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Theorem efltlemlt 12878
Description: Lemma for eflt 12879. The converse of efltim 11416 plus the epsilon-delta setup. (Contributed by Jim Kingdon, 22-May-2024.)
Hypotheses
Ref Expression
efltlemlt.a  |-  ( ph  ->  A  e.  RR )
efltlemlt.b  |-  ( ph  ->  B  e.  RR )
efltlemlt.lt  |-  ( ph  ->  ( exp `  A
)  <  ( exp `  B ) )
efltlemlt.d  |-  ( ph  ->  D  e.  RR+ )
efltlemlt.ed  |-  ( ph  ->  ( ( abs `  ( A  -  B )
)  <  D  ->  ( abs `  ( ( exp `  A )  -  ( exp `  B
) ) )  < 
( ( exp `  B
)  -  ( exp `  A ) ) ) )
Assertion
Ref Expression
efltlemlt  |-  ( ph  ->  A  <  B )

Proof of Theorem efltlemlt
StepHypRef Expression
1 efltlemlt.lt . . . . 5  |-  ( ph  ->  ( exp `  A
)  <  ( exp `  B ) )
21ad2antrr 479 . . . 4  |-  ( ( ( ph  /\  ( B  -  D )  <  A )  /\  B  <  A )  ->  ( exp `  A )  < 
( exp `  B
) )
3 efltlemlt.b . . . . . . 7  |-  ( ph  ->  B  e.  RR )
43ad2antrr 479 . . . . . 6  |-  ( ( ( ph  /\  ( B  -  D )  <  A )  /\  B  <  A )  ->  B  e.  RR )
54reefcld 11387 . . . . 5  |-  ( ( ( ph  /\  ( B  -  D )  <  A )  /\  B  <  A )  ->  ( exp `  B )  e.  RR )
6 efltlemlt.a . . . . . . 7  |-  ( ph  ->  A  e.  RR )
76ad2antrr 479 . . . . . 6  |-  ( ( ( ph  /\  ( B  -  D )  <  A )  /\  B  <  A )  ->  A  e.  RR )
87reefcld 11387 . . . . 5  |-  ( ( ( ph  /\  ( B  -  D )  <  A )  /\  B  <  A )  ->  ( exp `  A )  e.  RR )
96adantr 274 . . . . . . 7  |-  ( (
ph  /\  ( B  -  D )  <  A
)  ->  A  e.  RR )
10 efltim 11416 . . . . . . 7  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B  <  A  ->  ( exp `  B
)  <  ( exp `  A ) ) )
113, 9, 10syl2an2r 584 . . . . . 6  |-  ( (
ph  /\  ( B  -  D )  <  A
)  ->  ( B  <  A  ->  ( exp `  B )  <  ( exp `  A ) ) )
1211imp 123 . . . . 5  |-  ( ( ( ph  /\  ( B  -  D )  <  A )  /\  B  <  A )  ->  ( exp `  B )  < 
( exp `  A
) )
135, 8, 12ltnsymd 7894 . . . 4  |-  ( ( ( ph  /\  ( B  -  D )  <  A )  /\  B  <  A )  ->  -.  ( exp `  A )  <  ( exp `  B
) )
142, 13pm2.21dd 609 . . 3  |-  ( ( ( ph  /\  ( B  -  D )  <  A )  /\  B  <  A )  ->  A  <  B )
156reefcld 11387 . . . . . . 7  |-  ( ph  ->  ( exp `  A
)  e.  RR )
163reefcld 11387 . . . . . . 7  |-  ( ph  ->  ( exp `  B
)  e.  RR )
1715, 16, 1ltled 7893 . . . . . . 7  |-  ( ph  ->  ( exp `  A
)  <_  ( exp `  B ) )
1815, 16, 17abssuble0d 10961 . . . . . 6  |-  ( ph  ->  ( abs `  (
( exp `  A
)  -  ( exp `  B ) ) )  =  ( ( exp `  B )  -  ( exp `  A ) ) )
1918ad2antrr 479 . . . . 5  |-  ( ( ( ph  /\  ( B  -  D )  <  A )  /\  A  <  ( B  +  D
) )  ->  ( abs `  ( ( exp `  A )  -  ( exp `  B ) ) )  =  ( ( exp `  B )  -  ( exp `  A
) ) )
20 efltlemlt.d . . . . . . . . . 10  |-  ( ph  ->  D  e.  RR+ )
2120rpred 9495 . . . . . . . . 9  |-  ( ph  ->  D  e.  RR )
226, 3, 21absdifltd 10962 . . . . . . . 8  |-  ( ph  ->  ( ( abs `  ( A  -  B )
)  <  D  <->  ( ( B  -  D )  <  A  /\  A  < 
( B  +  D
) ) ) )
2322biimprd 157 . . . . . . 7  |-  ( ph  ->  ( ( ( B  -  D )  < 
A  /\  A  <  ( B  +  D ) )  ->  ( abs `  ( A  -  B
) )  <  D
) )
2423impl 377 . . . . . 6  |-  ( ( ( ph  /\  ( B  -  D )  <  A )  /\  A  <  ( B  +  D
) )  ->  ( abs `  ( A  -  B ) )  < 
D )
25 efltlemlt.ed . . . . . . 7  |-  ( ph  ->  ( ( abs `  ( A  -  B )
)  <  D  ->  ( abs `  ( ( exp `  A )  -  ( exp `  B
) ) )  < 
( ( exp `  B
)  -  ( exp `  A ) ) ) )
2625ad2antrr 479 . . . . . 6  |-  ( ( ( ph  /\  ( B  -  D )  <  A )  /\  A  <  ( B  +  D
) )  ->  (
( abs `  ( A  -  B )
