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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) ∈ S

Proof of Theorem nlelshi
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-hv0cl 29945 . . 3 0 ∈ ℋ
2 nlelsh.1 . . . 4 𝑇 ∈ LinFn
32lnfn0i 30984 . . 3 (𝑇‘0) = 0
42lnfnfi 30983 . . . 4 𝑇: ℋ⟶ℂ
5 elnlfn 30870 . . . 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 709 . 2 0 ∈ (null‘𝑇)
8 nlfnval 30823 . . . . . . . . . 10 (𝑇: ℋ⟶ℂ → (null‘𝑇) = (𝑇 “ {0}))
94, 8ax-mp 5 . . . . . . . . 9 (null‘𝑇) = (𝑇 “ {0})
10 cnvimass 6033 . . . . . . . . 9 (𝑇 “ {0}) ⊆ dom 𝑇
119, 10eqsstri 3978 . . . . . . . 8 (null‘𝑇) ⊆ dom 𝑇
124fdmi 6680 . . . . . . . 8 dom 𝑇 = ℋ
1311, 12sseqtri 3980 . . . . . . 7 (null‘𝑇) ⊆ ℋ
1413sseli 3940 . . . . . 6 (𝑥 ∈ (null‘𝑇) → 𝑥 ∈ ℋ)
1513sseli 3940 . . . . . 6 (𝑦 ∈ (null‘𝑇) → 𝑦 ∈ ℋ)
16 hvaddcl 29954 . . . . . 6 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 + 𝑦) ∈ ℋ)
1714, 15, 16syl2an 596 . . . . 5 ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 + 𝑦) ∈ ℋ)
182lnfnaddi 30985 . . . . . . . 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 30870 . . . . . . . . . 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 30870 . . . . . . . . . 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 7376 . . . . . . 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 11330 . . . . . 6 (0 + 0) = 0
2927, 28eqtrdi 2792 . . . . 5 ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 + 𝑦)) = 0)
30 elnlfn 30870 . . . . . 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 3194 . . 3 𝑥 ∈ (null‘𝑇)∀𝑦 ∈ (null‘𝑇)(𝑥 + 𝑦) ∈ (null‘𝑇)
34 hvmulcl 29955 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 · 𝑦) ∈ ℋ)
3515, 34sylan2 593 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 · 𝑦) ∈ ℋ)
362lnfnmuli 30986 . . . . . . 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 7373 . . . . . . 7 (𝑦 ∈ (null‘𝑇) → (𝑥 · (𝑇𝑦)) = (𝑥 · 0))
39 mul01 11334 . . . . . . 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 30870 . . . . . 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 3194 . . 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 30161 . . 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 709 1 (null‘𝑇) ∈ S
Colors of variables: wff setvar class
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  cc 11049  0cc0 11051   + caddc 11054   · cmul 11056  chba 29861   + cva 29862   · csm 29863  0c0v 29866   S 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
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