MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  restcnrm Structured version   Visualization version   GIF version
Theorem restcnrm 22865
Description: A subspace of a completely normal space is completely normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
restcnrm ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) β†’ (𝐽 β†Ύt 𝐴) ∈ CNrm)
Proof of Theorem restcnrm
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 eqid 2732 . . 3 βˆͺ 𝐽 = βˆͺ 𝐽
21restin 22669 . 2 ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) β†’ (𝐽 β†Ύt 𝐴) = (𝐽 β†Ύt (𝐴 ∩ βˆͺ 𝐽)))
3 simpll 765 . . . . . 6 (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ π‘₯ ∈ 𝒫 (𝐴 ∩ βˆͺ 𝐽)) β†’ 𝐽 ∈ CNrm)
4 elpwi 4609 . . . . . . 7 (π‘₯ ∈ 𝒫 (𝐴 ∩ βˆͺ 𝐽) β†’ π‘₯ βŠ† (𝐴 ∩ βˆͺ 𝐽))
54adantl 482 . . . . . 6 (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ π‘₯ ∈ 𝒫 (𝐴 ∩ βˆͺ 𝐽)) β†’ π‘₯ βŠ† (𝐴 ∩ βˆͺ 𝐽))
6 inex1g 5319 . . . . . . 7 (𝐴 ∈ 𝑉 β†’ (𝐴 ∩ βˆͺ 𝐽) ∈ V)
76ad2antlr 725 . . . . . 6 (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ π‘₯ ∈ 𝒫 (𝐴 ∩ βˆͺ 𝐽)) β†’ (𝐴 ∩ βˆͺ 𝐽) ∈ V)
8 restabs 22668 . . . . . 6 ((𝐽 ∈ CNrm ∧ π‘₯ βŠ† (𝐴 ∩ βˆͺ 𝐽) ∧ (𝐴 ∩ βˆͺ 𝐽) ∈ V) β†’ ((𝐽 β†Ύt (𝐴 ∩ βˆͺ 𝐽)) β†Ύt π‘₯) = (𝐽 β†Ύt π‘₯))
93, 5, 7, 8syl3anc 1371 . . . . 5 (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ π‘₯ ∈ 𝒫 (𝐴 ∩ βˆͺ 𝐽)) β†’ ((𝐽 β†Ύt (𝐴 ∩ βˆͺ 𝐽)) β†Ύt π‘₯) = (𝐽 β†Ύt π‘₯))
10 cnrmi 22863 . . . . . 6 ((𝐽 ∈ CNrm ∧ π‘₯ ∈ 𝒫 (𝐴 ∩ βˆͺ 𝐽)) β†’ (𝐽 β†Ύt π‘₯) ∈ Nrm)
1110adantlr 713 . . . . 5 (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ π‘₯ ∈ 𝒫 (𝐴 ∩ βˆͺ 𝐽)) β†’ (𝐽 β†Ύt π‘₯) ∈ Nrm)
129, 11eqeltrd 2833 . . . 4 (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ π‘₯ ∈ 𝒫 (𝐴 ∩ βˆͺ 𝐽)) β†’ ((𝐽 β†Ύt (𝐴 ∩ βˆͺ 𝐽)) β†Ύt π‘₯) ∈ Nrm)
1312ralrimiva 3146 . . 3 ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) β†’ βˆ€π‘₯ ∈ 𝒫 (𝐴 ∩ βˆͺ 𝐽)((𝐽 β†Ύt (𝐴 ∩ βˆͺ 𝐽)) β†Ύt π‘₯) ∈ Nrm)
14 cnrmtop 22840 . . . . . . 7 (𝐽 ∈ CNrm β†’ 𝐽 ∈ Top)
1514adantr 481 . . . . . 6 ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) β†’ 𝐽 ∈ Top)
16 toptopon2 22419 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
1715, 16sylib 217 . . . . 5 ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) β†’ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
18 inss2 4229 . . . . 5 (𝐴 ∩ βˆͺ 𝐽) βŠ† βˆͺ 𝐽
19 resttopon 22664 . . . . 5 ((𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽) ∧ (𝐴 ∩ βˆͺ 𝐽) βŠ† βˆͺ 𝐽) β†’ (𝐽 β†Ύt (𝐴 ∩ βˆͺ 𝐽)) ∈ (TopOnβ€˜(𝐴 ∩ βˆͺ 𝐽)))
2017, 18, 19sylancl 586 . . . 4 ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) β†’ (𝐽 β†Ύt (𝐴 ∩ βˆͺ 𝐽)) ∈ (TopOnβ€˜(𝐴 ∩ βˆͺ 𝐽)))
21 iscnrm2 22841 . . . 4 ((𝐽 β†Ύt (𝐴 ∩ βˆͺ 𝐽)) ∈ (TopOnβ€˜(𝐴 ∩ βˆͺ 𝐽)) β†’ ((𝐽 β†Ύt (𝐴 ∩ βˆͺ 𝐽)) ∈ CNrm ↔ βˆ€π‘₯ ∈ 𝒫 (𝐴 ∩ βˆͺ 𝐽)((𝐽 β†Ύt (𝐴 ∩ βˆͺ 𝐽)) β†Ύt π‘₯) ∈ Nrm))
2220, 21syl 17 . . 3 ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) β†’ ((𝐽 β†Ύt (𝐴 ∩ βˆͺ 𝐽)) ∈ CNrm ↔ βˆ€π‘₯ ∈ 𝒫 (𝐴 ∩ βˆͺ 𝐽)((𝐽 β†Ύt (𝐴 ∩ βˆͺ 𝐽)) β†Ύt π‘₯) ∈ Nrm))
2313, 22mpbird 256 . 2 ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) β†’ (𝐽 β†Ύt (𝐴 ∩ βˆͺ 𝐽)) ∈ CNrm)
242, 23eqeltrd 2833 1 ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) β†’ (𝐽 β†Ύt 𝐴) ∈ CNrm)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  Vcvv 3474   ∩ cin 3947   βŠ† wss 3948  π’« cpw 4602  βˆͺ cuni 4908  β€˜cfv 6543  (class class class)co 7408   β†Ύt crest 17365  Topctop 22394  TopOnctopon 22411  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-3or 1088  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-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  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-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  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-om 7855  df-1st 7974  df-2nd 7975  df-en 8939  df-fin 8942  df-fi 9405  df-rest 17367  df-topgen 17388  df-top 22395  df-topon 22412  df-bases 22448  df-cnrm 22821
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator