Theorem efltlemlt 14335

Description: Lemma for eflt 14336. The converse of efltim 11709 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

Step	Hyp	Ref	Expression
1		efltlemlt.lt	. . . . 5 |- ( ph -> ( exp ` A ) < ( exp ` B ) )
2	1	ad2antrr 488	. . . 4 |- ( ( ( ph /\ ( B - D ) < A ) /\ B < A ) -> ( exp ` A ) < ( exp ` B ) )
3		efltlemlt.b	. . . . . . 7 |- ( ph -> B e. RR )
4	3	ad2antrr 488	. . . . . 6 |- ( ( ( ph /\ ( B - D ) < A ) /\ B < A ) -> B e. RR )
5	4	reefcld 11680	. . . . 5 |- ( ( ( ph /\ ( B - D ) < A ) /\ B < A ) -> ( exp ` B ) e. RR )
6		efltlemlt.a	. . . . . . 7 |- ( ph -> A e. RR )
7	6	ad2antrr 488	. . . . . 6 |- ( ( ( ph /\ ( B - D ) < A ) /\ B < A ) -> A e. RR )
8	7	reefcld 11680	. . . . 5 |- ( ( ( ph /\ ( B - D ) < A ) /\ B < A ) -> ( exp ` A ) e. RR )
9	6	adantr 276	. . . . . . 7 |- ( ( ph /\ ( B - D ) < A ) -> A e. RR )
10		efltim 11709	. . . . . . 7 |- ( ( B e. RR /\ A e. RR ) -> ( B < A -> ( exp ` B ) < ( exp ` A ) ) )
11	3, 9, 10	syl2an2r 595	. . . . . 6 |- ( ( ph /\ ( B - D ) < A ) -> ( B < A -> ( exp ` B ) < ( exp ` A ) ) )
12	11	imp 124	. . . . 5 |- ( ( ( ph /\ ( B - D ) < A ) /\ B < A ) -> ( exp ` B ) < ( exp ` A ) )
13	5, 8, 12	ltnsymd 8080	. . . 4 |- ( ( ( ph /\ ( B - D ) < A ) /\ B < A ) -> -. ( exp ` A ) < ( exp ` B ) )
14	2, 13	pm2.21dd 620	. . 3 |- ( ( ( ph /\ ( B - D ) < A ) /\ B < A ) -> A < B )
15	6	reefcld 11680	. . . . . . 7 |- ( ph -> ( exp ` A ) e. RR )
16	3	reefcld 11680	. . . . . . 7 |- ( ph -> ( exp ` B ) e. RR )
17	15, 16, 1	ltled 8079	. . . . . . 7 |- ( ph -> ( exp ` A ) <_ ( exp ` B ) )
18	15, 16, 17	abssuble0d 11189	. . . . . 6 |- ( ph -> ( abs ` ( ( exp ` A ) - ( exp ` B ) ) ) = ( ( exp ` B ) - ( exp ` A ) ) )
19	18	ad2antrr 488	. . . . 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+ )
21	20	rpred 9699	. . . . . . . . 9 |- ( ph -> D e. RR )
22	6, 3, 21	absdifltd 11190	. . . . . . . 8 |- ( ph -> ( ( abs ` ( A - B ) ) < D <-> ( ( B - D ) < A /\ A < ( B + D ) ) ) )
23	22	biimprd 158	. . . . . . 7 |- ( ph -> ( ( ( B - D ) < A /\ A < ( B + D ) ) -> ( abs ` ( A - B ) ) < D ) )
24	23	impl 380	. . . . . 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 ) ) ) )
26	25	ad2antrr 488	. . . . . 6 |- ( ( ( ph /\ ( B - D ) < A ) /\ A < ( B + D ) ) -> ( ( abs ` ( A - B ) ) < D -> ( abs ` ( ( exp ` A ) - ( exp ` B ) ) ) < ( ( exp ` B ) - ( exp ` A ) ) ) )
27	24, 26	mpd 13	. . . . 5 |- ( ( ( ph /\ ( B - D ) < A ) /\ A < ( B + D ) ) -> ( abs ` ( ( exp ` A ) - ( exp ` B ) ) ) < ( ( exp ` B ) - ( exp ` A ) ) )
28	19, 27	eqbrtrrd 4029	. . . 4 |- ( ( ( ph /\ ( B - D ) < A ) /\ A < ( B + D ) ) -> ( ( exp ` B ) - ( exp ` A ) ) < ( ( exp ` B ) - ( exp ` A ) ) )
29	16, 15	resubcld 8341	. . . . . 6 |- ( ph -> ( ( exp ` B ) - ( exp ` A ) ) e. RR )
30	29	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ ( B - D ) < A ) /\ A < ( B + D ) ) -> ( ( exp ` B ) - ( exp ` A ) ) e. RR )
31	30	ltnrd 8072	. . . 4 |- ( ( ( ph /\ ( B - D ) < A ) /\ A < ( B + D ) ) -> -. ( ( exp ` B ) - ( exp ` A ) ) < ( ( exp ` B ) - ( exp ` A ) ) )
32	28, 31	pm2.21dd 620	. . 3 |- ( ( ( ph /\ ( B - D ) < A ) /\ A < ( B + D ) ) -> A < B )
33	3, 20	ltaddrpd 9733	. . . . 5 |- ( ph -> B < ( B + D ) )
34	3, 21	readdcld 7990	. . . . . 6 |- ( ph -> ( B + D ) e. RR )
35		axltwlin 8028	. . . . . 6 |- ( ( B e. RR /\ ( B + D ) e. RR /\ A e. RR ) -> ( B < ( B + D ) -> ( B < A \/ A < ( B + D ) ) ) )
36	3, 34, 6, 35	syl3anc 1238	. . . . 5 |- ( ph -> ( B < ( B + D ) -> ( B < A \/ A < ( B + D ) ) ) )
37	33, 36	mpd 13	. . . 4 |- ( ph -> ( B < A \/ A < ( B + D ) ) )
38	37	adantr 276	. . 3 |- ( ( ph /\ ( B - D ) < A ) -> ( B < A \/ A < ( B + D ) ) )
39	14, 32, 38	mpjaodan 798	. 2 |- ( ( ph /\ ( B - D ) < A ) -> A < B )
40		simpr 110	. 2 |- ( ( ph /\ A < B ) -> A < B )
41	3, 20	ltsubrpd 9732	. . 3 |- ( ph -> ( B - D ) < B )
42	3, 21	resubcld 8341	. . . 4 |- ( ph -> ( B - D ) e. RR )
43		axltwlin 8028	. . . 4 |- ( ( ( B - D ) e. RR /\ B e. RR /\ A e. RR ) -> ( ( B - D ) < B -> ( ( B - D ) < A \/ A < B ) ) )
44	42, 3, 6, 43	syl3anc 1238	. . 3 |- ( ph -> ( ( B - D ) < B -> ( ( B - D ) < A \/ A < B ) ) )
45	41, 44	mpd 13	. 2 |- ( ph -> ( ( B - D ) < A \/ A < B ) )
46	39, 40, 45	mpjaodan 798	1 |- ( ph -> A < B )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 708   = wceq 1353   e. wcel 2148   class class class wbr 4005   ` cfv 5218  (class class class)co 5878   RRcr 7813   + caddc 7817   < clt 7995   - cmin 8131   RR+crp 9656   abscabs 11009   expce 11653

