Theorem climcn1lem 12008

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 11971	. . 3 |- ( F ~~> A -> A e. CC )
5	3, 4	syl 14	. 2 |- ( ph -> A e. CC )
6		climcn1lem.7	. . . 4 |- H : CC --> CC
7	6	ffvelcdmi 5813	. . 3 |- ( z e. CC -> ( H ` z ) e. CC )
8	7	adantl 277	. 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 283	. 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 11997	1 |- ( ph -> G ~~> ( H ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   A.wral 2522   E.wrex 2523   class class class wbr 4111   -->wf 5350   ` cfv 5354  (class class class)co 6052   CCcc 8127   < clt 8310   - cmin 8446   ZZcz 9579   ZZ>=cuz 9856   RR+crp 9989   abscabs 11686   ~~> cli 11967

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-inn 9240  df-n0 9499  df-z 9580  df-uz 9857  df-clim 11968

This theorem is referenced by:  climabs  12009  climcj  12010  climre  12011  climim  12012