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Theorem addccncf 15014
Description: Adding a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypothesis
Ref Expression
addccncf.1  |-  F  =  ( x  e.  CC  |->  ( x  +  A
) )
Assertion
Ref Expression
addccncf  |-  ( A  e.  CC  ->  F  e.  ( CC -cn-> CC ) )
Distinct variable group:    x, A
Allowed substitution hint:    F( x)

Proof of Theorem addccncf
Dummy variables  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssid 3212 . 2  |-  CC  C_  CC
2 addcl 8049 . . . . 5  |-  ( ( x  e.  CC  /\  A  e.  CC )  ->  ( x  +  A
)  e.  CC )
32ancoms 268 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( x  +  A
)  e.  CC )
4 addccncf.1 . . . 4  |-  F  =  ( x  e.  CC  |->  ( x  +  A
) )
53, 4fmptd 5733 . . 3  |-  ( A  e.  CC  ->  F : CC --> CC )
6 simpr 110 . . . 4  |-  ( ( y  e.  CC  /\  w  e.  RR+ )  ->  w  e.  RR+ )
76a1i 9 . . 3  |-  ( A  e.  CC  ->  (
( y  e.  CC  /\  w  e.  RR+ )  ->  w  e.  RR+ )
)
8 oveq1 5950 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  +  A )  =  ( y  +  A ) )
9 simprll 537 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( ( y  e.  CC  /\  z  e.  CC )  /\  w  e.  RR+ ) )  -> 
y  e.  CC )
10 simpl 109 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( ( y  e.  CC  /\  z  e.  CC )  /\  w  e.  RR+ ) )  ->  A  e.  CC )
119, 10addcld 8091 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( ( y  e.  CC  /\  z  e.  CC )  /\  w  e.  RR+ ) )  -> 
( y  +  A
)  e.  CC )
124, 8, 9, 11fvmptd3 5672 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( ( y  e.  CC  /\  z  e.  CC )  /\  w  e.  RR+ ) )  -> 
( F `  y
)  =  ( y  +  A ) )
13 oveq1 5950 . . . . . . . . 9  |-  ( x  =  z  ->  (
x  +  A )  =  ( z  +  A ) )
14 simprlr 538 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( ( y  e.  CC  /\  z  e.  CC )  /\  w  e.  RR+ ) )  -> 
z  e.  CC )
1514, 10addcld 8091 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( ( y  e.  CC  /\  z  e.  CC )  /\  w  e.  RR+ ) )  -> 
( z  +  A
)  e.  CC )
164, 13, 14, 15fvmptd3 5672 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( ( y  e.  CC  /\  z  e.  CC )  /\  w  e.  RR+ ) )  -> 
( F `  z
)  =  ( z  +  A ) )
1712, 16oveq12d 5961 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( ( y  e.  CC  /\  z  e.  CC )  /\  w  e.  RR+ ) )  -> 
( ( F `  y )  -  ( F `  z )
)  =  ( ( y  +  A )  -  ( z  +  A ) ) )
189, 14, 10pnpcan2d 8420 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( ( y  e.  CC  /\  z  e.  CC )  /\  w  e.  RR+ ) )  -> 
( ( y  +  A )  -  (
z  +  A ) )  =  ( y  -  z ) )
1917, 18eqtrd 2237 . . . . . 6  |-  ( ( A  e.  CC  /\  ( ( y  e.  CC  /\  z  e.  CC )  /\  w  e.  RR+ ) )  -> 
( ( F `  y )  -  ( F `  z )
)  =  ( y  -  z ) )
2019fveq2d 5579 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( y  e.  CC  /\  z  e.  CC )  /\  w  e.  RR+ ) )  -> 
( abs `  (
( F `  y
)  -  ( F `
 z ) ) )  =  ( abs `  ( y  -  z
) ) )
2120breq1d 4053 . . . 4  |-  ( ( A  e.  CC  /\  ( ( y  e.  CC  /\  z  e.  CC )  /\  w  e.  RR+ ) )  -> 
( ( abs `  (
( F `  y
)  -  ( F `
 z ) ) )  <  w  <->  ( abs `  ( y  -  z
) )  <  w
) )
2221exbiri 382 . . 3  |-  ( A  e.  CC  ->  (
( ( y  e.  CC  /\  z  e.  CC )  /\  w  e.  RR+ )  ->  (
( abs `  (
y  -  z ) )  <  w  -> 
( abs `  (
( F `  y
)  -  ( F `
 z ) ) )  <  w ) ) )
235, 7, 22elcncf1di 14993 . 2  |-  ( A  e.  CC  ->  (
( CC  C_  CC  /\  CC  C_  CC )  ->  F  e.  ( CC
-cn-> CC ) ) )
241, 1, 23mp2ani 432 1  |-  ( A  e.  CC  ->  F  e.  ( CC -cn-> CC ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1372    e. wcel 2175    C_ wss 3165   class class class wbr 4043    |-> cmpt 4104   ` cfv 5270  (class class class)co 5943   CCcc 7922    + caddc 7927    < clt 8106    - cmin 8242   RR+crp 9774   abscabs 11250   -cn->ccncf 14984
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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-addass 8026  ax-distr 8028  ax-i2m1 8029  ax-0id 8032  ax-rnegex 8033  ax-cnre 8035
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-map 6736  df-sub 8244  df-cncf 14985
This theorem is referenced by: (None)
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