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Theorem elnlfn 31176
Description: Membership in the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
elnlfn (𝑇: β„‹βŸΆβ„‚ β†’ (𝐴 ∈ (nullβ€˜π‘‡) ↔ (𝐴 ∈ β„‹ ∧ (π‘‡β€˜π΄) = 0)))

Proof of Theorem elnlfn
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 nlfnval 31129 . . . . . 6 (𝑇: β„‹βŸΆβ„‚ β†’ (nullβ€˜π‘‡) = (◑𝑇 β€œ {0}))
2 cnvimass 6080 . . . . . 6 (◑𝑇 β€œ {0}) βŠ† dom 𝑇
31, 2eqsstrdi 4036 . . . . 5 (𝑇: β„‹βŸΆβ„‚ β†’ (nullβ€˜π‘‡) βŠ† dom 𝑇)
4 fdm 6726 . . . . 5 (𝑇: β„‹βŸΆβ„‚ β†’ dom 𝑇 = β„‹)
53, 4sseqtrd 4022 . . . 4 (𝑇: β„‹βŸΆβ„‚ β†’ (nullβ€˜π‘‡) βŠ† β„‹)
65sseld 3981 . . 3 (𝑇: β„‹βŸΆβ„‚ β†’ (𝐴 ∈ (nullβ€˜π‘‡) β†’ 𝐴 ∈ β„‹))
76pm4.71rd 563 . 2 (𝑇: β„‹βŸΆβ„‚ β†’ (𝐴 ∈ (nullβ€˜π‘‡) ↔ (𝐴 ∈ β„‹ ∧ 𝐴 ∈ (nullβ€˜π‘‡))))
81eleq2d 2819 . . . . 5 (𝑇: β„‹βŸΆβ„‚ β†’ (𝐴 ∈ (nullβ€˜π‘‡) ↔ 𝐴 ∈ (◑𝑇 β€œ {0})))
98adantr 481 . . . 4 ((𝑇: β„‹βŸΆβ„‚ ∧ 𝐴 ∈ β„‹) β†’ (𝐴 ∈ (nullβ€˜π‘‡) ↔ 𝐴 ∈ (◑𝑇 β€œ {0})))
10 ffn 6717 . . . . 5 (𝑇: β„‹βŸΆβ„‚ β†’ 𝑇 Fn β„‹)
11 eleq1 2821 . . . . . . . 8 (π‘₯ = 𝐴 β†’ (π‘₯ ∈ (◑𝑇 β€œ {0}) ↔ 𝐴 ∈ (◑𝑇 β€œ {0})))
12 fveqeq2 6900 . . . . . . . 8 (π‘₯ = 𝐴 β†’ ((π‘‡β€˜π‘₯) = 0 ↔ (π‘‡β€˜π΄) = 0))
1311, 12bibi12d 345 . . . . . . 7 (π‘₯ = 𝐴 β†’ ((π‘₯ ∈ (◑𝑇 β€œ {0}) ↔ (π‘‡β€˜π‘₯) = 0) ↔ (𝐴 ∈ (◑𝑇 β€œ {0}) ↔ (π‘‡β€˜π΄) = 0)))
1413imbi2d 340 . . . . . 6 (π‘₯ = 𝐴 β†’ ((𝑇 Fn β„‹ β†’ (π‘₯ ∈ (◑𝑇 β€œ {0}) ↔ (π‘‡β€˜π‘₯) = 0)) ↔ (𝑇 Fn β„‹ β†’ (𝐴 ∈ (◑𝑇 β€œ {0}) ↔ (π‘‡β€˜π΄) = 0))))
15 0cn 11205 . . . . . . . . 9 0 ∈ β„‚
16 vex 3478 . . . . . . . . . 10 π‘₯ ∈ V
1716eliniseg 6093 . . . . . . . . 9 (0 ∈ β„‚ β†’ (π‘₯ ∈ (◑𝑇 β€œ {0}) ↔ π‘₯𝑇0))
1815, 17ax-mp 5 . . . . . . . 8 (π‘₯ ∈ (◑𝑇 β€œ {0}) ↔ π‘₯𝑇0)
19 fnbrfvb 6944 . . . . . . . 8 ((𝑇 Fn β„‹ ∧ π‘₯ ∈ β„‹) β†’ ((π‘‡β€˜π‘₯) = 0 ↔ π‘₯𝑇0))
2018, 19bitr4id 289 . . . . . . 7 ((𝑇 Fn β„‹ ∧ π‘₯ ∈ β„‹) β†’ (π‘₯ ∈ (◑𝑇 β€œ {0}) ↔ (π‘‡β€˜π‘₯) = 0))
2120expcom 414 . . . . . 6 (π‘₯ ∈ β„‹ β†’ (𝑇 Fn β„‹ β†’ (π‘₯ ∈ (◑𝑇 β€œ {0}) ↔ (π‘‡β€˜π‘₯) = 0)))
2214, 21vtoclga 3565 . . . . 5 (𝐴 ∈ β„‹ β†’ (𝑇 Fn β„‹ β†’ (𝐴 ∈ (◑𝑇 β€œ {0}) ↔ (π‘‡β€˜π΄) = 0)))
2310, 22mpan9 507 . . . 4 ((𝑇: β„‹βŸΆβ„‚ ∧ 𝐴 ∈ β„‹) β†’ (𝐴 ∈ (◑𝑇 β€œ {0}) ↔ (π‘‡β€˜π΄) = 0))
249, 23bitrd 278 . . 3 ((𝑇: β„‹βŸΆβ„‚ ∧ 𝐴 ∈ β„‹) β†’ (𝐴 ∈ (nullβ€˜π‘‡) ↔ (π‘‡β€˜π΄) = 0))
2524pm5.32da 579 . 2 (𝑇: β„‹βŸΆβ„‚ β†’ ((𝐴 ∈ β„‹ ∧ 𝐴 ∈ (nullβ€˜π‘‡)) ↔ (𝐴 ∈ β„‹ ∧ (π‘‡β€˜π΄) = 0)))
267, 25bitrd 278 1 (𝑇: β„‹βŸΆβ„‚ β†’ (𝐴 ∈ (nullβ€˜π‘‡) ↔ (𝐴 ∈ β„‹ ∧ (π‘‡β€˜π΄) = 0)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {csn 4628   class class class wbr 5148  β—‘ccnv 5675  dom cdm 5676   β€œ cima 5679   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  β„‚cc 11107  0cc0 11109   β„‹chba 30167  nullcnl 30200
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-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-mulcl 11171  ax-i2m1 11177  ax-hilex 30247
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  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-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  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-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  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-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-map 8821  df-nlfn 31094
This theorem is referenced by:  elnlfn2  31177  nlelshi  31308  nlelchi  31309  riesz3i  31310
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