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Theorem cncfmptc 12495
Description: A constant function is a continuous function on  CC. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Sep-2015.)
Assertion
Ref Expression
cncfmptc  |-  ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  ->  (
x  e.  S  |->  A )  e.  ( S
-cn-> T ) )
Distinct variable groups:    x, A    x, S    x, T

Proof of Theorem cncfmptc
Dummy variables  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 3simpc 948 . 2  |-  ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  ->  ( S  C_  CC  /\  T  C_  CC ) )
2 simpl1 952 . . . 4  |-  ( ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  /\  x  e.  S )  ->  A  e.  T )
32fmpttd 5507 . . 3  |-  ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  ->  (
x  e.  S  |->  A ) : S --> T )
4 1rp 9295 . . . 4  |-  1  e.  RR+
542a1i 27 . . 3  |-  ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  ->  (
( y  e.  S  /\  z  e.  RR+ )  ->  1  e.  RR+ )
)
6 eqid 2100 . . . . . . . . . 10  |-  ( x  e.  S  |->  A )  =  ( x  e.  S  |->  A )
7 eqidd 2101 . . . . . . . . . 10  |-  ( x  =  y  ->  A  =  A )
8 simprll 507 . . . . . . . . . 10  |-  ( ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  /\  ( ( y  e.  S  /\  w  e.  S )  /\  z  e.  RR+ ) )  -> 
y  e.  S )
9 simpl1 952 . . . . . . . . . 10  |-  ( ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  /\  ( ( y  e.  S  /\  w  e.  S )  /\  z  e.  RR+ ) )  ->  A  e.  T )
106, 7, 8, 9fvmptd3 5446 . . . . . . . . 9  |-  ( ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  /\  ( ( y  e.  S  /\  w  e.  S )  /\  z  e.  RR+ ) )  -> 
( ( x  e.  S  |->  A ) `  y )  =  A )
11 eqidd 2101 . . . . . . . . . 10  |-  ( x  =  w  ->  A  =  A )
12 simprlr 508 . . . . . . . . . 10  |-  ( ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  /\  ( ( y  e.  S  /\  w  e.  S )  /\  z  e.  RR+ ) )  ->  w  e.  S )
136, 11, 12, 9fvmptd3 5446 . . . . . . . . 9  |-  ( ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  /\  ( ( y  e.  S  /\  w  e.  S )  /\  z  e.  RR+ ) )  -> 
( ( x  e.  S  |->  A ) `  w )  =  A )
1410, 13oveq12d 5724 . . . . . . . 8  |-  ( ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  /\  ( ( y  e.  S  /\  w  e.  S )  /\  z  e.  RR+ ) )  -> 
( ( ( x  e.  S  |->  A ) `
 y )  -  ( ( x  e.  S  |->  A ) `  w ) )  =  ( A  -  A
) )
15 simpl3 954 . . . . . . . . . 10  |-  ( ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  /\  ( ( y  e.  S  /\  w  e.  S )  /\  z  e.  RR+ ) )  ->  T  C_  CC )
1615, 9sseldd 3048 . . . . . . . . 9  |-  ( ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  /\  ( ( y  e.  S  /\  w  e.  S )  /\  z  e.  RR+ ) )  ->  A  e.  CC )
1716subidd 7932 . . . . . . . 8  |-  ( ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  /\  ( ( y  e.  S  /\  w  e.  S )  /\  z  e.  RR+ ) )  -> 
( A  -  A
)  =  0 )
1814, 17eqtrd 2132 . . . . . . 7  |-  ( ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  /\  ( ( y  e.  S  /\  w  e.  S )  /\  z  e.  RR+ ) )  -> 
( ( ( x  e.  S  |->  A ) `
 y )  -  ( ( x  e.  S  |->  A ) `  w ) )  =  0 )
1918abs00bd 10678 . . . . . 6  |-  ( ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  /\  ( ( y  e.  S  /\  w  e.  S )  /\  z  e.  RR+ ) )  -> 
( abs `  (
( ( x  e.  S  |->  A ) `  y )  -  (
( x  e.  S  |->  A ) `  w
) ) )  =  0 )
20 simprr 502 . . . . . . 7  |-  ( ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  /\  ( ( y  e.  S  /\  w  e.  S )  /\  z  e.  RR+ ) )  -> 
z  e.  RR+ )
2120rpgt0d 9333 . . . . . 6  |-  ( ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  /\  ( ( y  e.  S  /\  w  e.  S )  /\  z  e.  RR+ ) )  -> 
0  <  z )
2219, 21eqbrtrd 3895 . . . . 5  |-  ( ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  /\  ( ( y  e.  S  /\  w  e.  S )  /\  z  e.  RR+ ) )  -> 
( abs `  (
( ( x  e.  S  |->  A ) `  y )  -  (
( x  e.  S  |->  A ) `  w
) ) )  < 
z )
2322a1d 22 . . . 4  |-  ( ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  /\  ( ( y  e.  S  /\  w  e.  S )  /\  z  e.  RR+ ) )  -> 
( ( abs `  (
y  -  w ) )  <  1  -> 
( abs `  (
( ( x  e.  S  |->  A ) `  y )  -  (
( x  e.  S  |->  A ) `  w
) ) )  < 
z ) )
2423ex 114 . . 3  |-  ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  ->  (
( ( y  e.  S  /\  w  e.  S )  /\  z  e.  RR+ )  ->  (
( abs `  (
y  -  w ) )  <  1  -> 
( abs `  (
( ( x  e.  S  |->  A ) `  y )  -  (
( x  e.  S  |->  A ) `  w
) ) )  < 
z ) ) )
253, 5, 24elcncf1di 12479 . 2  |-  ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  ->  (
( S  C_  CC  /\  T  C_  CC )  ->  ( x  e.  S  |->  A )  e.  ( S -cn-> T ) ) )
261, 25mpd 13 1  |-  ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  ->  (
x  e.  S  |->  A )  e.  ( S
-cn-> T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 930    e. wcel 1448    C_ wss 3021   class class class wbr 3875    |-> cmpt 3929   ` cfv 5059  (class class class)co 5706   CCcc 7498   0cc0 7500   1c1 7501    < clt 7672    - cmin 7804   RR+crp 9291   abscabs 10609   -cn->ccncf 12470
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-map 6474  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-2 8637  df-n0 8830  df-z 8907  df-uz 9177  df-rp 9292  df-seqfrec 10060  df-exp 10134  df-cj 10455  df-rsqrt 10610  df-abs 10611  df-cncf 12471
This theorem is referenced by:  expcncf  12504  dvidlemap  12533
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