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Theorem dvcn 13304
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 991 . . 3 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ dom ( S _D F ) = A ) -> F : A --> CC )
2 eqid 2165 . . . . . 6 |- ( ( MetOpen ` ( abs o. - ) ) ↾t A ) = ( ( MetOpen ` ( abs o. - ) ) ↾t A )
3 eqid 2165 . . . . . 6 |- ( MetOpen ` ( abs o. - ) ) = ( MetOpen ` ( abs o. - ) )
42, 3dvcnp2cntop 13303 . . . . 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 2539 . . . 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 2661 . . . . 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 13172 . . . . 5 |- ( MetOpen ` ( abs o. - ) ) e. (TopOn ` CC )
10 simpl3 992 . . . . . 6 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ dom ( S _D F ) = A ) -> A C_ S )
11 simpl1 990 . . . . . 6 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ dom ( S _D F ) = A ) -> S C_ CC )
1210, 11sstrd 3152 . . . . 5 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ dom ( S _D F ) = A ) -> A C_ CC )
13 resttopon 12811 . . . . 5 |- ( ( ( MetOpen ` ( abs o. - ) ) e. (TopOn ` CC ) /\ A C_ CC ) -> ( ( MetOpen ` ( abs o. - ) ) ↾t A ) e. (TopOn ` A ) )
149, 12, 13sylancr 411 . . . 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 12870 . . . 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 410 . . 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 934 . 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 3162 . . 3 |- CC C_ CC
199toponrestid 12659 . . . 4 |- ( MetOpen ` ( abs o. - ) ) = ( ( MetOpen ` ( abs o. - ) ) ↾t CC )
203, 2, 19cncfcncntop 13220 . . 3 |- ( ( A C_ CC /\ CC C_ CC ) -> ( A -cn-> CC ) = ( ( ( MetOpen ` ( abs o. - ) ) ↾t A ) Cn ( MetOpen ` ( abs o. - ) ) ) )
2112, 18, 20sylancl 410 . 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 2246 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 968   = wceq 1343   e. wcel 2136   A.wral 2444   C_ wss 3116   dom cdm 4604   o. ccom 4608   -->wf 5184   ` cfv 5188  (class class class)co 5842   CCcc 7751   - cmin 8069   abscabs 10939   ↾t crest 12556   MetOpencmopn 12625  TopOnctopon 12648   Cn ccn 12825   CnP ccnp 12826   -cn->ccncf 13197   _D cdv 13264
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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873  ax-addf 7875  ax-mulf 7876
This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-map 6616  df-pm 6617  df-sup 6949  df-inf 6950  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-xneg 9708  df-xadd 9709  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-rest 12558  df-topgen 12577  df-psmet 12627  df-xmet 12628  df-met 12629  df-bl 12630  df-mopn 12631  df-top 12636  df-topon 12649  df-bases 12681  df-ntr 12736  df-cn 12828  df-cnp 12829  df-tx 12893  df-cncf 13198  df-limced 13265  df-dvap 13266
This theorem is referenced by:  efcn  13329
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