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Theorem restcnrm 21945
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 2821 . . 3 𝐽 = 𝐽
21restin 21749 . 2 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t 𝐴) = (𝐽t (𝐴 𝐽)))
3 simpll 766 . . . . . 6 (((𝐽 ∈ CNrm ∧ 𝐴𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 𝐽)) → 𝐽 ∈ CNrm)
4 elpwi 4521 . . . . . . 7 (𝑥 ∈ 𝒫 (𝐴 𝐽) → 𝑥 ⊆ (𝐴 𝐽))
54adantl 485 . . . . . 6 (((𝐽 ∈ CNrm ∧ 𝐴𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 𝐽)) → 𝑥 ⊆ (𝐴 𝐽))
6 inex1g 5196 . . . . . . 7 (𝐴𝑉 → (𝐴 𝐽) ∈ V)
76ad2antlr 726 . . . . . 6 (((𝐽 ∈ CNrm ∧ 𝐴𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 𝐽)) → (𝐴 𝐽) ∈ V)
8 restabs 21748 . . . . . 6 ((𝐽 ∈ CNrm ∧ 𝑥 ⊆ (𝐴 𝐽) ∧ (𝐴 𝐽) ∈ V) → ((𝐽t (𝐴 𝐽)) ↾t 𝑥) = (𝐽t 𝑥))
93, 5, 7, 8syl3anc 1368 . . . . 5 (((𝐽 ∈ CNrm ∧ 𝐴𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 𝐽)) → ((𝐽t (𝐴 𝐽)) ↾t 𝑥) = (𝐽t 𝑥))
10 cnrmi 21943 . . . . . 6 ((𝐽 ∈ CNrm ∧ 𝑥 ∈ 𝒫 (𝐴 𝐽)) → (𝐽t 𝑥) ∈ Nrm)
1110adantlr 714 . . . . 5 (((𝐽 ∈ CNrm ∧ 𝐴𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 𝐽)) → (𝐽t 𝑥) ∈ Nrm)
129, 11eqeltrd 2912 . . . 4 (((𝐽 ∈ CNrm ∧ 𝐴𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 𝐽)) → ((𝐽t (𝐴 𝐽)) ↾t 𝑥) ∈ Nrm)
1312ralrimiva 3170 . . 3 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → ∀𝑥 ∈ 𝒫 (𝐴 𝐽)((𝐽t (𝐴 𝐽)) ↾t 𝑥) ∈ Nrm)
14 cnrmtop 21920 . . . . . . 7 (𝐽 ∈ CNrm → 𝐽 ∈ Top)
1514adantr 484 . . . . . 6 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → 𝐽 ∈ Top)
16 toptopon2 21501 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
1715, 16sylib 221 . . . . 5 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → 𝐽 ∈ (TopOn‘ 𝐽))
18 inss2 4181 . . . . 5 (𝐴 𝐽) ⊆ 𝐽
19 resttopon 21744 . . . . 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 21921 . . . 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 2912 1 ((𝐽 ∈ CNrm ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ CNrm)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3126  Vcvv 3471  cin 3909  wss 3910  𝒫 cpw 4512   cuni 4811  cfv 6328  (class class class)co 7130  t crest 16672  Topctop 21476  TopOnctopon 21493  Nrmcnrm 21893  CNrmccnrm 21894
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-oadd 8081  df-er 8264  df-en 8485  df-fin 8488  df-fi 8851  df-rest 16674  df-topgen 16695  df-top 21477  df-topon 21494  df-bases 21529  df-cnrm 21901
This theorem is referenced by: (None)
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