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Theorem nlelshi 31869
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 30812 . . 3 0β„Ž ∈ β„‹
2 nlelsh.1 . . . 4 𝑇 ∈ LinFn
32lnfn0i 31851 . . 3 (π‘‡β€˜0β„Ž) = 0
42lnfnfi 31850 . . . 4 𝑇: β„‹βŸΆβ„‚
5 elnlfn 31737 . . . 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 710 . 2 0β„Ž ∈ (nullβ€˜π‘‡)
8 nlfnval 31690 . . . . . . . . . 10 (𝑇: β„‹βŸΆβ„‚ β†’ (nullβ€˜π‘‡) = (◑𝑇 β€œ {0}))
94, 8ax-mp 5 . . . . . . . . 9 (nullβ€˜π‘‡) = (◑𝑇 β€œ {0})
10 cnvimass 6085 . . . . . . . . 9 (◑𝑇 β€œ {0}) βŠ† dom 𝑇
119, 10eqsstri 4014 . . . . . . . 8 (nullβ€˜π‘‡) βŠ† dom 𝑇
124fdmi 6734 . . . . . . . 8 dom 𝑇 = β„‹
1311, 12sseqtri 4016 . . . . . . 7 (nullβ€˜π‘‡) βŠ† β„‹
1413sseli 3976 . . . . . 6 (π‘₯ ∈ (nullβ€˜π‘‡) β†’ π‘₯ ∈ β„‹)
1513sseli 3976 . . . . . 6 (𝑦 ∈ (nullβ€˜π‘‡) β†’ 𝑦 ∈ β„‹)
16 hvaddcl 30821 . . . . . 6 ((π‘₯ ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (π‘₯ +β„Ž 𝑦) ∈ β„‹)
1714, 15, 16syl2an 595 . . . . 5 ((π‘₯ ∈ (nullβ€˜π‘‡) ∧ 𝑦 ∈ (nullβ€˜π‘‡)) β†’ (π‘₯ +β„Ž 𝑦) ∈ β„‹)
182lnfnaddi 31852 . . . . . . . 8 ((π‘₯ ∈ β„‹ ∧ 𝑦 ∈ β„‹) β†’ (π‘‡β€˜(π‘₯ +β„Ž 𝑦)) = ((π‘‡β€˜π‘₯) + (π‘‡β€˜π‘¦)))
1914, 15, 18syl2an 595 . . . . . . 7 ((π‘₯ ∈ (nullβ€˜π‘‡) ∧ 𝑦 ∈ (nullβ€˜π‘‡)) β†’ (π‘‡β€˜(π‘₯ +β„Ž 𝑦)) = ((π‘‡β€˜π‘₯) + (π‘‡β€˜π‘¦)))
20 elnlfn 31737 . . . . . . . . . 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 496 . . . . . . . 8 (π‘₯ ∈ (nullβ€˜π‘‡) β†’ (π‘‡β€˜π‘₯) = 0)
23 elnlfn 31737 . . . . . . . . . 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 496 . . . . . . . 8 (𝑦 ∈ (nullβ€˜π‘‡) β†’ (π‘‡β€˜π‘¦) = 0)
2622, 25oveqan12d 7439 . . . . . . 7 ((π‘₯ ∈ (nullβ€˜π‘‡) ∧ 𝑦 ∈ (nullβ€˜π‘‡)) β†’ ((π‘‡β€˜π‘₯) + (π‘‡β€˜π‘¦)) = (0 + 0))
2719, 26eqtrd 2768 . . . . . 6 ((π‘₯ ∈ (nullβ€˜π‘‡) ∧ 𝑦 ∈ (nullβ€˜π‘‡)) β†’ (π‘‡β€˜(π‘₯ +β„Ž 𝑦)) = (0 + 0))
28 00id 11419 . . . . . 6 (0 + 0) = 0
2927, 28eqtrdi 2784 . . . . 5 ((π‘₯ ∈ (nullβ€˜π‘‡) ∧ 𝑦 ∈ (nullβ€˜π‘‡)) β†’ (π‘‡β€˜(π‘₯ +β„Ž 𝑦)) = 0)
30 elnlfn 31737 . . . . . 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 582 . . . 4 ((π‘₯ ∈ (nullβ€˜π‘‡) ∧ 𝑦 ∈ (nullβ€˜π‘‡)) β†’ (π‘₯ +β„Ž 𝑦) ∈ (nullβ€˜π‘‡))
3332rgen2 3194 . . 3 βˆ€π‘₯ ∈ (nullβ€˜π‘‡)βˆ€π‘¦ ∈ (nullβ€˜π‘‡)(π‘₯ +β„Ž 𝑦) ∈ (nullβ€˜π‘‡)
