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Theorem nlelshi 31807
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 30750 . . 3 0β„Ž ∈ β„‹
2 nlelsh.1 . . . 4 𝑇 ∈ LinFn
32lnfn0i 31789 . . 3 (π‘‡β€˜0β„Ž) = 0
42lnfnfi 31788 . . . 4 𝑇: β„‹βŸΆβ„‚
5 elnlfn 31675 . . . 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 708 . 2 0β„Ž ∈ (nullβ€˜π‘‡)
8 nlfnval 31628 . . . . . . . . . 10 (𝑇: β„‹βŸΆβ„‚ β†’ (nullβ€˜π‘‡) = (◑𝑇 β€œ {0}))
94, 8ax-mp 5 . . . . . . . . 9 (nullβ€˜π‘‡) = (◑𝑇 β€œ {0})
10 cnvimass 6071 . . . . . . . . 9 (◑𝑇 β€œ {0}) βŠ† dom 𝑇
119, 10eqsstri 4009 . . . . . . . 8 (nullβ€˜π‘‡) βŠ† dom 𝑇
124fdmi 6720 . . . . . . . 8 dom 𝑇 = β„‹
1311, 12sseqtri 4011 . . . . . . 7 (nullβ€˜π‘‡) βŠ† β„‹
1413sseli 3971 . . . . . 6 (π‘₯ ∈ (nullβ€˜π‘‡) β†’ π‘₯ ∈ β„‹)
1513sseli 3971 . . . . . 6 (𝑦 ∈ (nullβ€˜π‘‡) β†’ 𝑦 ∈ β„‹)
16 hvaddcl 30759 . . . . . 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 31790 . . . . . . . 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 31675 . . . . . . . . . 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 31675 . . . . . . . . . 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 7421 . . . . . . 7 ((π‘₯ ∈ (nullβ€˜π‘‡) ∧ 𝑦 ∈ (nullβ€˜π‘‡)) β†’ ((π‘‡β€˜π‘₯) + (π‘‡β€˜π‘¦)) = (0 + 0))
2719, 26eqtrd 2764 . . . . . 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 2780 . . . . 5 ((π‘₯ ∈ (nullβ€˜π‘‡) ∧ 𝑦 ∈ (nullβ€˜π‘‡)) β†’ (π‘‡β€˜(π‘₯ +β„Ž 𝑦)) = 0)
30 elnlfn 31675 . . . . . 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 3189 . . 3 βˆ€π‘₯ ∈ (nullβ€˜π‘‡)βˆ€π‘¦ ∈ (nullβ€˜π‘‡)(π‘₯ +β„Ž 𝑦) ∈ (nullβ€˜π‘‡)
34 hvmulcl 30760 . . . . . 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 31791 . . . . . . 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 7418 . . . . . . 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 2786 . . . . . 6 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ (nullβ€˜π‘‡)) β†’ (π‘₯ Β· (π‘‡β€˜π‘¦)) = 0)
4137, 40eqtrd 2764 . . . . 5 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ (nullβ€˜π‘‡)) β†’ (π‘‡β€˜(π‘₯ Β·β„Ž 𝑦)) = 0)
42 elnlfn 31675 . . . . . 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 3189 . . 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 30966 . . 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 708 1 (nullβ€˜π‘‡) ∈ Sβ„‹
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053   βŠ† wss 3941  {csn 4621  β—‘ccnv 5666  dom cdm 5667   β€œ cima 5670  βŸΆwf 6530  β€˜cfv 6534  (class class class)co 7402  β„‚cc 11105  0cc0 11107   + caddc 11110   Β· cmul 11112   β„‹chba 30666   +β„Ž cva 30667   Β·β„Ž csm 30668  0β„Žc0v 30671   Sβ„‹ csh 30675  nullcnl 30699  LinFnclf 30701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-hilex 30746  ax-hfvadd 30747  ax-hv0cl 30750  ax-hvaddid 30751  ax-hfvmul 30752  ax-hvmulid 30753
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-po 5579  df-so 5580  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-er 8700  df-map 8819  df-en 8937  df-dom 8938  df-sdom 8939  df-pnf 11249  df-mnf 11250  df-ltxr 11252  df-sub 11445  df-sh 30954  df-nlfn 31593  df-lnfn 31595
This theorem is referenced by:  nlelchi  31808
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