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Theorem cnrmi 22863
Description: A subspace of a completely normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
cnrmi ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Nrm)

Proof of Theorem cnrmi
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2732 . . 3 𝐽 = 𝐽
21restin 22669 . 2 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t 𝐴) = (𝐽t (𝐴 𝐽)))
3 oveq2 7416 . . . 4 (𝑥 = (𝐴 𝐽) → (𝐽t 𝑥) = (𝐽t (𝐴 𝐽)))
43eleq1d 2818 . . 3 (𝑥 = (𝐴 𝐽) → ((𝐽t 𝑥) ∈ Nrm ↔ (𝐽t (𝐴 𝐽)) ∈ Nrm))
51iscnrm 22826 . . . . 5 (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝐽(𝐽t 𝑥) ∈ Nrm))
65simprbi 497 . . . 4 (𝐽 ∈ CNrm → ∀𝑥 ∈ 𝒫 𝐽(𝐽t 𝑥) ∈ Nrm)
76adantr 481 . . 3 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → ∀𝑥 ∈ 𝒫 𝐽(𝐽t 𝑥) ∈ Nrm)
8 inss2 4229 . . . . 5 (𝐴 𝐽) ⊆ 𝐽
9 inex1g 5319 . . . . . 6 (𝐴𝑉 → (𝐴 𝐽) ∈ V)
10 elpwg 4605 . . . . . 6 ((𝐴 𝐽) ∈ V → ((𝐴 𝐽) ∈ 𝒫 𝐽 ↔ (𝐴 𝐽) ⊆ 𝐽))
119, 10syl 17 . . . . 5 (𝐴𝑉 → ((𝐴 𝐽) ∈ 𝒫 𝐽 ↔ (𝐴 𝐽) ⊆ 𝐽))
128, 11mpbiri 257 . . . 4 (𝐴𝑉 → (𝐴 𝐽) ∈ 𝒫 𝐽)
1312adantl 482 . . 3 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐴 𝐽) ∈ 𝒫 𝐽)
144, 7, 13rspcdva 3613 . 2 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t (𝐴 𝐽)) ∈ Nrm)
152, 14eqeltrd 2833 1 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Nrm)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3061  Vcvv 3474  cin 3947  wss 3948  𝒫 cpw 4602   cuni 4908  (class class class)co 7408  t crest 17365  Topctop 22394  Nrmcnrm 22813  CNrmccnrm 22814
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-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
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-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  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-iun 4999  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-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-rest 17367  df-cnrm 22821
This theorem is referenced by:  cnrmnrm  22864  restcnrm  22865
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