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Theorem negcncf 12771
Description: The negative function is continuous. (Contributed by Mario Carneiro, 30-Dec-2016.)
Hypothesis
Ref Expression
negcncf.1  |-  F  =  ( x  e.  A  |-> 
-u x )
Assertion
Ref Expression
negcncf  |-  ( A 
C_  CC  ->  F  e.  ( A -cn-> CC ) )
Distinct variable group:    x, A
Allowed substitution hint:    F( x)

Proof of Theorem negcncf
Dummy variables  e  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 19 . 2  |-  ( A 
C_  CC  ->  A  C_  CC )
2 ssidd 3118 . 2  |-  ( A 
C_  CC  ->  CC  C_  CC )
3 ssel2 3092 . . . . 5  |-  ( ( A  C_  CC  /\  x  e.  A )  ->  x  e.  CC )
43negcld 8072 . . . 4  |-  ( ( A  C_  CC  /\  x  e.  A )  ->  -u x  e.  CC )
5 negcncf.1 . . . 4  |-  F  =  ( x  e.  A  |-> 
-u x )
64, 5fmptd 5574 . . 3  |-  ( A 
C_  CC  ->  F : A
--> CC )
7 simpr 109 . . . 4  |-  ( ( u  e.  A  /\  e  e.  RR+ )  -> 
e  e.  RR+ )
87a1i 9 . . 3  |-  ( A 
C_  CC  ->  ( ( u  e.  A  /\  e  e.  RR+ )  -> 
e  e.  RR+ )
)
9 negeq 7967 . . . . . . . . . 10  |-  ( x  =  u  ->  -u x  =  -u u )
10 simprll 526 . . . . . . . . . 10  |-  ( ( A  C_  CC  /\  (
( u  e.  A  /\  v  e.  A
)  /\  e  e.  RR+ ) )  ->  u  e.  A )
11 simpl 108 . . . . . . . . . . . 12  |-  ( ( A  C_  CC  /\  (
( u  e.  A  /\  v  e.  A
)  /\  e  e.  RR+ ) )  ->  A  C_  CC )
1211, 10sseldd 3098 . . . . . . . . . . 11  |-  ( ( A  C_  CC  /\  (
( u  e.  A  /\  v  e.  A
)  /\  e  e.  RR+ ) )  ->  u  e.  CC )
1312negcld 8072 . . . . . . . . . 10  |-  ( ( A  C_  CC  /\  (
( u  e.  A  /\  v  e.  A
)  /\  e  e.  RR+ ) )  ->  -u u  e.  CC )
145, 9, 10, 13fvmptd3 5514 . . . . . . . . 9  |-  ( ( A  C_  CC  /\  (
( u  e.  A  /\  v  e.  A
)  /\  e  e.  RR+ ) )  ->  ( F `  u )  =  -u u )
15 negeq 7967 . . . . . . . . . 10  |-  ( x  =  v  ->  -u x  =  -u v )
16 simprlr 527 . . . . . . . . . 10  |-  ( ( A  C_  CC  /\  (
( u  e.  A  /\  v  e.  A
)  /\  e  e.  RR+ ) )  ->  v  e.  A )
1711, 16sseldd 3098 . . . . . . . . . . 11  |-  ( ( A  C_  CC  /\  (
( u  e.  A  /\  v  e.  A
)  /\  e  e.  RR+ ) )  ->  v  e.  CC )
1817negcld 8072 . . . . . . . . . 10  |-  ( ( A  C_  CC  /\  (
( u  e.  A  /\  v  e.  A
)  /\  e  e.  RR+ ) )  ->  -u v  e.  CC )
195, 15, 16, 18fvmptd3 5514 . . . . . . . . 9  |-  ( ( A  C_  CC  /\  (
( u  e.  A  /\  v  e.  A
)  /\  e  e.  RR+ ) )  ->  ( F `  v )  =  -u v )
2014, 19oveq12d 5792 . . . . . . . 8  |-  ( ( A  C_  CC  /\  (
( u  e.  A  /\  v  e.  A
)  /\  e  e.  RR+ ) )  ->  (
( F `  u
)  -  ( F `
 v ) )  =  ( -u u  -  -u v ) )
2112, 17neg2subd 8102 . . . . . . . 8  |-  ( ( A  C_  CC  /\  (
( u  e.  A  /\  v  e.  A
)  /\  e  e.  RR+ ) )  ->  ( -u u  -  -u v
)  =  ( v  -  u ) )
2220, 21eqtrd 2172 . . . . . . 7  |-  ( ( A  C_  CC  /\  (
( u  e.  A  /\  v  e.  A
)  /\  e  e.  RR+ ) )  ->  (
( F `  u
)  -  ( F `
 v ) )  =  ( v  -  u ) )
2322fveq2d 5425 . . . . . 6  |-  ( ( A  C_  CC  /\  (
( u  e.  A  /\  v  e.  A
)  /\  e  e.  RR+ ) )  ->  ( abs `  ( ( F `
 u )  -  ( F `  v ) ) )  =  ( abs `  ( v  -  u ) ) )
2417, 12abssubd 10977 . . . . . 6  |-  ( ( A  C_  CC  /\  (
( u  e.  A  /\  v  e.  A
)  /\  e  e.  RR+ ) )  ->  ( abs `  ( v  -  u ) )  =  ( abs `  (
u  -  v ) ) )
2523, 24eqtrd 2172 . . . . 5  |-  ( ( A  C_  CC  /\  (
( u  e.  A  /\  v  e.  A
)  /\  e  e.  RR+ ) )  ->  ( abs `  ( ( F `
 u )  -  ( F `  v ) ) )  =  ( abs `  ( u  -  v ) ) )
2625breq1d 3939 . . . 4  |-  ( ( A  C_  CC  /\  (
( u  e.  A  /\  v  e.  A
)  /\  e  e.  RR+ ) )  ->  (
( abs `  (
( F `  u
)  -  ( F `
 v ) ) )  <  e  <->  ( abs `  ( u  -  v
) )  <  e
) )
2726exbiri 379 . . 3  |-  ( A 
C_  CC  ->  ( ( ( u  e.  A  /\  v  e.  A
)  /\  e  e.  RR+ )  ->  ( ( abs `  ( u  -  v ) )  < 
e  ->  ( abs `  ( ( F `  u )  -  ( F `  v )
) )  <  e
) ) )
286, 8, 27elcncf1di 12749 . 2  |-  ( A 
C_  CC  ->  ( ( A  C_  CC  /\  CC  C_  CC )  ->  F  e.  ( A -cn-> CC ) ) )
291, 2, 28mp2and 429 1  |-  ( A 
C_  CC  ->  F  e.  ( A -cn-> CC ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480    C_ wss 3071   class class class wbr 3929    |-> cmpt 3989   ` cfv 5123  (class class class)co 5774   CCcc 7630    < clt 7812    - cmin 7945   -ucneg 7946   RR+crp 9453   abscabs 10781   -cn->ccncf 12740
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-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulrcl 7731  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-precex 7742  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748  ax-pre-mulgt0 7749  ax-pre-mulext 7750
This theorem depends on definitions:  df-bi 116  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-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-po 4218  df-iso 4219  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-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-map 6544  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-reap 8349  df-ap 8356  df-div 8445  df-2 8791  df-cj 10626  df-re 10627  df-im 10628  df-rsqrt 10782  df-abs 10783  df-cncf 12741
This theorem is referenced by:  negfcncf  12772
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