Theorem climcn1lem 11465

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 11428	. . 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 5693	. . 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 11454	1 |- ( ph -> G ~~> ( H ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   A.wral 2472   E.wrex 2473   class class class wbr 4030   -->wf 5251   ` cfv 5255  (class class class)co 5919   CCcc 7872   < clt 8056   - cmin 8192   ZZcz 9320   ZZ>=cuz 9595   RR+crp 9722   abscabs 11144   ~~> cli 11424

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-inn 8985  df-n0 9244  df-z 9321  df-uz 9596  df-clim 11425

This theorem is referenced by:  climabs  11466  climcj  11467  climre  11468  climim  11469