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Theorem cncfmptc 15068
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 999 . 2 |- ( ( A e. T /\ S C_ CC /\ T C_ CC ) -> ( S C_ CC /\ T C_ CC ) )
2 simpl1 1003 . . . 4 |- ( ( ( A e. T /\ S C_ CC /\ T C_ CC ) /\ x e. S ) -> A e. T )
32fmpttd 5735 . . 3 |- ( ( A e. T /\ S C_ CC /\ T C_ CC ) -> ( x e. S |-> A ) : S --> T )
4 1rp 9779 . . . 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 2205 . . . . . . . . . 10 |- ( x e. S |-> A ) = ( x e. S |-> A )
7 eqidd 2206 . . . . . . . . . 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 1003 . . . . . . . . . 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 5673 . . . . . . . . 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 2206 . . . . . . . . . 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 5673 . . . . . . . . 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 5962 . . . . . . . 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 1005 . . . . . . . . . 10 |- ( ( ( A e. T /\ S C_ CC /\ T C_ CC ) /\ ( ( y e. S /\ w e. S ) /\ z e. RR+ ) ) -> T C_ CC )
1615, 9sseldd 3194 . . . . . . . . 9 |- ( ( ( A e. T /\ S C_ CC /\ T C_ CC ) /\ ( ( y e. S /\ w e. S ) /\ z e. RR+ ) ) -> A e. CC )
1716subidd 8371 . . . . . . . 8 |- ( ( ( A e. T /\ S C_ CC /\ T C_ CC ) /\ ( ( y e. S /\ w e. S ) /\ z e. RR+ ) ) -> ( A - A ) = 0 )
1814, 17eqtrd 2238 . . . . . . 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 11377 . . . . . 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 9821 . . . . . 6 |- ( ( ( A e. T /\ S C_ CC /\ T C_ CC ) /\ ( ( y e. S /\ w e. S ) /\ z e. RR+ ) ) -> 0 < z )
2219, 21eqbrtrd 4066 . . . . 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 15051 . 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 981   e. wcel 2176   C_ wss 3166   class class class wbr 4044   |-> cmpt 4105   ` cfv 5271  (class class class)co 5944   CCcc 7923   0cc0 7925   1c1 7926   < clt 8107   - cmin 8243   RR+crp 9775   abscabs 11308   -cn->ccncf 15042
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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043
This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-frec 6477  df-map 6737  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-2 9095  df-n0 9296  df-z 9373  df-uz 9649  df-rp 9776  df-seqfrec 10593  df-exp 10684  df-cj 11153  df-rsqrt 11309  df-abs 11310  df-cncf 15043
This theorem is referenced by:  sub1cncf  15074  sub2cncf  15075  expcncf  15081  maxcncf  15087  mincncf  15088  ivthreinc  15117  hovercncf  15118  dvidlemap  15163  dvidrelem  15164  dvidsslem  15165  dvcnp2cntop  15171  dvmulxxbr  15174
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