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.

Step	Hyp	Ref	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 )
3	2	fmpttd 5507	. . 3 |- ( ( A e. T /\ S C_ CC /\ T C_ CC ) -> ( x e. S |-> A ) : S --> T )
4		1rp 9295	. . . 4 |- 1 e. RR+
5	4	2a1i 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 )
10	6, 7, 8, 9	fvmptd3 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 )
13	6, 11, 12, 9	fvmptd3 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 )
14	10, 13	oveq12d 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 )
16	15, 9	sseldd 3048	. . . . . . . . 9 |- ( ( ( A e. T /\ S C_ CC /\ T C_ CC ) /\ ( ( y e. S /\ w e. S ) /\ z e. RR+ ) ) -> A e. CC )
17	16	subidd 7932	. . . . . . . 8 |- ( ( ( A e. T /\ S C_ CC /\ T C_ CC ) /\ ( ( y e. S /\ w e. S ) /\ z e. RR+ ) ) -> ( A - A ) = 0 )
18	14, 17	eqtrd 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 )
19	18	abs00bd 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+ )
21	20	rpgt0d 9333	. . . . . 6 |- ( ( ( A e. T /\ S C_ CC /\ T C_ CC ) /\ ( ( y e. S /\ w e. S ) /\ z e. RR+ ) ) -> 0 < z )
22	19, 21	eqbrtrd 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 )
23	22	a1d 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 ) )
24	23	ex 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 ) ) )
25	3, 5, 24	elcncf1di 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 ) ) )
26	1, 25	mpd 13	1 |- ( ( A e. T /\ S C_ CC /\ T C_ CC ) -> ( x e. S |-> A ) e. ( S -cn-> T ) )

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