Theorem cnplimclemle 14904

Description: Lemma for cnplimccntop 14906. 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 5698	. . . . . . 7 |- ( ph -> ( F ` Z ) e. CC )
5		cnplimclemr.b	. . . . . . . 8 |- ( ph -> B e. A )
6	2, 5	ffvelcdmd 5698	. . . . . . 7 |- ( ph -> ( F ` B ) e. CC )
7	4, 6	subcld 8337	. . . . . 6 |- ( ph -> ( ( F ` Z ) - ( F ` B ) ) e. CC )
8	7	abscld 11346	. . . . 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 9784	. . . . . 6 |- ( ph -> ( E / 2 ) e. RR+ )
12	11	rpred 9771	. . . . 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 1249	. . . . . . . . . . 11 |- ( ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) /\ Z # B ) -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) < ( E / 2 ) )
24	17, 18, 23	ltnsymd 8146	. . . . . . . . . 10 |- ( ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) /\ Z # B ) -> -. ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) )
25	15, 24	pm2.65da 662	. . . . . . . . 9 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> -. Z # B )
26		cnplimclemr.a	. . . . . . . . . . 11 |- ( ph -> A C_ CC )
27	26, 3	sseldd 3184	. . . . . . . . . 10 |- ( ph -> Z e. CC )
28	26, 5	sseldd 3184	. . . . . . . . . . 11 |- ( ph -> B e. CC )
29	28	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> B e. CC )
30		apti 8649	. . . . . . . . . 10 |- ( ( Z e. CC /\ B e. CC ) -> ( Z = B <-> -. Z # B ) )
31	27, 29, 30	syl2an2r 595	. . . . . . . . 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 5562	. . . . . . 7 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> ( F ` Z ) = ( F ` B ) )
34	14, 33	subeq0bd 8405	. . . . . 6 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> ( ( F ` Z ) - ( F ` B ) ) = 0 )
35	34	abs00bd 11231	. . . . 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 9774	. . . . 5 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> 0 < ( E / 2 ) )
38	35, 37	eqbrtrd 4055	. . . 4 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) < ( E / 2 ) )
39	9, 13, 38	ltnsymd 8146	. . 3 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> -. ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) )
40	1, 39	pm2.21dd 621	. 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 9758	. . . 4 |- ( E e. RR+ -> ( E / 2 ) < E )
43	10, 42	syl 14	. . 3 |- ( ph -> ( E / 2 ) < E )
44	10	rpred 9771	. . . 4 |- ( ph -> E e. RR )
45		axltwlin 8094	. . . 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 1249	. . 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 799	1 |- ( ph -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) < E )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   /\ w3a 980   = wceq 1364   e. wcel 2167   C_ wss 3157   class class class wbr 4033   o. ccom 4667   -->wf 5254   ` cfv 5258  (class class class)co 5922   CCcc 7877   RRcr 7878   0cc0 7879   < clt 8061   - cmin 8197   # cap 8608   / cdiv 8699   2c2 9041   RR+crp 9728   abscabs 11162   ↾_t crest 12910   MetOpencmopn 14097   lim CC climc 14890

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-rp 9729  df-seqfrec 10540  df-exp 10631  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164

This theorem is referenced by:  cnplimclemr  14905