Theorem cnplimclemle 15342

Description: Lemma for cnplimccntop 15344. Satisfying the epsilon condition for continuity. (Contributed by Mario Carneiro and Jim Kingdon, 17-Nov-2023.)

Hypotheses

Ref	Expression
cnplimccntop.k	|- K = ( MetOpen ` ( abs o. - ) )
cnplimc.j	|- J = ( K ↾t A )
cnplimclemr.a	|- ( ph -> A C_ CC )
cnplimclemr.f	|- ( ph -> F : A --> CC )
cnplimclemr.b	|- ( ph -> B e. A )
cnplimclemr.l	|- ( ph -> ( F ` B ) e. ( F lim CC B ) )
cnplimclemle.e	|- ( ph -> E e. RR+ )
cnplimclemle.d	|- ( ph -> D e. RR+ )
cnplimclemle.z	|- ( ph -> Z e. A )
cnplimclemle.im	|- ( ( ph /\ Z # B /\ ( abs ` ( Z - B ) ) < D ) -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) < ( E / 2 ) )
cnplimclemle.zd	|- ( ph -> ( abs ` ( Z - B ) ) < D )

Assertion

Ref	Expression
cnplimclemle	|- ( ph -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) < E )

Proof of Theorem cnplimclemle

Step	Hyp	Ref	Expression
1		simpr 110	. . 3 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) )
2		cnplimclemr.f	. . . . . . . 8 |- ( ph -> F : A --> CC )
3		cnplimclemle.z	. . . . . . . 8 |- ( ph -> Z e. A )
4	2, 3	ffvelcdmd 5771	. . . . . . 7 |- ( ph -> ( F ` Z ) e. CC )
5		cnplimclemr.b	. . . . . . . 8 |- ( ph -> B e. A )
6	2, 5	ffvelcdmd 5771	. . . . . . 7 |- ( ph -> ( F ` B ) e. CC )
7	4, 6	subcld 8457	. . . . . 6 |- ( ph -> ( ( F ` Z ) - ( F ` B ) ) e. CC )
8	7	abscld 11692	. . . . 5 |- ( ph -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) e. RR )
9	8	adantr 276	. . . 4 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) e. RR )
10		cnplimclemle.e	. . . . . . 7 |- ( ph -> E e. RR+ )
11	10	rphalfcld 9905	. . . . . 6 |- ( ph -> ( E / 2 ) e. RR+ )
12	11	rpred 9892	. . . . 5 |- ( ph -> ( E / 2 ) e. RR )
13	12	adantr 276	. . . 4 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> ( E / 2 ) e. RR )
14	4	adantr 276	. . . . . . 7 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> ( F ` Z ) e. CC )
15	1	adantr 276	. . . . . . . . . 10 |- ( ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) /\ Z # B ) -> ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) )
16		simpll 527	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) /\ Z # B ) -> ph )
17	16, 8	syl 14	. . . . . . . . . . 11 |- ( ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) /\ Z # B ) -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) e. RR )
18	16, 12	syl 14	. . . . . . . . . . 11 |- ( ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) /\ Z # B ) -> ( E / 2 ) e. RR )
19		simpr 110	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) /\ Z # B ) -> Z # B )
20		cnplimclemle.zd	. . . . . . . . . . . . 13 |- ( ph -> ( abs ` ( Z - B ) ) < D )
21	16, 20	syl 14	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) /\ Z # B ) -> ( abs ` ( Z - B ) ) < D )
22		cnplimclemle.im	. . . . . . . . . . . 12 |- ( ( ph /\ Z # B /\ ( abs ` ( Z - B ) ) < D ) -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) < ( E / 2 ) )
23	16, 19, 21, 22	syl3anc 1271	. . . . . . . . . . 11 |- ( ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) /\ Z # B ) -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) < ( E / 2 ) )
24	17, 18, 23	ltnsymd 8266	. . . . . . . . . 10 |- ( ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) /\ Z # B ) -> -. ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) )
25	15, 24	pm2.65da 665	. . . . . . . . 9 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> -. Z # B )
26		cnplimclemr.a	. . . . . . . . . . 11 |- ( ph -> A C_ CC )
27	26, 3	sseldd 3225	. . . . . . . . . 10 |- ( ph -> Z e. CC )
28	26, 5	sseldd 3225	. . . . . . . . . . 11 |- ( ph -> B e. CC )
29	28	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> B e. CC )
30		apti 8769	. . . . . . . . . 10 |- ( ( Z e. CC /\ B e. CC ) -> ( Z = B <-> -. Z # B ) )
31	27, 29, 30	syl2an2r 597	. . . . . . . . 9 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> ( Z = B <-> -. Z # B ) )
32	25, 31	mpbird 167	. . . . . . . 8 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> Z = B )
33	32	fveq2d 5631	. . . . . . 7 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> ( F ` Z ) = ( F ` B ) )
34	14, 33	subeq0bd 8525	. . . . . 6 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> ( ( F ` Z ) - ( F ` B ) ) = 0 )
35	34	abs00bd 11577	. . . . 5 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) = 0 )
36	11	adantr 276	. . . . . 6 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> ( E / 2 ) e. RR+ )
37	36	rpgt0d 9895	. . . . 5 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> 0 < ( E / 2 ) )
38	35, 37	eqbrtrd 4105	. . . 4 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) < ( E / 2 ) )
39	9, 13, 38	ltnsymd 8266	. . 3 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> -. ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) )
40	1, 39	pm2.21dd 623	. 2 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) < E )
41		simpr 110	. 2 |- ( ( ph /\ ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) < E ) -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) < E )
42		rphalflt 9879	. . . 4 |- ( E e. RR+ -> ( E / 2 ) < E )
43	10, 42	syl 14	. . 3 |- ( ph -> ( E / 2 ) < E )
44	10	rpred 9892	. . . 4 |- ( ph -> E e. RR )
45		axltwlin 8214	. . . 4 |- ( ( ( E / 2 ) e. RR /\ E e. RR /\ ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) e. RR ) -> ( ( E / 2 ) < E -> ( ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) \/ ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) < E ) ) )
46	12, 44, 8, 45	syl3anc 1271	. . 3 |- ( ph -> ( ( E / 2 ) < E -> ( ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) \/ ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) < E ) ) )
47	43, 46	mpd 13	. 2 |- ( ph -> ( ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) \/ ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) < E ) )
48	40, 41, 47	mpjaodan 803	1 |- ( ph -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) < E )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   C_ wss 3197   class class class wbr 4083   o. ccom 4723   -->wf 5314   ` cfv 5318  (class class class)co 6001   CCcc 7997   RRcr 7998   0cc0 7999   < clt 8181   - cmin 8317   # cap 8728   / cdiv 8819   2c2 9161   RR+crp 9849   abscabs 11508   ↾_t crest 13272   MetOpencmopn 14505   lim CC climc 15328

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-n0 9370  df-z 9447  df-uz 9723  df-rp 9850  df-seqfrec 10670  df-exp 10761  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510

This theorem is referenced by:  cnplimclemr  15343