Theorem cnplimclemle 15255

Description: Lemma for cnplimccntop 15257. 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 5739	. . . . . . 7 |- ( ph -> ( F ` Z ) e. CC )
5		cnplimclemr.b	. . . . . . . 8 |- ( ph -> B e. A )
6	2, 5	ffvelcdmd 5739	. . . . . . 7 |- ( ph -> ( F ` B ) e. CC )
7	4, 6	subcld 8418	. . . . . 6 |- ( ph -> ( ( F ` Z ) - ( F ` B ) ) e. CC )
8	7	abscld 11607	. . . . 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 9866	. . . . . 6 |- ( ph -> ( E / 2 ) e. RR+ )
12	11	rpred 9853	. . . . 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 1250	. . . . . . . . . . 11 |- ( ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) /\ Z # B ) -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) < ( E / 2 ) )
24	17, 18, 23	ltnsymd 8227	. . . . . . . . . 10 |- ( ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) /\ Z # B ) -> -. ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) )
25	15, 24	pm2.65da 663	. . . . . . . . 9 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> -. Z # B )
26		cnplimclemr.a	. . . . . . . . . . 11 |- ( ph -> A C_ CC )
27	26, 3	sseldd 3202	. . . . . . . . . 10 |- ( ph -> Z e. CC )
28	26, 5	sseldd 3202	. . . . . . . . . . 11 |- ( ph -> B e. CC )
29	28	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> B e. CC )
30		apti 8730	. . . . . . . . . 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 5603	. . . . . . 7 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> ( F ` Z ) = ( F ` B ) )
34	14, 33	subeq0bd 8486	. . . . . 6 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> ( ( F ` Z ) - ( F ` B ) ) = 0 )
35	34	abs00bd 11492	. . . . 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 9856	. . . . 5 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> 0 < ( E / 2 ) )
38	35, 37	eqbrtrd 4081	. . . 4 |- ( ( ph /\ ( E / 2 ) < ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) ) -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) < ( E / 2 ) )
39	9, 13, 38	ltnsymd 8227	. . 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 9840	. . . 4 |- ( E e. RR+ -> ( E / 2 ) < E )
43	10, 42	syl 14	. . 3 |- ( ph -> ( E / 2 ) < E )
44	10	rpred 9853	. . . 4 |- ( ph -> E e. RR )
45		axltwlin 8175	. . . 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 1250	. . 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 800	1 |- ( ph -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) < E )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 710   /\ w3a 981   = wceq 1373   e. wcel 2178   C_ wss 3174   class class class wbr 4059   o. ccom 4697   -->wf 5286   ` cfv 5290  (class class class)co 5967   CCcc 7958   RRcr 7959   0cc0 7960   < clt 8142   - cmin 8278   # cap 8689   / cdiv 8780   2c2 9122   RR+crp 9810   abscabs 11423   ↾_t crest 13186   MetOpencmopn 14418   lim CC climc 15241

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-z 9408  df-uz 9684  df-rp 9811  df-seqfrec 10630  df-exp 10721  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425

This theorem is referenced by:  cnplimclemr  15256