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Theorem restcnrm 22736
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 2733 . . 3 βˆͺ 𝐽 = βˆͺ 𝐽
21restin 22540 . 2 ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) β†’ (𝐽 β†Ύt 𝐴) = (𝐽 β†Ύt (𝐴 ∩ βˆͺ 𝐽)))
3 simpll 766 . . . . . 6 (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ π‘₯ ∈ 𝒫 (𝐴 ∩ βˆͺ 𝐽)) β†’ 𝐽 ∈ CNrm)
4 elpwi 4571 . . . . . . 7 (π‘₯ ∈ 𝒫 (𝐴 ∩ βˆͺ 𝐽) β†’ π‘₯ βŠ† (𝐴 ∩ βˆͺ 𝐽))
54adantl 483 . . . . . 6 (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ π‘₯ ∈ 𝒫 (𝐴 ∩ βˆͺ 𝐽)) β†’ π‘₯ βŠ† (𝐴 ∩ βˆͺ 𝐽))
6 inex1g 5280 . . . . . . 7 (𝐴 ∈ 𝑉 β†’ (𝐴 ∩ βˆͺ 𝐽) ∈ V)
76ad2antlr 726 . . . . . 6 (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ π‘₯ ∈ 𝒫 (𝐴 ∩ βˆͺ 𝐽)) β†’ (𝐴 ∩ βˆͺ 𝐽) ∈ V)
8 restabs 22539 . . . . . 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 1372 . . . . 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 22734 . . . . . 6 ((𝐽 ∈ CNrm ∧ π‘₯ ∈ 𝒫 (𝐴 ∩ βˆͺ 𝐽)) β†’ (𝐽 β†Ύt π‘₯) ∈ Nrm)
1110adantlr 714 . . . . 5 (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ π‘₯ ∈ 𝒫 (𝐴 ∩ βˆͺ 𝐽)) β†’ (𝐽 β†Ύt π‘₯) ∈ Nrm)
129, 11eqeltrd 2834 . . . 4 (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ π‘₯ ∈ 𝒫 (𝐴 ∩ βˆͺ 𝐽)) β†’ ((𝐽 β†Ύt (𝐴 ∩ βˆͺ 𝐽)) β†Ύt π‘₯) ∈ Nrm)
1312ralrimiva 3140 . . 3 ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) β†’ βˆ€π‘₯ ∈ 𝒫 (𝐴 ∩ βˆͺ 𝐽)((𝐽 β†Ύt (𝐴 ∩ βˆͺ 𝐽)) β†Ύt π‘₯) ∈ Nrm)
14 cnrmtop 22711 . . . . . . 7 (𝐽 ∈ CNrm β†’ 𝐽 ∈ Top)
1514adantr 482 . . . . . 6 ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) β†’ 𝐽 ∈ Top)
16 toptopon2 22290 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
1715, 16sylib 217 . . . . 5 ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) β†’ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
18 inss2 4193 . . . . 5 (𝐴 ∩ βˆͺ 𝐽) βŠ† βˆͺ 𝐽
19 resttopon 22535 . . . . 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 587 . . . 4 ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) β†’ (𝐽 β†Ύt (𝐴 ∩ βˆͺ 𝐽)) ∈ (TopOnβ€˜(𝐴 ∩ βˆͺ 𝐽)))
21 iscnrm2 22712 . . . 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 257 . 2 ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) β†’ (𝐽 β†Ύt (𝐴 ∩ βˆͺ 𝐽)) ∈ CNrm)
242, 23eqeltrd 2834 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  Vcvv 3447   ∩ cin 3913   βŠ† wss 3914  π’« cpw 4564  βˆͺ cuni 4869  β€˜cfv 6500  (class class class)co 7361   β†Ύt crest 17310  Topctop 22265  TopOnctopon 22282  Nrmcnrm 22684  CNrmccnrm 22685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-en 8890  df-fin 8893  df-fi 9355  df-rest 17312  df-topgen 17333  df-top 22266  df-topon 22283  df-bases 22319  df-cnrm 22692
This theorem is referenced by: (None)
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