ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cncfmptc Unicode version
Theorem cncfmptc 14916
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 5720 . . 3 |- ( ( A e. T /\ S C_ CC /\ T C_ CC ) -> ( x e. S |-> A ) : S --> T )
4 1rp 9749 . . . 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 2196 . . . . . . . . . 10 |- ( x e. S |-> A ) = ( x e. S |-> A )
7 eqidd 2197 . . . . . . . . . 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 5658 . . . . . . . . 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 2197 . . . . . . . . . 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 5658 . . . . . . . . 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 5943 . . . . . . . 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 3185 . . . . . . . . 9 |- ( ( ( A e. T /\ S C_ CC /\ T C_ CC ) /\ ( ( y e. S /\ w e. S ) /\ z e. RR+ ) ) -> A e. CC )
1716subidd 8342 . . . . . . . 8 |- ( ( ( A e. T /\ S C_ CC /\ T C_ CC ) /\ ( ( y e. S /\ w e. S ) /\ z e. RR+ ) ) -> ( A - A ) = 0 )
1814, 17eqtrd 2229 . . . . . . 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 11248 . . . . . 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 9791 . . . . . 6 |- ( ( ( A e. T /\ S C_ CC /\ T C_ CC ) /\ ( ( y e. S /\ w e. S ) /\ z e. RR+ ) ) -> 0 < z )
2219, 21eqbrtrd 4056 . . . . 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 14899 . 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 2167   C_ wss 3157   class class class wbr 4034   |-> cmpt 4095   ` cfv 5259  (class class class)co 5925   CCcc 7894   0cc0 7896   1c1 7897   < clt 8078   - cmin 8214   RR+crp 9745   abscabs 11179   -cn->ccncf 14890
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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014
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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-map 6718  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-n0 9267  df-z 9344  df-uz 9619  df-rp 9746  df-seqfrec 10557  df-exp 10648  df-cj 11024  df-rsqrt 11180  df-abs 11181  df-cncf 14891
This theorem is referenced by:  sub1cncf  14922  sub2cncf  14923  expcncf  14929  maxcncf  14935  mincncf  14936  ivthreinc  14965  hovercncf  14966  dvidlemap  15011  dvidrelem  15012  dvidsslem  15013  dvcnp2cntop  15019  dvmulxxbr  15022
  Copyright terms: Public domain W3C validator