ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dvcn Unicode version
Theorem dvcn 13458
Description: A differentiable function is continuous. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-Sep-2015.)
Assertion
Ref Expression
dvcn |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ dom ( S _D F ) = A ) -> F e. ( A -cn-> CC ) )
Proof of Theorem dvcn
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 simpl2 996 . . 3 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ dom ( S _D F ) = A ) -> F : A --> CC )
2 eqid 2170 . . . . . 6 |- ( ( MetOpen ` ( abs o. - ) ) ↾t A ) = ( ( MetOpen ` ( abs o. - ) ) ↾t A )
3 eqid 2170 . . . . . 6 |- ( MetOpen ` ( abs o. - ) ) = ( MetOpen ` ( abs o. - ) )
42, 3dvcnp2cntop 13457 . . . . 5 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ x e. dom ( S _D F ) ) -> F e. ( ( ( ( MetOpen ` ( abs o. - ) ) ↾t A ) CnP ( MetOpen ` ( abs o. - ) ) ) ` x ) )
54ralrimiva 2543 . . . 4 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> A. x e. dom ( S _D F ) F e. ( ( ( ( MetOpen ` ( abs o. - ) ) ↾t A ) CnP ( MetOpen ` ( abs o. - ) ) ) ` x ) )
6 raleq 2665 . . . . 5 |- ( dom ( S _D F ) = A -> ( A. x e. dom ( S _D F ) F e. ( ( ( ( MetOpen ` ( abs o. - ) ) ↾t A ) CnP ( MetOpen ` ( abs o. - ) ) ) ` x ) <-> A. x e. A F e. ( ( ( ( MetOpen ` ( abs o. - ) ) ↾t A ) CnP ( MetOpen ` ( abs o. - ) ) ) ` x ) ) )
76biimpd 143 . . . 4 |- ( dom ( S _D F ) = A -> ( A. x e. dom ( S _D F ) F e. ( ( ( ( MetOpen ` ( abs o. - ) ) ↾t A ) CnP ( MetOpen ` ( abs o. - ) ) ) ` x ) -> A. x e. A F e. ( ( ( ( MetOpen ` ( abs o. - ) ) ↾t A ) CnP ( MetOpen ` ( abs o. - ) ) ) ` x ) ) )
85, 7mpan9 279 . . 3 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ dom ( S _D F ) = A ) -> A. x e. A F e. ( ( ( ( MetOpen ` ( abs o. - ) ) ↾t A ) CnP ( MetOpen ` ( abs o. - ) ) ) ` x ) )
93cntoptopon 13326 . . . . 5 |- ( MetOpen ` ( abs o. - ) ) e. (TopOn ` CC )
10 simpl3 997 . . . . . 6 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ dom ( S _D F ) = A ) -> A C_ S )
11 simpl1 995 . . . . . 6 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ dom ( S _D F ) = A ) -> S C_ CC )
1210, 11sstrd 3157 . . . . 5 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ dom ( S _D F ) = A ) -> A C_ CC )
13 resttopon 12965 . . . . 5 |- ( ( ( MetOpen ` ( abs o. - ) ) e. (TopOn ` CC ) /\ A C_ CC ) -> ( ( MetOpen ` ( abs o. - ) ) ↾t A ) e. (TopOn ` A ) )
149, 12, 13sylancr 412 . . . 4 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ dom ( S _D F ) = A ) -> ( ( MetOpen ` ( abs o. - ) ) ↾t A ) e. (TopOn ` A ) )
15 cncnp 13024 . . . 4 |- ( ( ( ( MetOpen ` ( abs o. - ) ) ↾t A ) e. (TopOn ` A ) /\ ( MetOpen ` ( abs o. - ) ) e. (TopOn ` CC ) ) -> ( F e. ( ( ( MetOpen ` ( abs o. - ) ) ↾t A ) Cn ( MetOpen ` ( abs o. - ) ) ) <-> ( F : A --> CC /\ A. x e. A F e. ( ( ( ( MetOpen ` ( abs o. - ) ) ↾t A ) CnP ( MetOpen ` ( abs o. - ) ) ) ` x ) ) ) )
1614, 9, 15sylancl 411 . . 3 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ dom ( S _D F ) = A ) -> ( F e. ( ( ( MetOpen ` ( abs o. - ) ) ↾t A ) Cn ( MetOpen ` ( abs o. - ) ) ) <-> ( F : A --> CC /\ A. x e. A F e. ( ( ( ( MetOpen ` ( abs o. - ) ) ↾t A ) CnP ( MetOpen ` ( abs o. - ) ) ) ` x ) ) ) )
171, 8, 16mpbir2and 939 . 2 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ dom ( S _D F ) = A ) -> F e. ( ( ( MetOpen ` ( abs o. - ) ) ↾t A ) Cn ( MetOpen ` ( abs o. - ) ) ) )
18 ssid 3167 . . 3 |- CC C_ CC
199toponrestid 12813 . . . 4 |- ( MetOpen ` ( abs o. - ) ) = ( ( MetOpen ` ( abs o. - ) ) ↾t CC )
203, 2, 19cncfcncntop 13374 . . 3 |- ( ( A C_ CC /\ CC C_ CC ) -> ( A -cn-> CC ) = ( ( ( MetOpen ` ( abs o. - ) ) ↾t A ) Cn ( MetOpen ` ( abs o. - ) ) ) )
2112, 18, 20sylancl 411 . 2 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ dom ( S _D F ) = A ) -> ( A -cn-> CC ) = ( ( ( MetOpen ` ( abs o. - ) ) ↾t A ) Cn ( MetOpen ` ( abs o. - ) ) ) )
2217, 21eleqtrrd 2250 1 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ dom ( S _D F ) = A ) -> F e. ( A -cn-> CC ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 973   = wceq 1348   e. wcel 2141   A.wral 2448   C_ wss 3121   dom cdm 4611   o. ccom 4615   -->wf 5194   ` cfv 5198  (class class class)co 5853   CCcc 7772   - cmin 8090   abscabs 10961   ↾t crest 12579   MetOpencmopn 12779  TopOnctopon 12802   Cn ccn 12979   CnP ccnp 12980   -cn->ccncf 13351   _D cdv 13418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894  ax-addf 7896  ax-mulf 7897
This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-map 6628  df-pm 6629  df-sup 6961  df-inf 6962  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-xneg 9729  df-xadd 9730  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-rest 12581  df-topgen 12600  df-psmet 12781  df-xmet 12782  df-met 12783  df-bl 12784  df-mopn 12785  df-top 12790  df-topon 12803  df-bases 12835  df-ntr 12890  df-cn 12982  df-cnp 12983  df-tx 13047  df-cncf 13352  df-limced 13419  df-dvap 13420
This theorem is referenced by:  efcn  13483
  Copyright terms: Public domain W3C validator