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Theorem nlelshi 31308
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 (nullβ€˜π‘‡) ∈ Sβ„‹
Proof of Theorem nlelshi
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-hv0cl 30251 . . 3 0β„Ž ∈ β„‹
2 nlelsh.1 . . . 4 𝑇 ∈ LinFn
32lnfn0i 31290 . . 3 (π‘‡β€˜0β„Ž) = 0
42lnfnfi 31289 . . . 4 𝑇: β„‹βŸΆβ„‚
5 elnlfn 31176 . . . 4 (𝑇: β„‹βŸΆβ„‚ β†’ (0β„Ž ∈ (nullβ€˜π‘‡) ↔ (0β„Ž ∈ β„‹ ∧ (π‘‡β€˜0β„Ž) = 0)))
64, 5ax-mp 5 . . 3 (0β„Ž ∈ (nullβ€˜π‘‡) ↔ (0β„Ž ∈ β„‹ ∧ (π‘‡β€˜0β„Ž) = 0))
71, 3, 6mpbir2an 709 . 2 0β„Ž ∈ (nullβ€˜π‘‡)
8 nlfnval 31129 . . . . . . . . . 10 (𝑇: β„‹βŸΆβ„‚ β†’ (nullβ€˜π‘‡) = (◑𝑇 β€œ {0}))
94, 8ax-mp 5 . . . . . . . . 9 (nullβ€˜π‘‡) = (◑𝑇 β€œ {0})
10 cnvimass 6080 . . . . . . . . 9 (◑𝑇 β€œ {0}) βŠ† dom 𝑇
119, 10eqsstri 4016 . . . . . . . 8 (nullβ€˜π‘‡) βŠ† dom 𝑇
124fdmi 6729 . . . . . . . 8 dom 𝑇 = β„‹
1311, 12sseqtri 4018 . . . . . . 7 (nullβ€˜π‘‡) βŠ† β„‹
1413sseli 3978 . . . . . 6 (π‘₯ ∈ (nullβ€˜π‘‡) β†’ π‘₯ ∈ β„‹)
1513sseli 3978 . . . . . 6 (𝑦 ∈ (nullβ€˜π‘‡) β†’ 𝑦 ∈ β„‹)
16 hvaddcl 30260 . . . . . 6 ((π‘₯ ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (π‘₯ +β„Ž 𝑦) ∈ β„‹)
1714, 15, 16syl2an 596 . . . . 5 ((π‘₯ ∈ (nullβ€˜π‘‡) ∧ 𝑦 ∈ (nullβ€˜π‘‡)) β†’ (π‘₯ +β„Ž 𝑦) ∈ β„‹)
182lnfnaddi 31291 . . . . . . . 8 ((π‘₯ ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (π‘‡β€˜(π‘₯ +β„Ž 𝑦)) = ((π‘‡β€˜π‘₯) + (π‘‡β€˜π‘¦)))
1914, 15, 18syl2an 596 . . . . . . 7 ((π‘₯ ∈ (nullβ€˜π‘‡) ∧ 𝑦 ∈ (nullβ€˜π‘‡)) β†’ (π‘‡β€˜(π‘₯ +β„Ž 𝑦)) = ((π‘‡β€˜π‘₯) + (π‘‡β€˜π‘¦)))
20 elnlfn 31176 . . . . . . . . . 10 (𝑇: β„‹βŸΆβ„‚ β†’ (π‘₯ ∈ (nullβ€˜π‘‡) ↔ (π‘₯ ∈ β„‹ ∧ (π‘‡β€˜π‘₯) = 0)))
214, 20ax-mp 5 . . . . . . . . 9 (π‘₯ ∈ (nullβ€˜π‘‡) ↔ (π‘₯ ∈ β„‹ ∧ (π‘‡β€˜π‘₯) = 0))
2221simprbi 497 . . . . . . . 8 (π‘₯ ∈ (nullβ€˜π‘‡) β†’ (π‘‡β€˜π‘₯) = 0)
23 elnlfn 31176 . . . . . . . . . 10 (𝑇: β„‹βŸΆβ„‚ β†’ (𝑦 ∈ (nullβ€˜π‘‡) ↔ (𝑦 ∈ β„‹ ∧ (π‘‡β€˜π‘¦) = 0)))
244, 23ax-mp 5 . . . . . . . . 9 (𝑦 ∈ (nullβ€˜π‘‡) ↔ (𝑦 ∈ β„‹ ∧ (π‘‡β€˜π‘¦) = 0))
