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Theorem restcnrm 23386
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 2735 . . 3 𝐽 = 𝐽
21restin 23190 . 2 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t 𝐴) = (𝐽t (𝐴 𝐽)))
3 simpll 767 . . . . . 6 (((𝐽 ∈ CNrm ∧ 𝐴𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 𝐽)) → 𝐽 ∈ CNrm)
4 elpwi 4612 . . . . . . 7 (𝑥 ∈ 𝒫 (𝐴 𝐽) → 𝑥 ⊆ (𝐴 𝐽))
54adantl 481 . . . . . 6 (((𝐽 ∈ CNrm ∧ 𝐴𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 𝐽)) → 𝑥 ⊆ (𝐴 𝐽))
6 inex1g 5325 . . . . . . 7 (𝐴𝑉 → (𝐴 𝐽) ∈ V)
76ad2antlr 727 . . . . . 6 (((𝐽 ∈ CNrm ∧ 𝐴𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 𝐽)) → (𝐴 𝐽) ∈ V)
8 restabs 23189 . . . . . 6 ((𝐽 ∈ CNrm ∧ 𝑥 ⊆ (𝐴 𝐽) ∧ (𝐴 𝐽) ∈ V) → ((𝐽t (𝐴 𝐽)) ↾t 𝑥) = (𝐽t 𝑥))
93, 5, 7, 8syl3anc 1370 . . . . 5 (((𝐽 ∈ CNrm ∧ 𝐴𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 𝐽)) → ((𝐽t (𝐴 𝐽)) ↾t 𝑥) = (𝐽t 𝑥))
10 cnrmi 23384 . . . . . 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 2839 . . . 4 (((𝐽 ∈ CNrm ∧ 𝐴𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 𝐽)) → ((𝐽t (𝐴 𝐽)) ↾t 𝑥) ∈ Nrm)
1312ralrimiva 3144 . . 3 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → ∀𝑥 ∈ 𝒫 (𝐴 𝐽)((𝐽t (𝐴 𝐽)) ↾t 𝑥) ∈ Nrm)
14 cnrmtop 23361 . . . . . . 7 (𝐽 ∈ CNrm → 𝐽 ∈ Top)
1514adantr 480 . . . . . 6 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → 𝐽 ∈ Top)
16 toptopon2 22940 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
1715, 16sylib 218 . . . . 5 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → 𝐽 ∈ (TopOn‘ 𝐽))
18 inss2 4246 . . . . 5 (𝐴 𝐽) ⊆ 𝐽
19 resttopon 23185 . . . . 5 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ (𝐴 𝐽) ⊆ 𝐽) → (𝐽t (𝐴 𝐽)) ∈ (TopOn‘(𝐴 𝐽)))
2017, 18, 19sylancl 586 . . . 4 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t (𝐴 𝐽)) ∈ (TopOn‘(𝐴 𝐽)))
21 iscnrm2 23362 . . . 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 257 . 2 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t (𝐴 𝐽)) ∈ CNrm)
242, 23eqeltrd 2839 1 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ CNrm)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  cin 3962  wss 3963  𝒫 cpw 4605   cuni 4912  cfv 6563  (class class class)co 7431  t crest 17467  Topctop 22915  TopOnctopon 22932  Nrmcnrm 23334  CNrmccnrm 23335
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-en 8985  df-fin 8988  df-fi 9449  df-rest 17469  df-topgen 17490  df-top 22916  df-topon 22933  df-bases 22969  df-cnrm 23342
This theorem is referenced by: (None)
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