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Theorem addccncf 14089
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 3176 . 2  |-  CC  C_  CC
2 addcl 7936 . . . . 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 5671 . . 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 5882 . . . . . . . . 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 7977 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( ( y  e.  CC  /\  z  e.  CC )  /\  w  e.  RR+ ) )  -> 
( y  +  A
)  e.  CC )
124, 8, 9, 11fvmptd3 5610 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( ( y  e.  CC  /\  z  e.  CC )  /\  w  e.  RR+ ) )  -> 
( F `  y
)  =  ( y  +  A ) )
13 oveq1 5882 . . . . . . . . 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 7977 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( ( y  e.  CC  /\  z  e.  CC )  /\  w  e.  RR+ ) )  -> 
( z  +  A
)  e.  CC )
164, 13, 14, 15fvmptd3 5610 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( ( y  e.  CC  /\  z  e.  CC )  /\  w  e.  RR+ ) )  -> 
( F `  z
)  =  ( z  +  A ) )
1712, 16oveq12d 5893 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( ( y  e.  CC  /\  z  e.  CC )  /\  w  e.  RR+ ) )  -> 
( ( F `  y )  -  ( F `  z )
)  =  ( ( y  +  A )  -  ( z  +  A ) ) )
189, 14, 10pnpcan2d 8306 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( ( y  e.  CC  /\  z  e.  CC )  /\  w  e.  RR+ ) )  -> 
( ( y  +  A )  -  (
z  +  A ) )  =  ( y  -  z ) )
1917, 18eqtrd 2210 . . . . . 6  |-  ( ( A  e.  CC  /\  ( ( y  e.  CC  /\  z  e.  CC )  /\  w  e.  RR+ ) )  -> 
( ( F `  y )  -  ( F `  z )
)  =  ( y  -  z ) )
2019fveq2d 5520 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( y  e.  CC  /\  z  e.  CC )  /\  w  e.  RR+ ) )  -> 
( abs `  (
( F `  y
)  -  ( F `
 z ) ) )  =  ( abs `  ( y  -  z
) ) )
2120breq1d 4014 . . . 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 14069 . 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 1353    e. wcel 2148    C_ wss 3130   class class class wbr 4004    |-> cmpt 4065   ` cfv 5217  (class class class)co 5875   CCcc 7809    + caddc 7814    < clt 7992    - cmin 8128   RR+crp 9653   abscabs 11006   -cn->ccncf 14060
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-distr 7915  ax-i2m1 7916  ax-0id 7919  ax-rnegex 7920  ax-cnre 7922
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-map 6650  df-sub 8130  df-cncf 14061
This theorem is referenced by: (None)
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