Theorem cn1lem 11255

Description: A sufficient condition for a function to be continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)

Hypotheses

Ref	Expression
cn1lem.1	|- F : CC --> CC
cn1lem.2	|- ( ( z e. CC /\ A e. CC ) -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) <_ ( abs ` ( z - A ) ) )

Assertion

Ref	Expression
cn1lem	|- ( ( A e. CC /\ x e. RR+ ) -> E. y e. RR+ A. z e. CC ( ( abs ` ( z - A ) ) < y -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) < x ) )

Distinct variable groups:   x, y, z   y, A, z   y, F

Allowed substitution hints:   A( x)   F( x, z)

Proof of Theorem cn1lem

Step	Hyp	Ref	Expression
1		simpr 109	. 2 |- ( ( A e. CC /\ x e. RR+ ) -> x e. RR+ )
2		simpr 109	. . . . 5 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> z e. CC )
3		simpll 519	. . . . 5 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> A e. CC )
4		cn1lem.2	. . . . 5 |- ( ( z e. CC /\ A e. CC ) -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) <_ ( abs ` ( z - A ) ) )
5	2, 3, 4	syl2anc 409	. . . 4 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) <_ ( abs ` ( z - A ) ) )
6		cn1lem.1	. . . . . . . . 9 |- F : CC --> CC
7	6	ffvelrni 5619	. . . . . . . 8 |- ( z e. CC -> ( F ` z ) e. CC )
8	2, 7	syl 14	. . . . . . 7 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> ( F ` z ) e. CC )
9	6	ffvelrni 5619	. . . . . . . 8 |- ( A e. CC -> ( F ` A ) e. CC )
10	3, 9	syl 14	. . . . . . 7 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> ( F ` A ) e. CC )
11	8, 10	subcld 8209	. . . . . 6 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> ( ( F ` z ) - ( F ` A ) ) e. CC )
12	11	abscld 11123	. . . . 5 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) e. RR )
13	2, 3	subcld 8209	. . . . . 6 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> ( z - A ) e. CC )
14	13	abscld 11123	. . . . 5 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> ( abs ` ( z - A ) ) e. RR )
15		rpre 9596	. . . . . 6 |- ( x e. RR+ -> x e. RR )
16	15	ad2antlr 481	. . . . 5 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> x e. RR )
17		lelttr 7987	. . . . 5 |- ( ( ( abs ` ( ( F ` z ) - ( F ` A ) ) ) e. RR /\ ( abs ` ( z - A ) ) e. RR /\ x e. RR ) -> ( ( ( abs ` ( ( F ` z ) - ( F ` A ) ) ) <_ ( abs ` ( z - A ) ) /\ ( abs ` ( z - A ) ) < x ) -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) < x ) )
18	12, 14, 16, 17	syl3anc 1228	. . . 4 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> ( ( ( abs ` ( ( F ` z ) - ( F ` A ) ) ) <_ ( abs ` ( z - A ) ) /\ ( abs ` ( z - A ) ) < x ) -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) < x ) )
19	5, 18	mpand 426	. . 3 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> ( ( abs ` ( z - A ) ) < x -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) < x ) )
20	19	ralrimiva 2539	. 2 |- ( ( A e. CC /\ x e. RR+ ) -> A. z e. CC ( ( abs ` ( z - A ) ) < x -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) < x ) )
21		breq2 3986	. . . . 5 |- ( y = x -> ( ( abs ` ( z - A ) ) < y <-> ( abs ` ( z - A ) ) < x ) )
22	21	imbi1d 230	. . . 4 |- ( y = x -> ( ( ( abs ` ( z - A ) ) < y -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) < x ) <-> ( ( abs ` ( z - A ) ) < x -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) < x ) ) )
23	22	ralbidv 2466	. . 3 |- ( y = x -> ( A. z e. CC ( ( abs ` ( z - A ) ) < y -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) < x ) <-> A. z e. CC ( ( abs ` ( z - A ) ) < x -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) < x ) ) )
24	23	rspcev 2830	. 2 |- ( ( x e. RR+ /\ A. z e. CC ( ( abs ` ( z - A ) ) < x -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) < x ) ) -> E. y e. RR+ A. z e. CC ( ( abs ` ( z - A ) ) < y -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) < x ) )
25	1, 20, 24	syl2anc 409	1 |- ( ( A e. CC /\ x e. RR+ ) -> E. y e. RR+ A. z e. CC ( ( abs ` ( z - A ) ) < y -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) < x ) )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2136   A.wral 2444   E.wrex 2445   class class class wbr 3982   -->wf 5184   ` cfv 5188  (class class class)co 5842   CCcc 7751   RRcr 7752   < clt 7933   <_ cle 7934   - cmin 8069   RR+crp 9589   abscabs 10939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-rp 9590  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941

This theorem is referenced by:  abscn2  11256  cjcn2  11257  recn2  11258  imcn2  11259