Theorem cnplimclemle 15894

Description: Lemma for cnplimccntop 15896. 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 5847	. . . . . . 7 |- ( ph -> ( F ` Z ) e. CC )
5		cnplimclemr.b	. . . . . . . 8 |- ( ph -> B e. A )
6	2, 5	ffvelcdmd 5847	. . . . . . 7 |- ( ph -> ( F ` B ) e. CC )
7	4, 6	subcld 8644	. . . . . 6 |- ( ph -> ( ( F ` Z ) - ( F ` B ) ) e. CC )
8	7	abscld 11970	. . . . 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 10127	. . . . . 6 |- ( ph -> ( E / 2 ) e. RR+ )
12	11	rpred 10114	. . . . 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 531	. . . . . . . . . . . 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 1278	. . . . . . . . . . 11 |- ( ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) /\ Z # B ) -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) < ( E / 2 ) )
24	17, 18, 23	ltnsymd 8453	. . . . . . . . . 10 |- ( ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) /\ Z # B ) -> -. ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) )
25	15, 24	pm2.65da 671	. . . . . . . . 9 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> -. Z # B )
26		cnplimclemr.a	. . . . . . . . . . 11 |- ( ph -> A C_ CC )
27	26, 3	sseldd 3249	. . . . . . . . . 10 |- ( ph -> Z e. CC )
28	26, 5	sseldd 3249	. . . . . . . . . . 11 |- ( ph -> B e. CC )
29	28	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> B e. CC )
30		apti 8958	. . . . . . . . . 10 |- ( ( Z e. CC /\ B e. CC ) -> ( Z = B <-> -. Z # B ) )
31	27, 29, 30	syl2an2r 603	. . . . . . . . 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 5702	. . . . . . 7 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> ( F ` Z ) = ( F ` B ) )
34	14, 33	subeq0bd 8713	. . . . . 6 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> ( ( F ` Z ) - ( F ` B ) ) = 0 )
35	34	abs00bd 11854	. . . . 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 10117	. . . . 5 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> 0 < ( E / 2 ) )
38	35, 37	eqbrtrd 4153	. . . 4 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) < ( E / 2 ) )
39	9, 13, 38	ltnsymd 8453	. . 3 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> -. ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) )
40	1, 39	pm2.21dd 629	. 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 10101	. . . 4 |- ( E e. RR+ -> ( E / 2 ) < E )
43	10, 42	syl 14	. . 3 |- ( ph -> ( E / 2 ) < E )
44	10	rpred 10114	. . . 4 |- ( ph -> E e. RR )
45		axltwlin 8399	. . . 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 1278	. . 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 810	1 |- ( ph -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) < E )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   /\ w3a 1009   = wceq 1402   e. wcel 2209   C_ wss 3220   class class class wbr 4131   o. ccom 4780   -->wf 5376   ` cfv 5380  (class class class)co 6088   CCcc 8183   RRcr 8184   0cc0 8185   < clt 8366   - cmin 8504   # cap 8917   / cdiv 9010   2c2 9364   RR+crp 10071   abscabs 11785   ↾_t crest 13652   MetOpencmopn 14987   lim CC climc 15880

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-n0 9575  df-z 9656  df-uz 9938  df-rp 10072  df-seqfrec 10906  df-exp 10997  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787

This theorem is used by:  cnplimclemr  15895