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-mulrcl 7913  ax-addcom 7914  ax-mulcom 7915  ax-addass 7916  ax-mulass 7917  ax-distr 7918  ax-i2m1 7919  ax-0lt1 7920  ax-1rid 7921  ax-0id 7922  ax-rnegex 7923  ax-precex 7924  ax-cnre 7925  ax-pre-ltirr 7926  ax-pre-ltwlin 7927  ax-pre-lttrn 7928  ax-pre-apti 7929  ax-pre-ltadd 7930  ax-pre-mulgt0 7931  ax-pre-mulext 7932  ax-arch 7933  ax-caucvg 7934

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-disj 3983  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-recs 6309  df-irdg 6374  df-frec 6395  df-1o 6420  df-oadd 6424  df-er 6538  df-en 6744  df-dom 6745  df-fin 6746  df-sup 6986  df-pnf 7997  df-mnf 7998  df-xr 7999  df-ltxr 8000  df-le 8001  df-sub 8133  df-neg 8134  df-reap 8535  df-ap 8542  df-div 8633  df-inn 8923  df-2 8981  df-3 8982  df-4 8983  df-n0 9180  df-z 9257  df-uz 9532  df-q 9623  df-rp 9657  df-ico 9897  df-fz 10012  df-fzo 10146  df-seqfrec 10449  df-exp 10523  df-fac 10709  df-bc 10731  df-ihash 10759  df-cj 10854  df-re 10855  df-im 10856  df-rsqrt 11010  df-abs 11011  df-clim 11290  df-sumdc 11365  df-ef 11659

This theorem is referenced by:  eflt  14336