ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dvcn Unicode version
Theorem dvcn 14641
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 1003 . . 3 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ dom ( S _D F ) = A ) -> F : A --> CC )
2 eqid 2189 . . . . . 6 |- ( ( MetOpen ` ( abs o. - ) ) ↾t A ) = ( ( MetOpen ` ( abs o. - ) ) ↾t A )
3 eqid 2189 . . . . . 6 |- ( MetOpen ` ( abs o. - ) ) = ( MetOpen ` ( abs o. - ) )
42, 3dvcnp2cntop 14640 . . . . 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 2563 . . . 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 2686 . . . . 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 144 . . . 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 281 . . 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 14509 . . . . 5 |- ( MetOpen ` ( abs o. - ) ) e. (TopOn ` CC )
10 simpl3 1004 . . . . . 6 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ dom ( S _D F ) = A ) -> A C_ S )
11 simpl1 1002 . . . . . 6 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ dom ( S _D F ) = A ) -> S C_ CC )
1210, 11sstrd 3180 . . . . 5 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ dom ( S _D F ) = A ) -> A C_ CC )
13 resttopon 14148 . . . . 5 |- ( ( ( MetOpen ` ( abs o. - ) ) e. (TopOn ` CC ) /\ A C_ CC ) -> ( ( MetOpen ` ( abs o. - ) ) ↾t A ) e. (TopOn ` A ) )
149, 12, 13sylancr 414 . . . 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 14207 . . . 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 413 . . 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 946 . 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 3190 . . 3 |- CC C_ CC
199toponrestid 13998 . . . 4 |- ( MetOpen ` ( abs o. - ) ) = ( ( MetOpen ` ( abs o. - ) ) ↾t CC )
203, 2, 19cncfcncntop 14557 . . 3 |- ( ( A C_ CC /\ CC C_ CC ) -> ( A -cn-> CC ) = ( ( ( MetOpen ` ( abs o. - ) ) ↾t A ) Cn ( MetOpen ` ( abs o. - ) ) ) )
2112, 18, 20sylancl 413 . 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 2269 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 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2160   A.wral 2468   C_ wss 3144   dom cdm 4644   o. ccom 4648   -->wf 5231   ` cfv 5235  (class class class)co 5897   CCcc 7840   - cmin 8159   abscabs 11041   ↾t crest 12747   MetOpencmopn 13871  TopOnctopon 13987   Cn ccn 14162   CnP ccnp 14163   -cn->ccncf 14534   _D cdv 14601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-mulrcl 7941  ax-addcom 7942  ax-mulcom 7943  ax-addass 7944  ax-mulass 7945  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-1rid 7949  ax-0id 7950  ax-rnegex 7951  ax-precex 7952  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958  ax-pre-mulgt0 7959  ax-pre-mulext 7960  ax-arch 7961  ax-caucvg 7962  ax-addf 7964  ax-mulf 7965
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-isom 5244  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-frec 6417  df-map 6677  df-pm 6678  df-sup 7014  df-inf 7015  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-reap 8563  df-ap 8570  df-div 8661  df-inn 8951  df-2 9009  df-3 9010  df-4 9011  df-n0 9208  df-z 9285  df-uz 9560  df-q 9652  df-rp 9686  df-xneg 9804  df-xadd 9805  df-seqfrec 10479  df-exp 10554  df-cj 10886  df-re 10887  df-im 10888  df-rsqrt 11042  df-abs 11043  df-rest 12749  df-topgen 12768  df-psmet 13873  df-xmet 13874  df-met 13875  df-bl 13876  df-mopn 13877  df-top 13975  df-topon 13988  df-bases 14020  df-ntr 14073  df-cn 14165  df-cnp 14166  df-tx 14230  df-cncf 14535  df-limced 14602  df-dvap 14603
This theorem is referenced by:  efcn  14666
  Copyright terms: Public domain W3C validator