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Theorem nlelshi 32092
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 31035 . . 3 0 ∈ ℋ
2 nlelsh.1 . . . 4 𝑇 ∈ LinFn
32lnfn0i 32074 . . 3 (𝑇‘0) = 0
42lnfnfi 32073 . . . 4 𝑇: ℋ⟶ℂ
5 elnlfn 31960 . . . 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 710 . 2 0 ∈ (null‘𝑇)
8 nlfnval 31913 . . . . . . . . . 10 (𝑇: ℋ⟶ℂ → (null‘𝑇) = (𝑇 “ {0}))
94, 8ax-mp 5 . . . . . . . . 9 (null‘𝑇) = (𝑇 “ {0})
10 cnvimass 6111 . . . . . . . . 9 (𝑇 “ {0}) ⊆ dom 𝑇
119, 10eqsstri 4043 . . . . . . . 8 (null‘𝑇) ⊆ dom 𝑇
124fdmi 6758 . . . . . . . 8 dom 𝑇 = ℋ
1311, 12sseqtri 4045 . . . . . . 7 (null‘𝑇) ⊆ ℋ
1413sseli 4004 . . . . . 6 (𝑥 ∈ (null‘𝑇) → 𝑥 ∈ ℋ)
1513sseli 4004 . . . . . 6 (𝑦 ∈ (null‘𝑇) → 𝑦 ∈ ℋ)
16 hvaddcl 31044 . . . . . 6 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 + 𝑦) ∈ ℋ)
1714, 15, 16syl2an 595 . . . . 5 ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 + 𝑦) ∈ ℋ)
182lnfnaddi 32075 . . . . . . . 8 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑥 + 𝑦)) = ((𝑇𝑥) + (𝑇𝑦)))
1914, 15, 18syl2an 595 . . . . . . 7 ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 + 𝑦)) = ((𝑇𝑥) + (𝑇𝑦)))
20 elnlfn 31960 . . . . . . . . . 10 (𝑇: ℋ⟶ℂ → (𝑥 ∈ (null‘𝑇) ↔ (𝑥 ∈ ℋ ∧ (𝑇𝑥) = 0)))
214, 20ax-mp 5 . . . . . . . . 9 (𝑥 ∈ (null‘𝑇) ↔ (𝑥 ∈ ℋ ∧ (𝑇𝑥) = 0))
2221simprbi 496 . . . . . . . 8 (𝑥 ∈ (null‘𝑇) → (𝑇𝑥) = 0)
23 elnlfn 31960 . . . . . . . . . 10 (𝑇: ℋ⟶ℂ → (𝑦 ∈ (null‘𝑇) ↔ (𝑦 ∈ ℋ ∧ (𝑇𝑦) = 0)))
244, 23ax-mp 5 . . . . . . . . 9 (𝑦 ∈ (null‘𝑇) ↔ (𝑦 ∈ ℋ ∧ (𝑇𝑦) = 0))
2524simprbi 496 . . . . . . . 8 (𝑦 ∈ (null‘𝑇) → (𝑇𝑦) = 0)
2622, 25oveqan12d 7467 . . . . . . 7 ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → ((𝑇𝑥) + (𝑇𝑦)) = (0 + 0))
2719, 26eqtrd 2780 . . . . . 6 ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 + 𝑦)) = (0 + 0))
28 00id 11465 . . . . . 6 (0 + 0) = 0
2927, 28eqtrdi 2796 . . . . 5 ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 + 𝑦)) = 0)
30 elnlfn 31960 . . . . . 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 582 . . . 4 ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 + 𝑦) ∈ (null‘𝑇))
3332rgen2 3205 . . 3 𝑥 ∈ (null‘𝑇)∀𝑦 ∈ (null‘𝑇)(𝑥 + 𝑦) ∈ (null‘𝑇)
34 hvmulcl 31045 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 · 𝑦) ∈ ℋ)
3515, 34sylan2 592 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 · 𝑦) ∈ ℋ)
362lnfnmuli 32076 . . . . . . 7 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑥 · 𝑦)) = (𝑥 · (𝑇𝑦)))
3715, 36sylan2 592 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 · 𝑦)) = (𝑥 · (𝑇𝑦)))
3825oveq2d 7464 . . . . . . 7 (𝑦 ∈ (null‘𝑇) → (𝑥 · (𝑇𝑦)) = (𝑥 · 0))
39 mul01 11469 . . . . . . 7 (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
4038, 39sylan9eqr 2802 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 · (𝑇𝑦)) = 0)
4137, 40eqtrd 2780 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 · 𝑦)) = 0)
42 elnlfn 31960 . . . . . 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 582 . . . 4 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 · 𝑦) ∈ (null‘𝑇))
4544rgen2 3205 . . 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 31251 . . 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 710 1 (null‘𝑇) ∈ S
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2108  wral 3067  wss 3976  {csn 4648  ccnv 5699  dom cdm 5700  cima 5703  wf 6569  cfv 6573  (class class class)co 7448  cc 11182  0cc0 11184   + caddc 11187   · cmul 11189  chba 30951   + cva 30952   · csm 30953  0c0v 30956   S csh 30960  nullcnl 30984  LinFnclf 30986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-hilex 31031  ax-hfvadd 31032  ax-hv0cl 31035  ax-hvaddid 31036  ax-hfvmul 31037  ax-hvmulid 31038
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-po 5607  df-so 5608  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-ltxr 11329  df-sub 11522  df-sh 31239  df-nlfn 31878  df-lnfn 31880
This theorem is referenced by:  nlelchi  32093
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