ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cncfmptc Unicode version
Theorem cncfmptc 15339
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 1022 . 2 |- ( ( A e. T /\ S C_ CC /\ T C_ CC ) -> ( S C_ CC /\ T C_ CC ) )
2 simpl1 1026 . . . 4 |- ( ( ( A e. T /\ S C_ CC /\ T C_ CC ) /\ x e. S ) -> A e. T )
32fmpttd 5802 . . 3 |- ( ( A e. T /\ S C_ CC /\ T C_ CC ) -> ( x e. S |-> A ) : S --> T )
4 1rp 9892 . . . 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 2231 . . . . . . . . . 10 |- ( x e. S |-> A ) = ( x e. S |-> A )
7 eqidd 2232 . . . . . . . . . 10 |- ( x = y -> A = A )
8 simprll 539 . . . . . . . . . 10 |- ( ( ( A e. T /\ S C_ CC /\ T C_ CC ) /\ ( ( y e. S /\ w e. S ) /\ z e. RR+ ) ) -> y e. S )
9 simpl1 1026 . . . . . . . . . 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 5740 . . . . . . . . 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 2232 . . . . . . . . . 10 |- ( x = w -> A = A )
12 simprlr 540 . . . . . . . . . 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 5740 . . . . . . . . 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 6036 . . . . . . . 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 1028 . . . . . . . . . 10 |- ( ( ( A e. T /\ S C_ CC /\ T C_ CC ) /\ ( ( y e. S /\ w e. S ) /\ z e. RR+ ) ) -> T C_ CC )
1615, 9sseldd 3228 . . . . . . . . 9 |- ( ( ( A e. T /\ S C_ CC /\ T C_ CC ) /\ ( ( y e. S /\ w e. S ) /\ z e. RR+ ) ) -> A e. CC )
1716subidd 8478 . . . . . . . 8 |- ( ( ( A e. T /\ S C_ CC /\ T C_ CC ) /\ ( ( y e. S /\ w e. S ) /\ z e. RR+ ) ) -> ( A - A ) = 0 )
1814, 17eqtrd 2264 . . . . . . 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 11644 . . . . . 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 533 . . . . . . 7 |- ( ( ( A e. T /\ S C_ CC /\ T C_ CC ) /\ ( ( y e. S /\ w e. S ) /\ z e. RR+ ) ) -> z e. RR+ )
2120rpgt0d 9934 . . . . . 6 |- ( ( ( A e. T /\ S C_ CC /\ T C_ CC ) /\ ( ( y e. S /\ w e. S ) /\ z e. RR+ ) ) -> 0 < z )
2219, 21eqbrtrd 4110 . . . . 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 15322 . 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 1004   e. wcel 2202   C_ wss 3200   class class class wbr 4088   |-> cmpt 4150   ` cfv 5326  (class class class)co 6018   CCcc 8030   0cc0 8032   1c1 8033   < clt 8214   - cmin 8350   RR+crp 9888   abscabs 11575   -cn->ccncf 15313
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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150
This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-map 6819  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-n0 9403  df-z 9480  df-uz 9756  df-rp 9889  df-seqfrec 10711  df-exp 10802  df-cj 11420  df-rsqrt 11576  df-abs 11577  df-cncf 15314
This theorem is referenced by:  sub1cncf  15345  sub2cncf  15346  expcncf  15352  maxcncf  15358  mincncf  15359  ivthreinc  15388  hovercncf  15389  dvidlemap  15434  dvidrelem  15435  dvidsslem  15436  dvcnp2cntop  15442  dvmulxxbr  15445
  Copyright terms: Public domain W3C validator