Theorem cncfmptc 13633

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 996	. 2 |- ( ( A e. T /\ S C_ CC /\ T C_ CC ) -> ( S C_ CC /\ T C_ CC ) )
2		simpl1 1000	. . . 4 |- ( ( ( A e. T /\ S C_ CC /\ T C_ CC ) /\ x e. S ) -> A e. T )
3	2	fmpttd 5663	. . 3 |- ( ( A e. T /\ S C_ CC /\ T C_ CC ) -> ( x e. S |-> A ) : S --> T )
4		1rp 9626	. . . 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 2175	. . . . . . . . . 10 |- ( x e. S |-> A ) = ( x e. S |-> A )
7		eqidd 2176	. . . . . . . . . 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 1000	. . . . . . . . . 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 5601	. . . . . . . . 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 2176	. . . . . . . . . 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 )
13	6, 11, 12, 9	fvmptd3 5601	. . . . . . . . 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 5883	. . . . . . . 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 1002	. . . . . . . . . 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 3154	. . . . . . . . 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 8230	. . . . . . . 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 2208	. . . . . . 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 11041	. . . . . 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+ )
21	20	rpgt0d 9668	. . . . . 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 4020	. . . . 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 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 ) ) )
25	3, 5, 24	elcncf1di 13617	. 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 104   /\ w3a 978   e. wcel 2146   C_ wss 3127   class class class wbr 3998   |-> cmpt 4059   ` cfv 5208  (class class class)co 5865   CCcc 7784   0cc0 7786   1c1 7787   < clt 7966   - cmin 8102   RR+crp 9622   abscabs 10972   -cn->ccncf 13608

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-map 6640  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8602  df-inn 8891  df-2 8949  df-n0 9148  df-z 9225  df-uz 9500  df-rp 9623  df-seqfrec 10414  df-exp 10488  df-cj 10817  df-rsqrt 10973  df-abs 10974  df-cncf 13609

This theorem is referenced by:  expcncf  13643  dvidlemap  13711  dvcnp2cntop  13714  dvmulxxbr  13717