Theorem climcncf 15307

Description: Image of a limit under a continuous map. (Contributed by Mario Carneiro, 7-Apr-2015.)

Hypotheses

Ref	Expression
climcncf.1	|- Z = ( ZZ>= ` M )
climcncf.2	|- ( ph -> M e. ZZ )
climcncf.4	|- ( ph -> F e. ( A -cn-> B ) )
climcncf.5	|- ( ph -> G : Z --> A )
climcncf.6	|- ( ph -> G ~~> D )
climcncf.7	|- ( ph -> D e. A )

Assertion

Ref	Expression
climcncf	|- ( ph -> ( F o. G ) ~~> ( F ` D ) )

Proof of Theorem climcncf

Dummy variables y z x k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		climcncf.1	. 2 |- Z = ( ZZ>= ` M )
2		climcncf.2	. 2 |- ( ph -> M e. ZZ )
3		climcncf.7	. 2 |- ( ph -> D e. A )
4		climcncf.4	. . . . 5 |- ( ph -> F e. ( A -cn-> B ) )
5		cncff 15300	. . . . 5 |- ( F e. ( A -cn-> B ) -> F : A --> B )
6	4, 5	syl 14	. . . 4 |- ( ph -> F : A --> B )
7	6	ffvelcdmda 5782	. . 3 |- ( ( ph /\ z e. A ) -> ( F ` z ) e. B )
8		cncfrss2 15299	. . . . 5 |- ( F e. ( A -cn-> B ) -> B C_ CC )
9	4, 8	syl 14	. . . 4 |- ( ph -> B C_ CC )
10	9	sselda 3227	. . 3 |- ( ( ph /\ ( F ` z ) e. B ) -> ( F ` z ) e. CC )
11	7, 10	syldan 282	. 2 |- ( ( ph /\ z e. A ) -> ( F ` z ) e. CC )
12		climcncf.6	. 2 |- ( ph -> G ~~> D )
13		climcncf.5	. . . 4 |- ( ph -> G : Z --> A )
14		zex 9487	. . . . . 6 |- ZZ e. _V
15		uzssz 9775	. . . . . 6 |- ( ZZ>= ` M ) C_ ZZ
16	14, 15	ssexi 4227	. . . . 5 |- ( ZZ>= ` M ) e. _V
17	1, 16	eqeltri 2304	. . . 4 |- Z e. _V
18		fex 5882	. . . 4 |- ( ( G : Z --> A /\ Z e. _V ) -> G e. _V )
19	13, 17, 18	sylancl 413	. . 3 |- ( ph -> G e. _V )
20		coexg 5281	. . 3 |- ( ( F e. ( A -cn-> B ) /\ G e. _V ) -> ( F o. G ) e. _V )
21	4, 19, 20	syl2anc 411	. 2 |- ( ph -> ( F o. G ) e. _V )
22		cncfi 15301	. . . . 5 |- ( ( F e. ( A -cn-> B ) /\ D e. A /\ x e. RR+ ) -> E. y e. RR+ A. z e. A ( ( abs ` ( z - D ) ) < y -> ( abs ` ( ( F ` z ) - ( F ` D ) ) ) < x ) )
23	22	3expia 1231	. . . 4 |- ( ( F e. ( A -cn-> B ) /\ D e. A ) -> ( x e. RR+ -> E. y e. RR+ A. z e. A ( ( abs ` ( z - D ) ) < y -> ( abs ` ( ( F ` z ) - ( F ` D ) ) ) < x ) ) )
24	4, 3, 23	syl2anc 411	. . 3 |- ( ph -> ( x e. RR+ -> E. y e. RR+ A. z e. A ( ( abs ` ( z - D ) ) < y -> ( abs ` ( ( F ` z ) - ( F ` D ) ) ) < x ) ) )
25	24	imp 124	. 2 |- ( ( ph /\ x e. RR+ ) -> E. y e. RR+ A. z e. A ( ( abs ` ( z - D ) ) < y -> ( abs ` ( ( F ` z ) - ( F ` D ) ) ) < x ) )
26	13	ffvelcdmda 5782	. 2 |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. A )
27		fvco3 5717	. . 3 |- ( ( G : Z --> A /\ k e. Z ) -> ( ( F o. G ) ` k ) = ( F ` ( G ` k ) ) )
28	13, 27	sylan 283	. 2 |- ( ( ph /\ k e. Z ) -> ( ( F o. G ) ` k ) = ( F ` ( G ` k ) ) )
29	1, 2, 3, 11, 12, 21, 25, 26, 28	climcn1 11868	1 |- ( ph -> ( F o. G ) ~~> ( F ` D ) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   A.wral 2510   E.wrex 2511   _Vcvv 2802   C_ wss 3200   class class class wbr 4088   o. ccom 4729   -->wf 5322   ` cfv 5326  (class class class)co 6017   CCcc 8029   < clt 8213   - cmin 8349   ZZcz 9478   ZZ>=cuz 9754   RR+crp 9887   abscabs 11557   ~~> cli 11838   -cn->ccncf 15293

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-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149

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-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-id 4390  df-po 4393  df-iso 4394  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 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-map 6818  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-n0 9402  df-z 9479  df-uz 9755  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-clim 11839  df-cncf 15294

This theorem is referenced by: (None)