass 11037  ax-mulass 11038  ax-distr 11039  ax-i2m1 11040  ax-1ne0 11041  ax-1rid 11042  ax-rnegex 11043  ax-rrecex 11044  ax-cnre 11045  ax-pre-lttri 11046  ax-pre-lttrn 11047  ax-pre-ltadd 11048  ax-hilex 29649  ax-hfvadd 29650  ax-hv0cl 29653  ax-hvaddid 29654  ax-hfvmul 29655  ax-hvmulid 29656
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-po 5532  df-so 5533  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-er 8569  df-map 8688  df-en 8805  df-dom 8806  df-sdom 8807  df-pnf 11112  df-mnf 11113  df-ltxr 11115  df-sub 11308  df-sh 29857  df-nlfn 30496  df-lnfn 30498
This theorem is referenced by:  nlelchi  30711
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