2524simprbi 497 . . . . . . . 8 (𝑦 ∈ (nullβ€˜π‘‡) β†’ (π‘‡β€˜π‘¦) = 0)
2622, 25oveqan12d 7427 . . . . . . 7 ((π‘₯ ∈ (nullβ€˜π‘‡) ∧ 𝑦 ∈ (nullβ€˜π‘‡)) β†’ ((π‘‡β€˜π‘₯) + (π‘‡β€˜π‘¦)) = (0 + 0))
2719, 26eqtrd 2772 . . . . . 6 ((π‘₯ ∈ (nullβ€˜π‘‡) ∧ 𝑦 ∈ (nullβ€˜π‘‡)) β†’ (π‘‡β€˜(π‘₯ +β„Ž 𝑦)) = (0 + 0))
28 00id 11388 . . . . . 6 (0 + 0) = 0
2927, 28eqtrdi 2788 . . . . 5 ((π‘₯ ∈ (nullβ€˜π‘‡) ∧ 𝑦 ∈ (nullβ€˜π‘‡)) β†’ (π‘‡β€˜(π‘₯ +β„Ž 𝑦)) = 0)
30 elnlfn 31176 . . . . . 6 (𝑇: β„‹βŸΆβ„‚ β†’ ((π‘₯ +β„Ž 𝑦) ∈ (nullβ€˜π‘‡) ↔ ((π‘₯ +β„Ž 𝑦) ∈ β„‹ ∧ (π‘‡β€˜(π‘₯ +β„Ž 𝑦)) = 0)))
314, 30ax-mp 5 . . . . 5 ((π‘₯ +β„Ž 𝑦) ∈ (nullβ€˜π‘‡) ↔ ((π‘₯ +β„Ž 𝑦) ∈ β„‹ ∧ (π‘‡β€˜(π‘₯ +β„Ž 𝑦)) = 0))
3217, 29, 31sylanbrc 583 . . . 4 ((π‘₯ ∈ (nullβ€˜π‘‡) ∧ 𝑦 ∈ (nullβ€˜π‘‡)) β†’ (π‘₯ +β„Ž 𝑦) ∈ (nullβ€˜π‘‡))
3332rgen2 3197 . . 3 βˆ€π‘₯ ∈ (nullβ€˜π‘‡)βˆ€π‘¦ ∈ (nullβ€˜π‘‡)(π‘₯ +β„Ž 𝑦) ∈ (nullβ€˜π‘‡)
34 hvmulcl 30261 . . . . . 6 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‹) β†’ (π‘₯ Β·β„Ž 𝑦) ∈ β„‹)
3515, 34sylan2 593 . . . . 5 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ (nullβ€˜π‘‡)) β†’ (π‘₯ Β·β„Ž 𝑦) ∈ β„‹)
362lnfnmuli 31292 . . . . . . 7 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‹) β†’ (π‘‡β€˜(π‘₯ Β·β„Ž 𝑦)) = (π‘₯ Β· (π‘‡β€˜π‘¦)))
3715, 36sylan2 593 . . . . . 6 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ (nullβ€˜π‘‡)) β†’ (π‘‡β€˜(π‘₯ Β·β„Ž 𝑦)) = (π‘₯ Β· (π‘‡β€˜π‘¦)))
3825oveq2d 7424 . . . . . . 7 (𝑦 ∈ (nullβ€˜π‘‡) β†’ (π‘₯ Β· (π‘‡β€˜π‘¦)) = (π‘₯ Β· 0))
39 mul01 11392 . . . . . . 7 (π‘₯ ∈ β„‚ β†’ (π‘₯ Β· 0) = 0)
4038, 39sylan9eqr 2794 . . . . . 6 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ (nullβ€˜π‘‡)) β†’ (π‘₯ Β· (π‘‡β€˜π‘¦)) = 0)
4137, 40eqtrd 2772 . . . . 5 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ (nullβ€˜π‘‡)) β†’ (π‘‡β€˜(π‘₯ Β·β„Ž 𝑦)) = 0)
42 elnlfn 31176 . . . . . 6 (𝑇: β„‹βŸΆβ„‚ β†’ ((π‘₯ Β·β„Ž 𝑦) ∈ (nullβ€˜π‘‡) ↔ ((π‘₯ Β·β„Ž 𝑦) ∈ β„‹ ∧ (π‘‡β€˜(π‘₯ Β·β„Ž 𝑦)) = 0)))
