Theorem cnplimclemle 12984

Description: Lemma for cnplimccntop 12986. 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 109	. . 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	ffvelrnd 5596	. . . . . . 7 |- ( ph -> ( F ` Z ) e. CC )
5		cnplimclemr.b	. . . . . . . 8 |- ( ph -> B e. A )
6	2, 5	ffvelrnd 5596	. . . . . . 7 |- ( ph -> ( F ` B ) e. CC )
7	4, 6	subcld 8165	. . . . . 6 |- ( ph -> ( ( F ` Z ) - ( F ` B ) ) e. CC )
8	7	abscld 11058	. . . . 5 |- ( ph -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) e. RR )
9	8	adantr 274	. . . 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 9594	. . . . . 6 |- ( ph -> ( E / 2 ) e. RR+ )
12	11	rpred 9581	. . . . 5 |- ( ph -> ( E / 2 ) e. RR )
13	12	adantr 274	. . . 4 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> ( E / 2 ) e. RR )
14	4	adantr 274	. . . . . . 7 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> ( F ` Z ) e. CC )
15	1	adantr 274	. . . . . . . . . 10 |- ( ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) /\ Z # B ) -> ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) )
16		simpll 519	. . . . . . . . . . . 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 109	. . . . . . . . . . . 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 1217	. . . . . . . . . . 11 |- ( ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) /\ Z # B ) -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) < ( E / 2 ) )
24	17, 18, 23	ltnsymd 7974	. . . . . . . . . 10 |- ( ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) /\ Z # B ) -> -. ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) )
25	15, 24	pm2.65da 651	. . . . . . . . 9 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> -. Z # B )
26		cnplimclemr.a	. . . . . . . . . . 11 |- ( ph -> A C_ CC )
27	26, 3	sseldd 3125	. . . . . . . . . 10 |- ( ph -> Z e. CC )
28	26, 5	sseldd 3125	. . . . . . . . . . 11 |- ( ph -> B e. CC )
29	28	adantr 274	. . . . . . . . . 10 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> B e. CC )
30		apti 8476	. . . . . . . . . 10 |- ( ( Z e. CC /\ B e. CC ) -> ( Z = B <-> -. Z # B ) )
31	27, 29, 30	syl2an2r 585	. . . . . . . . 9 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> ( Z = B <-> -. Z # B ) )
32	25, 31	mpbird 166	. . . . . . . 8 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> Z = B )
33	32	fveq2d 5465	. . . . . . 7 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> ( F ` Z ) = ( F ` B ) )
34	14, 33	subeq0bd 8233	. . . . . 6 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> ( ( F ` Z ) - ( F ` B ) ) = 0 )
35	34	abs00bd 10943	. . . . 5 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) = 0 )
36	11	adantr 274	. . . . . 6 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> ( E / 2 ) e. RR+ )
37	36	rpgt0d 9584	. . . . 5 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> 0 < ( E / 2 ) )
38	35, 37	eqbrtrd 3982	. . . 4 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) < ( E / 2 ) )
39	9, 13, 38	ltnsymd 7974	. . 3 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> -. ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) )
40	1, 39	pm2.21dd 610	. 2 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) < E )
41		simpr 109	. 2 |- ( ( ph /\ ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) < E ) -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) < E )
42		rphalflt 9568	. . . 4 |- ( E e. RR+ -> ( E / 2 ) < E )
43	10, 42	syl 14	. . 3 |- ( ph -> ( E / 2 ) < E )
44	10	rpred 9581	. . . 4 |- ( ph -> E e. RR )
45		axltwlin 7924	. . . 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 1217	. . 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 788	1 |- ( ph -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) < E )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   /\ w3a 963   = wceq 1332   e. wcel 2125   C_ wss 3098   class class class wbr 3961   o. ccom 4583   -->wf 5159   ` cfv 5163  (class class class)co 5814   CCcc 7709   RRcr 7710   0cc0 7711   < clt 7891   - cmin 8025   # cap 8435   / cdiv 8524   2c2 8863   RR+crp 9538   abscabs 10874   ↾_t crest 12298   MetOpencmopn 12332   lim CC climc 12970

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-coll 4075  ax-sep 4078  ax-nul 4086  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-setind 4490  ax-iinf 4541  ax-cnex 7802  ax-resscn 7803  ax-1cn 7804  ax-1re 7805  ax-icn 7806  ax-addcl 7807  ax-addrcl 7808  ax-mulcl 7809  ax-mulrcl 7810  ax-addcom 7811  ax-mulcom 7812  ax-addass 7813  ax-mulass 7814  ax-distr 7815  ax-i2m1 7816  ax-0lt1 7817  ax-1rid 7818  ax-0id 7819  ax-rnegex 7820  ax-precex 7821  ax-cnre 7822  ax-pre-ltirr 7823  ax-pre-ltwlin 7824  ax-pre-lttrn 7825  ax-pre-apti 7826  ax-pre-ltadd 7827  ax-pre-mulgt0 7828  ax-pre-mulext 7829  ax-arch 7830  ax-caucvg 7831

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-nel 2420  df-ral 2437  df-rex 2438  df-reu 2439  df-rmo 2440  df-rab 2441  df-v 2711  df-sbc 2934  df-csb 3028  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-nul 3391  df-if 3502  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-int 3804  df-iun 3847  df-br 3962  df-opab 4022  df-mpt 4023  df-tr 4059  df-id 4248  df-po 4251  df-iso 4252  df-iord 4321  df-on 4323  df-ilim 4324  df-suc 4326  df-iom 4544  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592  df-iota 5128  df-fun 5165  df-fn 5166  df-f 5167  df-f1 5168  df-fo 5169  df-f1o 5170  df-fv 5171  df-riota 5770  df-ov 5817  df-oprab 5818  df-mpo 5819  df-1st 6078  df-2nd 6079  df-recs 6242  df-frec 6328  df-pnf 7893  df-mnf 7894  df-xr 7895  df-ltxr 7896  df-le 7897  df-sub 8027  df-neg 8028  df-reap 8429  df-ap 8436  df-div 8525  df-inn 8813  df-2 8871  df-3 8872  df-4 8873  df-n0 9070  df-z 9147  df-uz 9419  df-rp 9539  df-seqfrec 10323  df-exp 10397  df-cj 10719  df-re 10720  df-im 10721  df-rsqrt 10875  df-abs 10876

This theorem is referenced by:  cnplimclemr  12985