Theorem efltlemlt 15656

Description: Lemma for eflt 15657. The converse of efltim 12388 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 12359	. . . . 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 12359	. . . . 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 12388	. . . . . . 7 |- ( ( B e. RR /\ A e. RR ) -> ( B < A -> ( exp ` B ) < ( exp ` A ) ) )
11	3, 9, 10	syl2an2r 599	. . . . . 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 8395	. . . 4 |- ( ( ( ph /\ ( B - D ) < A ) /\ B < A ) -> -. ( exp ` A ) < ( exp ` B ) )
14	2, 13	pm2.21dd 625	. . 3 |- ( ( ( ph /\ ( B - D ) < A ) /\ B < A ) -> A < B )
15	6	reefcld 12359	. . . . . . 7 |- ( ph -> ( exp ` A ) e. RR )
16	3	reefcld 12359	. . . . . . 7 |- ( ph -> ( exp ` B ) e. RR )
17	15, 16, 1	ltled 8394	. . . . . . 7 |- ( ph -> ( exp ` A ) <_ ( exp ` B ) )
18	15, 16, 17	abssuble0d 11866	. . . . . 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 10032	. . . . . . . . 9 |- ( ph -> D e. RR )
22	6, 3, 21	absdifltd 11867	. . . . . . . 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 4135	. . . 4 |- ( ( ( ph /\ ( B - D ) < A ) /\ A < ( B + D ) ) -> ( ( exp ` B ) - ( exp ` A ) ) < ( ( exp ` B ) - ( exp ` A ) ) )
29	16, 15	resubcld 8656	. . . . . 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 8387	. . . 4 |- ( ( ( ph /\ ( B - D ) < A ) /\ A < ( B + D ) ) -> -. ( ( exp ` B ) - ( exp ` A ) ) < ( ( exp ` B ) - ( exp ` A ) ) )
32	28, 31	pm2.21dd 625	. . 3 |- ( ( ( ph /\ ( B - D ) < A ) /\ A < ( B + D ) ) -> A < B )
33	3, 20	ltaddrpd 10066	. . . . 5 |- ( ph -> B < ( B + D ) )
34	3, 21	readdcld 8305	. . . . . 6 |- ( ph -> ( B + D ) e. RR )
35		axltwlin 8343	. . . . . 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 1274	. . . . 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 806	. 2 |- ( ( ph /\ ( B - D ) < A ) -> A < B )
40		simpr 110	. 2 |- ( ( ph /\ A < B ) -> A < B )
41	3, 20	ltsubrpd 10065	. . 3 |- ( ph -> ( B - D ) < B )
42	3, 21	resubcld 8656	. . . 4 |- ( ph -> ( B - D ) e. RR )
43		axltwlin 8343	. . . 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 1274	. . 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 806	1 |- ( ph -> A < B )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 716   = wceq 1398   e. wcel 2205   class class class wbr 4111   ` cfv 5354  (class class class)co 6052   RRcr 8128   + caddc 8132   < clt 8310   - cmin 8446   RR+crp 9989   abscabs 11686   expce 12332

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-mulrcl 8228  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-precex 8239  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245  ax-pre-mulgt0 8246  ax-pre-mulext 8247  ax-arch 8248  ax-caucvg 8249

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-disj 4088  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-isom 5363  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-frec 6624  df-1o 6649  df-oadd 6653  df-er 6769  df-en 6978  df-dom 6979  df-fin 6980  df-sup 7277  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-reap 8851  df-ap 8858  df-div 8949  df-inn 9240  df-2 9298  df-3 9299  df-4 9300  df-n0 9499  df-z 9580  df-uz 9857  df-q 9955  df-rp 9990  df-ico 10230  df-fz 10346  df-fzo 10481  df-seqfrec 10814  df-exp 10905  df-fac 11092  df-bc 11114  df-ihash 11143  df-cj 11531  df-re 11532  df-im 11533  df-rsqrt 11687  df-abs 11688  df-clim 11968  df-sumdc 12043  df-ef 12338

This theorem is referenced by:  eflt  15657