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Theorem elnlfn 30919
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 30872 . . . . . 6 (𝑇: β„‹βŸΆβ„‚ β†’ (nullβ€˜π‘‡) = (◑𝑇 β€œ {0}))
2 cnvimass 6037 . . . . . 6 (◑𝑇 β€œ {0}) βŠ† dom 𝑇
31, 2eqsstrdi 4002 . . . . 5 (𝑇: β„‹βŸΆβ„‚ β†’ (nullβ€˜π‘‡) βŠ† dom 𝑇)
4 fdm 6681 . . . . 5 (𝑇: β„‹βŸΆβ„‚ β†’ dom 𝑇 = β„‹)
53, 4sseqtrd 3988 . . . 4 (𝑇: β„‹βŸΆβ„‚ β†’ (nullβ€˜π‘‡) βŠ† β„‹)
65sseld 3947 . . 3 (𝑇: β„‹βŸΆβ„‚ β†’ (𝐴 ∈ (nullβ€˜π‘‡) β†’ 𝐴 ∈ β„‹))
76pm4.71rd 564 . 2 (𝑇: β„‹βŸΆβ„‚ β†’ (𝐴 ∈ (nullβ€˜π‘‡) ↔ (𝐴 ∈ β„‹ ∧ 𝐴 ∈ (nullβ€˜π‘‡))))
81eleq2d 2820 . . . . 5 (𝑇: β„‹βŸΆβ„‚ β†’ (𝐴 ∈ (nullβ€˜π‘‡) ↔ 𝐴 ∈ (◑𝑇 β€œ {0})))
98adantr 482 . . . 4 ((𝑇: β„‹βŸΆβ„‚ ∧ 𝐴 ∈ β„‹) β†’ (𝐴 ∈ (nullβ€˜π‘‡) ↔ 𝐴 ∈ (◑𝑇 β€œ {0})))
10 ffn 6672 . . . . 5 (𝑇: β„‹βŸΆβ„‚ β†’ 𝑇 Fn β„‹)
11 eleq1 2822 . . . . . . . 8 (π‘₯ = 𝐴 β†’ (π‘₯ ∈ (◑𝑇 β€œ {0}) ↔ 𝐴 ∈ (◑𝑇 β€œ {0})))
12 fveqeq2 6855 . . . . . . . 8 (π‘₯ = 𝐴 β†’ ((π‘‡β€˜π‘₯) = 0 ↔ (π‘‡β€˜π΄) = 0))
1311, 12bibi12d 346 . . . . . . 7 (π‘₯ = 𝐴 β†’ ((π‘₯ ∈ (◑𝑇 β€œ {0}) ↔ (π‘‡β€˜π‘₯) = 0) ↔ (𝐴 ∈ (◑𝑇 β€œ {0}) ↔ (π‘‡β€˜π΄) = 0)))
1413imbi2d 341 . . . . . 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 11155 . . . . . . . . 9 0 ∈ β„‚
16 vex 3451 . . . . . . . . . 10 π‘₯ ∈ V
1716eliniseg 6050 . . . . . . . . 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 6899 . . . . . . . 8 ((𝑇 Fn β„‹ ∧ π‘₯ ∈ β„‹) β†’ ((π‘‡β€˜π‘₯) = 0 ↔ π‘₯𝑇0))
2018, 19bitr4id 290 . . . . . . 7 ((𝑇 Fn β„‹ ∧ π‘₯ ∈ β„‹) β†’ (π‘₯ ∈ (◑𝑇 β€œ {0}) ↔ (π‘‡β€˜π‘₯) = 0))
2120expcom 415 . . . . . 6 (π‘₯ ∈ β„‹ β†’ (𝑇 Fn β„‹ β†’ (π‘₯ ∈ (◑𝑇 β€œ {0}) ↔ (π‘‡β€˜π‘₯) = 0)))
2214, 21vtoclga 3536 . . . . 5 (𝐴 ∈ β„‹ β†’ (𝑇 Fn β„‹ β†’ (𝐴 ∈ (◑𝑇 β€œ {0}) ↔ (π‘‡β€˜π΄) = 0)))
2310, 22mpan9 508 . . . 4 ((𝑇: β„‹βŸΆβ„‚ ∧ 𝐴 ∈ β„‹) β†’ (𝐴 ∈ (◑𝑇 β€œ {0}) ↔ (π‘‡β€˜π΄) = 0))
249, 23bitrd 279 . . 3 ((𝑇: β„‹βŸΆβ„‚ ∧ 𝐴 ∈ β„‹) β†’ (𝐴 ∈ (nullβ€˜π‘‡) ↔ (π‘‡β€˜π΄) = 0))
2524pm5.32da 580 . 2 (𝑇: β„‹βŸΆβ„‚ β†’ ((𝐴 ∈ β„‹ ∧ 𝐴 ∈ (nullβ€˜π‘‡)) ↔ (𝐴 ∈ β„‹ ∧ (π‘‡β€˜π΄) = 0)))
267, 25bitrd 279 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 397   = wceq 1542   ∈ wcel 2107  {csn 4590   class class class wbr 5109  β—‘ccnv 5636  dom cdm 5637   β€œ cima 5640   Fn wfn 6495  βŸΆwf 6496  β€˜cfv 6500  β„‚cc 11057  0cc0 11059   β„‹chba 29910  nullcnl 29943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-mulcl 11121  ax-i2m1 11127  ax-hilex 29990
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-map 8773  df-nlfn 30837
This theorem is referenced by:  elnlfn2  30920  nlelshi  31051  nlelchi  31052  riesz3i  31053
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