Theorem restcnrm 21166

Description: A subspace of a completely normal space is completely normal. (Contributed by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
restcnrm	|- ( ( J e. CNrm /\ A e. V ) -> ( J ↾t A ) e. CNrm )

Proof of Theorem restcnrm

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- U. J = U. J
2	1	restin 20970	. 2 |- ( ( J e. CNrm /\ A e. V ) -> ( J ↾t A ) = ( J ↾t ( A i^i U. J ) ) )
3		simpll 790	. . . . . 6 |- ( ( ( J e. CNrm /\ A e. V ) /\ x e. ~P ( A i^i U. J ) ) -> J e. CNrm )
4		elpwi 4168	. . . . . . 7 |- ( x e. ~P ( A i^i U. J ) -> x C_ ( A i^i U. J ) )
5	4	adantl 482	. . . . . 6 |- ( ( ( J e. CNrm /\ A e. V ) /\ x e. ~P ( A i^i U. J ) ) -> x C_ ( A i^i U. J ) )
6		inex1g 4801	. . . . . . 7 |- ( A e. V -> ( A i^i U. J ) e. _V )
7	6	ad2antlr 763	. . . . . 6 |- ( ( ( J e. CNrm /\ A e. V ) /\ x e. ~P ( A i^i U. J ) ) -> ( A i^i U. J ) e. _V )
8		restabs 20969	. . . . . 6 |- ( ( J e. CNrm /\ x C_ ( A i^i U. J ) /\ ( A i^i U. J ) e. _V ) -> ( ( J ↾t ( A i^i U. J ) ) ↾t x ) = ( J ↾t x ) )
9	3, 5, 7, 8	syl3anc 1326	. . . . 5 |- ( ( ( J e. CNrm /\ A e. V ) /\ x e. ~P ( A i^i U. J ) ) -> ( ( J ↾t ( A i^i U. J ) ) ↾t x ) = ( J ↾t x ) )
10		cnrmi 21164	. . . . . 6 |- ( ( J e. CNrm /\ x e. ~P ( A i^i U. J ) ) -> ( J ↾t x ) e. Nrm )
11	10	adantlr 751	. . . . 5 |- ( ( ( J e. CNrm /\ A e. V ) /\ x e. ~P ( A i^i U. J ) ) -> ( J ↾t x ) e. Nrm )
12	9, 11	eqeltrd 2701	. . . 4 |- ( ( ( J e. CNrm /\ A e. V ) /\ x e. ~P ( A i^i U. J ) ) -> ( ( J ↾t ( A i^i U. J ) ) ↾t x ) e. Nrm )
13	12	ralrimiva 2966	. . 3 |- ( ( J e. CNrm /\ A e. V ) -> A. x e. ~P ( A i^i U. J ) ( ( J ↾t ( A i^i U. J ) ) ↾t x ) e. Nrm )
14		cnrmtop 21141	. . . . . . 7 |- ( J e. CNrm -> J e. Top )
15	14	adantr 481	. . . . . 6 |- ( ( J e. CNrm /\ A e. V ) -> J e. Top )
16	1	toptopon 20722	. . . . . 6 |- ( J e. Top <-> J e. (TopOn ` U. J ) )
17	15, 16	sylib 208	. . . . 5 |- ( ( J e. CNrm /\ A e. V ) -> J e. (TopOn ` U. J ) )
18		inss2 3834	. . . . 5 |- ( A i^i U. J ) C_ U. J
19		resttopon 20965	. . . . 5 |- ( ( J e. (TopOn ` U. J ) /\ ( A i^i U. J ) C_ U. J ) -> ( J ↾t ( A i^i U. J ) ) e. (TopOn ` ( A i^i U. J ) ) )
20	17, 18, 19	sylancl 694	. . . 4 |- ( ( J e. CNrm /\ A e. V ) -> ( J ↾t ( A i^i U. J ) ) e. (TopOn ` ( A i^i U. J ) ) )
21		iscnrm2 21142	. . . 4 |- ( ( J ↾t ( A i^i U. J ) ) e. (TopOn ` ( A i^i U. J ) ) -> ( ( J ↾t ( A i^i U. J ) ) e. CNrm <-> A. x e. ~P ( A i^i U. J ) ( ( J ↾t ( A i^i U. J ) ) ↾t x ) e. Nrm ) )
22	20, 21	syl 17	. . 3 |- ( ( J e. CNrm /\ A e. V ) -> ( ( J ↾t ( A i^i U. J ) ) e. CNrm <-> A. x e. ~P ( A i^i U. J ) ( ( J ↾t ( A i^i U. J ) ) ↾t x ) e. Nrm ) )
23	13, 22	mpbird 247	. 2 |- ( ( J e. CNrm /\ A e. V ) -> ( J ↾t ( A i^i U. J ) ) e. CNrm )
24	2, 23	eqeltrd 2701	1 |- ( ( J e. CNrm /\ A e. V ) -> ( J ↾t A ) e. CNrm )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   ` cfv 5888  (class class class)co 6650   ↾_t crest 16081   Topctop 20698  TopOnctopon 20715   Nrmcnrm 21114  CNrmccnrm 21115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-er 7742  df-en 7956  df-fin 7959  df-fi 8317  df-rest 16083  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-cnrm 21122

This theorem is referenced by: (None)