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Theorem dvcn 15374
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 1025 . . 3 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ dom ( S _D F ) = A ) -> F : A --> CC )
2 eqid 2229 . . . . . 6 |- ( ( MetOpen ` ( abs o. - ) ) ↾t A ) = ( ( MetOpen ` ( abs o. - ) ) ↾t A )
3 eqid 2229 . . . . . 6 |- ( MetOpen ` ( abs o. - ) ) = ( MetOpen ` ( abs o. - ) )
42, 3dvcnp2cntop 15373 . . . . 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 2603 . . . 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 2728 . . . . 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 15206 . . . . 5 |- ( MetOpen ` ( abs o. - ) ) e. (TopOn ` CC )
10 simpl3 1026 . . . . . 6 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ dom ( S _D F ) = A ) -> A C_ S )
11 simpl1 1024 . . . . . 6 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ dom ( S _D F ) = A ) -> S C_ CC )
1210, 11sstrd 3234 . . . . 5 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ dom ( S _D F ) = A ) -> A C_ CC )
13 resttopon 14845 . . . . 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 14904 . . . 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 950 . 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 3244 . . 3 |- CC C_ CC
199toponrestid 14695 . . . 4 |- ( MetOpen ` ( abs o. - ) ) = ( ( MetOpen ` ( abs o. - ) ) ↾t CC )
203, 2, 19cncfcncntop 15267 . . 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 2309 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 1002   = wceq 1395   e. wcel 2200   A.wral 2508   C_ wss 3197   dom cdm 4719   o. ccom 4723   -->wf 5314   ` cfv 5318  (class class class)co 6001   CCcc 7997   - cmin 8317   abscabs 11508   ↾t crest 13272   MetOpencmopn 14505  TopOnctopon 14684   Cn ccn 14859   CnP ccnp 14860   -cn->ccncf 15244   _D cdv 15329
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119  ax-addf 8121  ax-mulf 8122
This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-map 6797  df-pm 6798  df-sup 7151  df-inf 7152  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-n0 9370  df-z 9447  df-uz 9723  df-q 9815  df-rp 9850  df-xneg 9968  df-xadd 9969  df-seqfrec 10670  df-exp 10761  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510  df-rest 13274  df-topgen 13293  df-psmet 14507  df-xmet 14508  df-met 14509  df-bl 14510  df-mopn 14511  df-top 14672  df-topon 14685  df-bases 14717  df-ntr 14770  df-cn 14862  df-cnp 14863  df-tx 14927  df-cncf 15245  df-limced 15330  df-dvap 15331
This theorem is referenced by:  efcn  15442
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