)  <  D  ->  ( abs `  ( ( exp `  A )  -  ( exp `  B
) ) )  < 
( ( exp `  B
)  -  ( exp `  A ) ) ) )
2724, 26mpd 13 . . . . 5  |-  ( ( ( ph  /\  ( B  -  D )  <  A )  /\  A  <  ( B  +  D
) )  ->  ( abs `  ( ( exp `  A )  -  ( exp `  B ) ) )  <  ( ( exp `  B )  -  ( exp `  A
) ) )
2819, 27eqbrtrrd 3952 . . . 4  |-  ( ( ( ph  /\  ( B  -  D )  <  A )  /\  A  <  ( B  +  D
) )  ->  (
( exp `  B
)  -  ( exp `  A ) )  < 
( ( exp `  B
)  -  ( exp `  A ) ) )
2916, 15resubcld 8155 . . . . . 6  |-  ( ph  ->  ( ( exp `  B
)  -  ( exp `  A ) )  e.  RR )
3029ad2antrr 479 . . . . 5  |-  ( ( ( ph  /\  ( B  -  D )  <  A )  /\  A  <  ( B  +  D
) )  ->  (
( exp `  B
)  -  ( exp `  A ) )  e.  RR )
3130ltnrd 7887 . . . 4  |-  ( ( ( ph  /\  ( B  -  D )  <  A )  /\  A  <  ( B  +  D
) )  ->  -.  ( ( exp `  B
)  -  ( exp `  A ) )  < 
( ( exp `  B
)  -  ( exp `  A ) ) )
3228, 31pm2.21dd 609 . . 3  |-  ( ( ( ph  /\  ( B  -  D )  <  A )  /\  A  <  ( B  +  D
) )  ->  A  <  B )
333, 20ltaddrpd 9529 . . . . 5  |-  ( ph  ->  B  <  ( B  +  D ) )
343, 21readdcld 7807 . . . . . 6  |-  ( ph  ->  ( B  +  D
)  e.  RR )
35 axltwlin 7844 . . . . . 6  |-  ( ( B  e.  RR  /\  ( B  +  D
)  e.  RR  /\  A  e.  RR )  ->  ( B  <  ( B  +  D )  ->  ( B  <  A  \/  A  <  ( B  +  D ) ) ) )
363, 34, 6, 35syl3anc 1216 . . . . 5  |-  ( ph  ->  ( B  <  ( B  +  D )  ->  ( B  <  A  \/  A  <  ( B  +  D ) ) ) )
3733, 36mpd 13 . . . 4  |-  ( ph  ->  ( B  <  A  \/  A  <  ( B  +  D ) ) )
3837adantr 274 . . 3  |-  ( (
ph  /\  ( B  -  D )  <  A
)  ->  ( B  <  A  \/  A  < 
( B  +  D
) ) )
3914, 32, 38mpjaodan 787 . 2  |-  ( (
ph  /\  ( B  -  D )  <  A
)  ->  A  <  B )
40 simpr 109 . 2  |-  ( (
ph  /\  A  <  B )  ->  A  <  B )
413, 20ltsubrpd 9528 . . 3  |-  ( ph  ->  ( B  -  D
)  <  B )
423, 21resubcld 8155 . . . 4  |-  ( ph  ->  ( B  -  D
)  e.  RR )
43 axltwlin 7844 . . . 4  |-  ( ( ( B  -  D
)  e.  RR  /\  B  e.  RR  /\  A  e.  RR )  ->  (
( B  -  D
)  <  B  ->  ( ( B  -  D
)  <  A  \/  A  <  B ) ) )
4442, 3, 6, 43syl3anc 1216 . . 3  |-  ( ph  ->  ( ( B  -  D )  <  B  ->  ( ( B  -  D )  <  A  \/  A  <  B ) ) )
4541, 44mpd 13 . 2  |-  ( ph  ->  ( ( B  -  D )  <  A  \/  A  <  B ) )
4639, 40, 45mpjaodan 787 1  |-  ( ph  ->  A  <  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 697    = wceq 1331    e. wcel 1480   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   RRcr 7631    + caddc 7635    < clt 7812    - cmin 7945   RR+crp 9453   abscabs 10781   expce 11360
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulrcl 7731  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-precex 7742  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748  ax-pre-mulgt0 7749  ax-pre-mulext 7750  ax-arch 7751  ax-caucvg 7752
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-disj 3907  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-frec 6288  df-1o 6313  df-oadd 6317  df-er 6429  df-en 6635  df-dom 6636  df-fin 6637  df-sup 6871  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-reap 8349  df-ap 8356  df-div 8445  df-inn 8733  df-2 8791  df-3 8792  df-4 8793  df-n0 8990  df-z 9067  df-uz 9339  df-q 9424  df-rp 9454  df-ico 9689  df-fz 9803  df-fzo 9932  df-seqfrec 10231  df-exp 10305  df-fac 10484  df-bc 10506  df-ihash 10534  df-cj 10626  df-re 10627  df-im 10628  df-rsqrt 10782  df-abs 10783  df-clim 11060  df-sumdc 11135  df-ef 11366
This theorem is referenced by:  eflt  12879
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