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Theorem restcnrm 20976
 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 2610 . . 3 𝐽 = 𝐽
21restin 20780 . 2 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t 𝐴) = (𝐽t (𝐴 𝐽)))
3 simpll 786 . . . . . 6 (((𝐽 ∈ CNrm ∧ 𝐴𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 𝐽)) → 𝐽 ∈ CNrm)
4 elpwi 4117 . . . . . . 7 (𝑥 ∈ 𝒫 (𝐴 𝐽) → 𝑥 ⊆ (𝐴 𝐽))
54adantl 481 . . . . . 6 (((𝐽 ∈ CNrm ∧ 𝐴𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 𝐽)) → 𝑥 ⊆ (𝐴 𝐽))
6 inex1g 4729 . . . . . . 7 (𝐴𝑉 → (𝐴 𝐽) ∈ V)
76ad2antlr 759 . . . . . 6 (((𝐽 ∈ CNrm ∧ 𝐴𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 𝐽)) → (𝐴 𝐽) ∈ V)
8 restabs 20779 . . . . . 6 ((𝐽 ∈ CNrm ∧ 𝑥 ⊆ (𝐴 𝐽) ∧ (𝐴 𝐽) ∈ V) → ((𝐽t (𝐴 𝐽)) ↾t 𝑥) = (𝐽t 𝑥))
93, 5, 7, 8syl3anc 1318 . . . . 5 (((𝐽 ∈ CNrm ∧ 𝐴𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 𝐽)) → ((𝐽t (𝐴 𝐽)) ↾t 𝑥) = (𝐽t 𝑥))
10 cnrmi 20974 . . . . . 6 ((𝐽 ∈ CNrm ∧ 𝑥 ∈ 𝒫 (𝐴 𝐽)) → (𝐽t 𝑥) ∈ Nrm)
1110adantlr 747 . . . . 5 (((𝐽 ∈ CNrm ∧ 𝐴𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 𝐽)) → (𝐽t 𝑥) ∈ Nrm)
129, 11eqeltrd 2688 . . . 4 (((𝐽 ∈ CNrm ∧ 𝐴𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 𝐽)) → ((𝐽t (𝐴 𝐽)) ↾t 𝑥) ∈ Nrm)
1312ralrimiva 2949 . . 3 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → ∀𝑥 ∈ 𝒫 (𝐴 𝐽)((𝐽t (𝐴 𝐽)) ↾t 𝑥) ∈ Nrm)
14 cnrmtop 20951 . . . . . . 7 (𝐽 ∈ CNrm → 𝐽 ∈ Top)
1514adantr 480 . . . . . 6 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → 𝐽 ∈ Top)
161toptopon 20548 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
1715, 16sylib 207 . . . . 5 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → 𝐽 ∈ (TopOn‘ 𝐽))
18 inss2 3796 . . . . 5 (𝐴 𝐽) ⊆ 𝐽
19 resttopon 20775 . . . . 5 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ (𝐴 𝐽) ⊆ 𝐽) → (𝐽t (𝐴 𝐽)) ∈ (TopOn‘(𝐴 𝐽)))
2017, 18, 19sylancl 693 . . . 4 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t (𝐴 𝐽)) ∈ (TopOn‘(𝐴 𝐽)))
21 iscnrm2 20952 . . . 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 246 . 2 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t (𝐴 𝐽)) ∈ CNrm)
242, 23eqeltrd 2688 1 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ CNrm)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  ‘cfv 5804  (class class class)co 6549   ↾t crest 15904  Topctop 20517  TopOnctopon 20518  Nrmcnrm 20924  CNrmccnrm 20925 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-er 7629  df-en 7842  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-cnrm 20932 This theorem is referenced by: (None)
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