34 hvmulcl 30822 . . . . . 6 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‹) β†’ (π‘₯ Β·β„Ž 𝑦) ∈ β„‹)
3515, 34sylan2 592 . . . . 5 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ (nullβ€˜π‘‡)) β†’ (π‘₯ Β·β„Ž 𝑦) ∈ β„‹)
362lnfnmuli 31853 . . . . . . 7 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‹) β†’ (π‘‡β€˜(π‘₯ Β·β„Ž 𝑦)) = (π‘₯ Β· (π‘‡β€˜π‘¦)))
3715, 36sylan2 592 . . . . . 6 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ (nullβ€˜π‘‡)) β†’ (π‘‡β€˜(π‘₯ Β·β„Ž 𝑦)) = (π‘₯ Β· (π‘‡β€˜π‘¦)))
3825oveq2d 7436 . . . . . . 7 (𝑦 ∈ (nullβ€˜π‘‡) β†’ (π‘₯ Β· (π‘‡β€˜π‘¦)) = (π‘₯ Β· 0))
39 mul01 11423 . . . . . . 7 (π‘₯ ∈ β„‚ β†’ (π‘₯ Β· 0) = 0)
4038, 39sylan9eqr 2790 . . . . . 6 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ (nullβ€˜π‘‡)) β†’ (π‘₯ Β· (π‘‡β€˜π‘¦)) = 0)
4137, 40eqtrd 2768 . . . . 5 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ (nullβ€˜π‘‡)) β†’ (π‘‡β€˜(π‘₯ Β·β„Ž 𝑦)) = 0)
42 elnlfn 31737 . . . . . 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 582 . . . 4 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ (nullβ€˜π‘‡)) β†’ (π‘₯ Β·β„Ž 𝑦) ∈ (nullβ€˜π‘‡))
4544rgen2 3194 . . 3 βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ (nullβ€˜π‘‡)(π‘₯ Β·β„Ž 𝑦) ∈ (nullβ€˜π‘‡)
4633, 45pm3.2i 470 . 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 31028 . . 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 710 1 (nullβ€˜π‘‡) ∈ Sβ„‹
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βˆ€wral 3058   βŠ† wss 3947  {csn 4629  β—‘ccnv 5677  dom cdm 5678   β€œ cima 5681  βŸΆwf 6544  β€˜cfv 6548  (class class class)co 7420  β„‚cc 11136  0cc0 11138   + caddc 11141   Β· cmul 11143   β„‹chba 30728   +β„Ž cva 30729   Β·β„Ž csm 30730  0β„Žc0v 30733   Sβ„‹ csh 30737  nullcnl 30761  LinFnclf 30763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-hilex 30808  ax-hfvadd 30809  ax-hv0cl 30812  ax-hvaddid 30813  ax-hfvmul 30814  ax-hvmulid 30815
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-po 5590  df-so 5591  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-er 8724  df-map 8846  df-en 8964  df-dom 8965  df-sdom 8966  df-pnf 11280  df-mnf 11281  df-ltxr 11283  df-sub 11476  df-sh 31016  df-nlfn 31655  df-lnfn 31657
This theorem is referenced by:  nlelchi  31870
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