MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  restcnrm Structured version   Visualization version   GIF version

Theorem restcnrm 22106
Description: A subspace of a completely normal space is completely normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
restcnrm ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ CNrm)

Proof of Theorem restcnrm
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2738 . . 3 𝐽 = 𝐽
21restin 21910 . 2 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t 𝐴) = (𝐽t (𝐴 𝐽)))
3 simpll 767 . . . . . 6 (((𝐽 ∈ CNrm ∧ 𝐴𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 𝐽)) → 𝐽 ∈ CNrm)
4 elpwi 4494 . . . . . . 7 (𝑥 ∈ 𝒫 (𝐴 𝐽) → 𝑥 ⊆ (𝐴 𝐽))
54adantl 485 . . . . . 6 (((𝐽 ∈ CNrm ∧ 𝐴𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 𝐽)) → 𝑥 ⊆ (𝐴 𝐽))
6 inex1g 5184 . . . . . . 7 (𝐴𝑉 → (𝐴 𝐽) ∈ V)
76ad2antlr 727 . . . . . 6 (((𝐽 ∈ CNrm ∧ 𝐴𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 𝐽)) → (𝐴 𝐽) ∈ V)
8 restabs 21909 . . . . . 6 ((𝐽 ∈ CNrm ∧ 𝑥 ⊆ (𝐴 𝐽) ∧ (𝐴 𝐽) ∈ V) → ((𝐽t (𝐴 𝐽)) ↾t 𝑥) = (𝐽t 𝑥))
93, 5, 7, 8syl3anc 1372 . . . . 5 (((𝐽 ∈ CNrm ∧ 𝐴𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 𝐽)) → ((𝐽t (𝐴 𝐽)) ↾t 𝑥) = (𝐽t 𝑥))
10 cnrmi 22104 . . . . . 6 ((𝐽 ∈ CNrm ∧ 𝑥 ∈ 𝒫 (𝐴 𝐽)) → (𝐽t 𝑥) ∈ Nrm)
1110adantlr 715 . . . . 5 (((𝐽 ∈ CNrm ∧ 𝐴𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 𝐽)) → (𝐽t 𝑥) ∈ Nrm)
129, 11eqeltrd 2833 . . . 4 (((𝐽 ∈ CNrm ∧ 𝐴𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 𝐽)) → ((𝐽t (𝐴 𝐽)) ↾t 𝑥) ∈ Nrm)
1312ralrimiva 3096 . . 3 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → ∀𝑥 ∈ 𝒫 (𝐴 𝐽)((𝐽t (𝐴 𝐽)) ↾t 𝑥) ∈ Nrm)
14 cnrmtop 22081 . . . . . . 7 (𝐽 ∈ CNrm → 𝐽 ∈ Top)
1514adantr 484 . . . . . 6 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → 𝐽 ∈ Top)
16 toptopon2 21662 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
1715, 16sylib 221 . . . . 5 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → 𝐽 ∈ (TopOn‘ 𝐽))
18 inss2 4118 . . . . 5 (𝐴 𝐽) ⊆ 𝐽
19 resttopon 21905 . . . . 5 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ (𝐴 𝐽) ⊆ 𝐽) → (𝐽t (𝐴 𝐽)) ∈ (TopOn‘(𝐴 𝐽)))
2017, 18, 19sylancl 589 . . . 4 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t (𝐴 𝐽)) ∈ (TopOn‘(𝐴 𝐽)))
21 iscnrm2 22082 . . . 4 ((𝐽t (𝐴 𝐽)) ∈ (TopOn‘(𝐴 𝐽)) → ((𝐽t (𝐴 𝐽)) ∈ CNrm ↔ ∀𝑥 ∈ 𝒫 (𝐴 𝐽)((𝐽t (𝐴 𝐽)) ↾t 𝑥) ∈ Nrm))
2220, 21syl 17 . . 3 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → ((𝐽t (𝐴 𝐽)) ∈ CNrm ↔ ∀𝑥 ∈ 𝒫 (𝐴 𝐽)((𝐽t (𝐴 𝐽)) ↾t 𝑥) ∈ Nrm))
2313, 22mpbird 260 . 2 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t (𝐴 𝐽)) ∈ CNrm)
242, 23eqeltrd 2833 1 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ CNrm)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2113  wral 3053  Vcvv 3397  cin 3840  wss 3841  𝒫 cpw 4485   cuni 4793  cfv 6333  (class class class)co 7164  t crest 16790  Topctop 21637  TopOnctopon 21654  Nrmcnrm 22054  CNrmccnrm 22055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-uni 4794  df-int 4834  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-tr 5134  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-ov 7167  df-oprab 7168  df-mpo 7169  df-om 7594  df-1st 7707  df-2nd 7708  df-en 8549  df-fin 8552  df-fi 8941  df-rest 16792  df-topgen 16813  df-top 21638  df-topon 21655  df-bases 21690  df-cnrm 22062
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator