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Theorem addccncf 12769
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 3117 . 2  |-  CC  C_  CC
2 addcl 7757 . . . . 5  |-  ( ( x  e.  CC  /\  A  e.  CC )  ->  ( x  +  A
)  e.  CC )
32ancoms 266 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( x  +  A
)  e.  CC )
4 addccncf.1 . . . 4  |-  F  =  ( x  e.  CC  |->  ( x  +  A
) )
53, 4fmptd 5574 . . 3  |-  ( A  e.  CC  ->  F : CC --> CC )
6 simpr 109 . . . 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 5781 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  +  A )  =  ( y  +  A ) )
9 simprll 526 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( ( y  e.  CC  /\  z  e.  CC )  /\  w  e.  RR+ ) )  -> 
y  e.  CC )
10 simpl 108 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( ( y  e.  CC  /\  z  e.  CC )  /\  w  e.  RR+ ) )  ->  A  e.  CC )
119, 10addcld 7797 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( ( y  e.  CC  /\  z  e.  CC )  /\  w  e.  RR+ ) )  -> 
( y  +  A
)  e.  CC )
124, 8, 9, 11fvmptd3 5514 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( ( y  e.  CC  /\  z  e.  CC )  /\  w  e.  RR+ ) )  -> 
( F `  y
)  =  ( y  +  A ) )
13 oveq1 5781 . . . . . . . . 9  |-  ( x  =  z  ->  (
x  +  A )  =  ( z  +  A ) )
14 simprlr 527 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( ( y  e.  CC  /\  z  e.  CC )  /\  w  e.  RR+ ) )  -> 
z  e.  CC )
1514, 10addcld 7797 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( ( y  e.  CC  /\  z  e.  CC )  /\  w  e.  RR+ ) )  -> 
( z  +  A
)  e.  CC )
164, 13, 14, 15fvmptd3 5514 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( ( y  e.  CC  /\  z  e.  CC )  /\  w  e.  RR+ ) )  -> 
( F `  z
)  =  ( z  +  A ) )
1712, 16oveq12d 5792 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( ( y  e.  CC  /\  z  e.  CC )  /\  w  e.  RR+ ) )  -> 
( ( F `  y )  -  ( F `  z )
)  =  ( ( y  +  A )  -  ( z  +  A ) ) )
189, 14, 10pnpcan2d 8123 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( ( y  e.  CC  /\  z  e.  CC )  /\  w  e.  RR+ ) )  -> 
( ( y  +  A )  -  (
z  +  A ) )  =  ( y  -  z ) )
1917, 18eqtrd 2172 . . . . . 6  |-  ( ( A  e.  CC  /\  ( ( y  e.  CC  /\  z  e.  CC )  /\  w  e.  RR+ ) )  -> 
( ( F `  y )  -  ( F `  z )
)  =  ( y  -  z ) )
2019fveq2d 5425 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( y  e.  CC  /\  z  e.  CC )  /\  w  e.  RR+ ) )  -> 
( abs `  (
( F `  y
)  -  ( F `
 z ) ) )  =  ( abs `  ( y  -  z
) ) )
2120breq1d 3939 . . . 4  |-  ( ( A  e.  CC  /\  ( ( y  e.  CC  /\  z  e.  CC )  /\  w  e.  RR+ ) )  -> 
( ( abs `  (
( F `  y
)  -  ( F `
 z ) ) )  <  w  <->  ( abs `  ( y  -  z
) )  <  w
) )
2221exbiri 379 . . 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 12749 . 2  |-  ( A  e.  CC  ->  (
( CC  C_  CC  /\  CC  C_  CC )  ->  F  e.  ( CC
-cn-> CC ) ) )
241, 1, 23mp2ani 428 1  |-  ( A  e.  CC  ->  F  e.  ( CC -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    + caddc 7635    < clt 7812    - cmin 7945   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-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-addcom 7732  ax-addass 7734  ax-distr 7736  ax-i2m1 7737  ax-0id 7740  ax-rnegex 7741  ax-cnre 7743
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-ral 2421  df-rex 2422  df-reu 2423  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-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  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-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-map 6544  df-sub 7947  df-cncf 12741
This theorem is referenced by: (None)
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