Theorem climcn1lem 14333

Description: The limit of a continuous function, theorem form. (Contributed by Mario Carneiro, 9-Feb-2014.)

Hypotheses

Ref	Expression
climcn1lem.1	|- Z = ( ZZ>= ` M )
climcn1lem.2	|- ( ph -> F ~~> A )
climcn1lem.4	|- ( ph -> G e. W )
climcn1lem.5	|- ( ph -> M e. ZZ )
climcn1lem.6	|- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
climcn1lem.7	|- H : CC --> CC
climcn1lem.8	|- ( ( A e. CC /\ x e. RR+ ) -> E. y e. RR+ A. z e. CC ( ( abs ` ( z - A ) ) < y -> ( abs ` ( ( H ` z ) - ( H ` A ) ) ) < x ) )
climcn1lem.9	|- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( H ` ( F ` k ) ) )

Assertion

Ref	Expression
climcn1lem	|- ( ph -> G ~~> ( H ` A ) )

Distinct variable groups:   x, k, y, z, A   k, F, y, z   k, G, x   ph, k, x, y, z   k, Z, y   k, H, x, y, z   k, M

Allowed substitution hints:   F( x)   G( y, z)   M( x, y, z)   W( x, y, z, k)   Z( x, z)

Proof of Theorem climcn1lem

Step	Hyp	Ref	Expression
1		climcn1lem.1	. 2 |- Z = ( ZZ>= ` M )
2		climcn1lem.5	. 2 |- ( ph -> M e. ZZ )
3		climcn1lem.2	. . 3 |- ( ph -> F ~~> A )
4		climcl 14230	. . 3 |- ( F ~~> A -> A e. CC )
5	3, 4	syl 17	. 2 |- ( ph -> A e. CC )
6		climcn1lem.7	. . . 4 |- H : CC --> CC
7	6	ffvelrni 6358	. . 3 |- ( z e. CC -> ( H ` z ) e. CC )
8	7	adantl 482	. 2 |- ( ( ph /\ z e. CC ) -> ( H ` z ) e. CC )
9		climcn1lem.4	. 2 |- ( ph -> G e. W )
10		climcn1lem.8	. . 3 |- ( ( A e. CC /\ x e. RR+ ) -> E. y e. RR+ A. z e. CC ( ( abs ` ( z - A ) ) < y -> ( abs ` ( ( H ` z ) - ( H ` A ) ) ) < x ) )
11	5, 10	sylan 488	. 2 |- ( ( ph /\ x e. RR+ ) -> E. y e. RR+ A. z e. CC ( ( abs ` ( z - A ) ) < y -> ( abs ` ( ( H ` z ) - ( H ` A ) ) ) < x ) )
12		climcn1lem.6	. 2 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
13		climcn1lem.9	. 2 |- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( H ` ( F ` k ) ) )
14	1, 2, 5, 8, 3, 9, 11, 12, 13	climcn1 14322	1 |- ( ph -> G ~~> ( H ` A ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   class class class wbr 4653   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   < clt 10074   - cmin 10266   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   abscabs 13974   ~~> cli 14215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-neg 10269  df-z 11378  df-uz 11688  df-clim 14219

This theorem is referenced by:  climabs  14334  climcj  14335  climre  14336  climim  14337  sinccvglem  31566