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Theorem dvcn 12833
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 985 . . 3 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ dom ( S _D F ) = A ) -> F : A --> CC )
2 eqid 2139 . . . . . 6 |- ( ( MetOpen ` ( abs o. - ) ) ↾t A ) = ( ( MetOpen ` ( abs o. - ) ) ↾t A )
3 eqid 2139 . . . . . 6 |- ( MetOpen ` ( abs o. - ) ) = ( MetOpen ` ( abs o. - ) )
42, 3dvcnp2cntop 12832 . . . . 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 2505 . . . 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 2626 . . . . 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 12701 . . . . 5 |- ( MetOpen ` ( abs o. - ) ) e. (TopOn ` CC )
10 simpl3 986 . . . . . 6 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ dom ( S _D F ) = A ) -> A C_ S )
11 simpl1 984 . . . . . 6 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ dom ( S _D F ) = A ) -> S C_ CC )
1210, 11sstrd 3107 . . . . 5 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ dom ( S _D F ) = A ) -> A C_ CC )
13 resttopon 12340 . . . . 5 |- ( ( ( MetOpen ` ( abs o. - ) ) e. (TopOn ` CC ) /\ A C_ CC ) -> ( ( MetOpen ` ( abs o. - ) ) ↾t A ) e. (TopOn ` A ) )
149, 12, 13sylancr 410 . . . 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 12399 . . . 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 409 . . 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 928 . 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 3117 . . 3 |- CC C_ CC
199toponrestid 12188 . . . 4 |- ( MetOpen ` ( abs o. - ) ) = ( ( MetOpen ` ( abs o. - ) ) ↾t CC )
203, 2, 19cncfcncntop 12749 . . 3 |- ( ( A C_ CC /\ CC C_ CC ) -> ( A -cn-> CC ) = ( ( ( MetOpen ` ( abs o. - ) ) ↾t A ) Cn ( MetOpen ` ( abs o. - ) ) ) )
2112, 18, 20sylancl 409 . 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 2219 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 962   = wceq 1331   e. wcel 1480   A.wral 2416   C_ wss 3071   dom cdm 4539   o. ccom 4543   -->wf 5119   ` cfv 5123  (class class class)co 5774   CCcc 7618   - cmin 7933   abscabs 10769   ↾t crest 12120   MetOpencmopn 12154  TopOnctopon 12177   Cn ccn 12354   CnP ccnp 12355   -cn->ccncf 12726   _D cdv 12793
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740  ax-addf 7742  ax-mulf 7743
This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-map 6544  df-pm 6545  df-sup 6871  df-inf 6872  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-xneg 9559  df-xadd 9560  df-seqfrec 10219  df-exp 10293  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-rest 12122  df-topgen 12141  df-psmet 12156  df-xmet 12157  df-met 12158  df-bl 12159  df-mopn 12160  df-top 12165  df-topon 12178  df-bases 12210  df-ntr 12265  df-cn 12357  df-cnp 12358  df-tx 12422  df-cncf 12727  df-limced 12794  df-dvap 12795
This theorem is referenced by:  efcn  12857
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