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Theorem cncfmptc 14479
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 998 . 2  |-  ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  ->  ( S  C_  CC  /\  T  C_  CC ) )
2 simpl1 1002 . . . 4  |-  ( ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  /\  x  e.  S )  ->  A  e.  T )
32fmpttd 5687 . . 3  |-  ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  ->  (
x  e.  S  |->  A ) : S --> T )
4 1rp 9675 . . . 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 2189 . . . . . . . . . 10  |-  ( x  e.  S  |->  A )  =  ( x  e.  S  |->  A )
7 eqidd 2190 . . . . . . . . . 10  |-  ( x  =  y  ->  A  =  A )
8 simprll 537 . . . . . . . . . 10  |-  ( ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  /\  ( ( y  e.  S  /\  w  e.  S )  /\  z  e.  RR+ ) )  -> 
y  e.  S )
9 simpl1 1002 . . . . . . . . . 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 5625 . . . . . . . . 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 2190 . . . . . . . . . 10  |-  ( x  =  w  ->  A  =  A )
12 simprlr 538 . . . . . . . . . 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 5625 . . . . . . . . 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 5909 . . . . . . . 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 1004 . . . . . . . . . 10  |-  ( ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  /\  ( ( y  e.  S  /\  w  e.  S )  /\  z  e.  RR+ ) )  ->  T  C_  CC )
1615, 9sseldd 3171 . . . . . . . . 9  |-  ( ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  /\  ( ( y  e.  S  /\  w  e.  S )  /\  z  e.  RR+ ) )  ->  A  e.  CC )
1716subidd 8274 . . . . . . . 8  |-  ( ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  /\  ( ( y  e.  S  /\  w  e.  S )  /\  z  e.  RR+ ) )  -> 
( A  -  A
)  =  0 )
1814, 17eqtrd 2222 . . . . . . 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 11093 . . . . . 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 531 . . . . . . 7  |-  ( ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  /\  ( ( y  e.  S  /\  w  e.  S )  /\  z  e.  RR+ ) )  -> 
z  e.  RR+ )
2120rpgt0d 9717 . . . . . 6  |-  ( ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  /\  ( ( y  e.  S  /\  w  e.  S )  /\  z  e.  RR+ ) )  -> 
0  <  z )
2219, 21eqbrtrd 4040 . . . . 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 115 . . 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 14463 . 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 104    /\ w3a 980    e. wcel 2160    C_ wss 3144   class class class wbr 4018    |-> cmpt 4079   ` cfv 5231  (class class class)co 5891   CCcc 7827   0cc0 7829   1c1 7830    < clt 8010    - cmin 8146   RR+crp 9671   abscabs 11024   -cn->ccncf 14454
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7920  ax-resscn 7921  ax-1cn 7922  ax-1re 7923  ax-icn 7924  ax-addcl 7925  ax-addrcl 7926  ax-mulcl 7927  ax-mulrcl 7928  ax-addcom 7929  ax-mulcom 7930  ax-addass 7931  ax-mulass 7932  ax-distr 7933  ax-i2m1 7934  ax-0lt1 7935  ax-1rid 7936  ax-0id 7937  ax-rnegex 7938  ax-precex 7939  ax-cnre 7940  ax-pre-ltirr 7941  ax-pre-ltwlin 7942  ax-pre-lttrn 7943  ax-pre-apti 7944  ax-pre-ltadd 7945  ax-pre-mulgt0 7946  ax-pre-mulext 7947
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-frec 6410  df-map 6668  df-pnf 8012  df-mnf 8013  df-xr 8014  df-ltxr 8015  df-le 8016  df-sub 8148  df-neg 8149  df-reap 8550  df-ap 8557  df-div 8648  df-inn 8938  df-2 8996  df-n0 9195  df-z 9272  df-uz 9547  df-rp 9672  df-seqfrec 10464  df-exp 10538  df-cj 10869  df-rsqrt 11025  df-abs 11026  df-cncf 14455
This theorem is referenced by:  expcncf  14489  dvidlemap  14557  dvcnp2cntop  14560  dvmulxxbr  14563
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