434, 42ax-mp 5 . . . . 5 ((π‘₯ Β·β„Ž 𝑦) ∈ (nullβ€˜π‘‡) ↔ ((π‘₯ Β·β„Ž 𝑦) ∈ β„‹ ∧ (π‘‡β€˜(π‘₯ Β·β„Ž 𝑦)) = 0))
4435, 41, 43sylanbrc 583 . . . 4 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ (nullβ€˜π‘‡)) β†’ (π‘₯ Β·β„Ž 𝑦) ∈ (nullβ€˜π‘‡))
4544rgen2 3197 . . 3 βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ (nullβ€˜π‘‡)(π‘₯ Β·β„Ž 𝑦) ∈ (nullβ€˜π‘‡)
4633, 45pm3.2i 471 . 2 (βˆ€π‘₯ ∈ (nullβ€˜π‘‡)βˆ€π‘¦ ∈ (nullβ€˜π‘‡)(π‘₯ +β„Ž 𝑦) ∈ (nullβ€˜π‘‡) ∧ βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ (nullβ€˜π‘‡)(π‘₯ Β·β„Ž 𝑦) ∈ (nullβ€˜π‘‡))
47 issh3 30467 . . 3 ((nullβ€˜π‘‡) βŠ† β„‹ β†’ ((nullβ€˜π‘‡) ∈ Sβ„‹ ↔ (0β„Ž ∈ (nullβ€˜π‘‡) ∧ (βˆ€π‘₯ ∈ (nullβ€˜π‘‡)βˆ€π‘¦ ∈ (nullβ€˜π‘‡)(π‘₯ +β„Ž 𝑦) ∈ (nullβ€˜π‘‡) ∧ βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ (nullβ€˜π‘‡)(π‘₯ Β·β„Ž 𝑦) ∈ (nullβ€˜π‘‡)))))
4813, 47ax-mp 5 . 2 ((nullβ€˜π‘‡) ∈ Sβ„‹ ↔ (0β„Ž ∈ (nullβ€˜π‘‡) ∧ (βˆ€π‘₯ ∈ (nullβ€˜π‘‡)βˆ€π‘¦ ∈ (nullβ€˜π‘‡)(π‘₯ +β„Ž 𝑦) ∈ (nullβ€˜π‘‡) ∧ βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ (nullβ€˜π‘‡)(π‘₯ Β·β„Ž 𝑦) ∈ (nullβ€˜π‘‡))))
497, 46, 48mpbir2an 709 1 (nullβ€˜π‘‡) ∈ Sβ„‹
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061   βŠ† wss 3948  {csn 4628  β—‘ccnv 5675  dom cdm 5676   β€œ cima 5679  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408  β„‚cc 11107  0cc0 11109   + caddc 11112   Β· cmul 11114   β„‹chba 30167   +β„Ž cva 30168   Β·β„Ž csm 30169  0β„Žc0v 30172   Sβ„‹ csh 30176  nullcnl 30200  LinFnclf 30202
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 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-hilex 30247  ax-hfvadd 30248  ax-hv0cl 30251  ax-hvaddid 30252  ax-hfvmul 30253  ax-hvmulid 30254
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-po 5588  df-so 5589  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11249  df-mnf 11250  df-ltxr 11252  df-sub 11445  df-sh 30455  df-nlfn 31094  df-lnfn 31096
This theorem is referenced by:  nlelchi  31309
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