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Theorem elnlfn 29127
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: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ (𝐴 ∈ ℋ ∧ (𝑇𝐴) = 0)))

Proof of Theorem elnlfn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 nlfnval 29080 . . . . . 6 (𝑇: ℋ⟶ℂ → (null‘𝑇) = (𝑇 “ {0}))
2 cnvimass 5625 . . . . . 6 (𝑇 “ {0}) ⊆ dom 𝑇
31, 2syl6eqss 3804 . . . . 5 (𝑇: ℋ⟶ℂ → (null‘𝑇) ⊆ dom 𝑇)
4 fdm 6192 . . . . 5 (𝑇: ℋ⟶ℂ → dom 𝑇 = ℋ)
53, 4sseqtrd 3790 . . . 4 (𝑇: ℋ⟶ℂ → (null‘𝑇) ⊆ ℋ)
65sseld 3751 . . 3 (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) → 𝐴 ∈ ℋ))
76pm4.71rd 552 . 2 (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ∈ (null‘𝑇))))
81eleq2d 2836 . . . . 5 (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ 𝐴 ∈ (𝑇 “ {0})))
98adantr 466 . . . 4 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ (null‘𝑇) ↔ 𝐴 ∈ (𝑇 “ {0})))
10 ffn 6184 . . . . 5 (𝑇: ℋ⟶ℂ → 𝑇 Fn ℋ)
11 eleq1 2838 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 ∈ (𝑇 “ {0}) ↔ 𝐴 ∈ (𝑇 “ {0})))
12 fveq2 6333 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑇𝑥) = (𝑇𝐴))
1312eqeq1d 2773 . . . . . . . 8 (𝑥 = 𝐴 → ((𝑇𝑥) = 0 ↔ (𝑇𝐴) = 0))
1411, 13bibi12d 334 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥 ∈ (𝑇 “ {0}) ↔ (𝑇𝑥) = 0) ↔ (𝐴 ∈ (𝑇 “ {0}) ↔ (𝑇𝐴) = 0)))
1514imbi2d 329 . . . . . 6 (𝑥 = 𝐴 → ((𝑇 Fn ℋ → (𝑥 ∈ (𝑇 “ {0}) ↔ (𝑇𝑥) = 0)) ↔ (𝑇 Fn ℋ → (𝐴 ∈ (𝑇 “ {0}) ↔ (𝑇𝐴) = 0))))
16 fnbrfvb 6379 . . . . . . . 8 ((𝑇 Fn ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇𝑥) = 0 ↔ 𝑥𝑇0))
17 0cn 10238 . . . . . . . . 9 0 ∈ ℂ
18 vex 3354 . . . . . . . . . 10 𝑥 ∈ V
1918eliniseg 5634 . . . . . . . . 9 (0 ∈ ℂ → (𝑥 ∈ (𝑇 “ {0}) ↔ 𝑥𝑇0))
2017, 19ax-mp 5 . . . . . . . 8 (𝑥 ∈ (𝑇 “ {0}) ↔ 𝑥𝑇0)
2116, 20syl6rbbr 279 . . . . . . 7 ((𝑇 Fn ℋ ∧ 𝑥 ∈ ℋ) → (𝑥 ∈ (𝑇 “ {0}) ↔ (𝑇𝑥) = 0))
2221expcom 398 . . . . . 6 (𝑥 ∈ ℋ → (𝑇 Fn ℋ → (𝑥 ∈ (𝑇 “ {0}) ↔ (𝑇𝑥) = 0)))
2315, 22vtoclga 3423 . . . . 5 (𝐴 ∈ ℋ → (𝑇 Fn ℋ → (𝐴 ∈ (𝑇 “ {0}) ↔ (𝑇𝐴) = 0)))
2410, 23mpan9 496 . . . 4 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ (𝑇 “ {0}) ↔ (𝑇𝐴) = 0))
259, 24bitrd 268 . . 3 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ (null‘𝑇) ↔ (𝑇𝐴) = 0))
2625pm5.32da 568 . 2 (𝑇: ℋ⟶ℂ → ((𝐴 ∈ ℋ ∧ 𝐴 ∈ (null‘𝑇)) ↔ (𝐴 ∈ ℋ ∧ (𝑇𝐴) = 0)))
277, 26bitrd 268 1 (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ (𝐴 ∈ ℋ ∧ (𝑇𝐴) = 0)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  {csn 4317   class class class wbr 4787  ccnv 5249  dom cdm 5250  cima 5253   Fn wfn 6025  wf 6026  cfv 6030  cc 10140  0cc0 10142  chil 28116  nullcnl 28149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-mulcl 10204  ax-i2m1 10210  ax-hilex 28196
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-map 8015  df-nlfn 29045
This theorem is referenced by:  elnlfn2  29128  nlelshi  29259  nlelchi  29260  riesz3i  29261
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