Theorem efltlemlt 13034

Description: Lemma for eflt 13035. The converse of efltim 11572 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 480	. . . 4 |- ( ( ( ph /\ ( B - D ) < A ) /\ B < A ) -> ( exp ` A ) < ( exp ` B ) )
3		efltlemlt.b	. . . . . . 7 |- ( ph -> B e. RR )
4	3	ad2antrr 480	. . . . . 6 |- ( ( ( ph /\ ( B - D ) < A ) /\ B < A ) -> B e. RR )
5	4	reefcld 11543	. . . . 5 |- ( ( ( ph /\ ( B - D ) < A ) /\ B < A ) -> ( exp ` B ) e. RR )
6		efltlemlt.a	. . . . . . 7 |- ( ph -> A e. RR )
7	6	ad2antrr 480	. . . . . 6 |- ( ( ( ph /\ ( B - D ) < A ) /\ B < A ) -> A e. RR )
8	7	reefcld 11543	. . . . 5 |- ( ( ( ph /\ ( B - D ) < A ) /\ B < A ) -> ( exp ` A ) e. RR )
9	6	adantr 274	. . . . . . 7 |- ( ( ph /\ ( B - D ) < A ) -> A e. RR )
10		efltim 11572	. . . . . . 7 |- ( ( B e. RR /\ A e. RR ) -> ( B < A -> ( exp ` B ) < ( exp ` A ) ) )
11	3, 9, 10	syl2an2r 585	. . . . . 6 |- ( ( ph /\ ( B - D ) < A ) -> ( B < A -> ( exp ` B ) < ( exp ` A ) ) )
12	11	imp 123	. . . . 5 |- ( ( ( ph /\ ( B - D ) < A ) /\ B < A ) -> ( exp ` B ) < ( exp ` A ) )
13	5, 8, 12	ltnsymd 7974	. . . 4 |- ( ( ( ph /\ ( B - D ) < A ) /\ B < A ) -> -. ( exp ` A ) < ( exp ` B ) )
14	2, 13	pm2.21dd 610	. . 3 |- ( ( ( ph /\ ( B - D ) < A ) /\ B < A ) -> A < B )
15	6	reefcld 11543	. . . . . . 7 |- ( ph -> ( exp ` A ) e. RR )
16	3	reefcld 11543	. . . . . . 7 |- ( ph -> ( exp ` B ) e. RR )
17	15, 16, 1	ltled 7973	. . . . . . 7 |- ( ph -> ( exp ` A ) <_ ( exp ` B ) )
18	15, 16, 17	abssuble0d 11054	. . . . . 6 |- ( ph -> ( abs ` ( ( exp ` A ) - ( exp ` B ) ) ) = ( ( exp ` B ) - ( exp ` A ) ) )
19	18	ad2antrr 480	. . . . 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 9581	. . . . . . . . 9 |- ( ph -> D e. RR )
22	6, 3, 21	absdifltd 11055	. . . . . . . 8 |- ( ph -> ( ( abs ` ( A - B ) ) < D <-> ( ( B - D ) < A /\ A < ( B + D ) ) ) )
23	22	biimprd 157	. . . . . . 7 |- ( ph -> ( ( ( B - D ) < A /\ A < ( B + D ) ) -> ( abs ` ( A - B ) ) < D ) )
24	23	impl 378	. . . . . 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 480	. . . . . 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 3984	. . . 4 |- ( ( ( ph /\ ( B - D ) < A ) /\ A < ( B + D ) ) -> ( ( exp ` B ) - ( exp ` A ) ) < ( ( exp ` B ) - ( exp ` A ) ) )
29	16, 15	resubcld 8235	. . . . . 6 |- ( ph -> ( ( exp ` B ) - ( exp ` A ) ) e. RR )
30	29	ad2antrr 480	. . . . 5 |- ( ( ( ph /\ ( B - D ) < A ) /\ A < ( B + D ) ) -> ( ( exp ` B ) - ( exp ` A ) ) e. RR )
31	30	ltnrd 7967	. . . 4 |- ( ( ( ph /\ ( B - D ) < A ) /\ A < ( B + D ) ) -> -. ( ( exp ` B ) - ( exp ` A ) ) < ( ( exp ` B ) - ( exp ` A ) ) )
32	28, 31	pm2.21dd 610	. . 3 |- ( ( ( ph /\ ( B - D ) < A ) /\ A < ( B + D ) ) -> A < B )
33	3, 20	ltaddrpd 9615	. . . . 5 |- ( ph -> B < ( B + D ) )
34	3, 21	readdcld 7886	. . . . . 6 |- ( ph -> ( B + D ) e. RR )
35		axltwlin 7924	. . . . . 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 1217	. . . . 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 274	. . 3 |- ( ( ph /\ ( B - D ) < A ) -> ( B < A \/ A < ( B + D ) ) )
39	14, 32, 38	mpjaodan 788	. 2 |- ( ( ph /\ ( B - D ) < A ) -> A < B )
40		simpr 109	. 2 |- ( ( ph /\ A < B ) -> A < B )
41	3, 20	ltsubrpd 9614	. . 3 |- ( ph -> ( B - D ) < B )
42	3, 21	resubcld 8235	. . . 4 |- ( ph -> ( B - D ) e. RR )
43		axltwlin 7924	. . . 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 1217	. . 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 788	1 |- ( ph -> A < B )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 698   = wceq 1332   e. wcel 2125   class class class wbr 3961   ` cfv 5163  (class class class)co 5814   RRcr 7710   + caddc 7714   < clt 7891   - cmin 8025   RR+crp 9538   abscabs 10874   expce 11516

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-coll 4075  ax-sep 4078  ax-nul 4086  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-setind 4490  ax-iinf 4541  ax-cnex 7802  ax-resscn 7803  ax-1cn 7804  ax-1re 7805  ax-icn 7806  ax-addcl 7807  ax-addrcl 7808  ax-mulcl 7809  ax-mulrcl 7810  ax-addcom 7811  ax-mulcom 7812  ax-addass 7813  ax-mulass 7814  ax-distr 7815  ax-i2m1 7816  ax-0lt1 7817  ax-1rid 7818  ax-0id 7819  ax-rnegex 7820  ax-precex 7821  ax-cnre 7822  ax-pre-ltirr 7823  ax-pre-ltwlin 7824  ax-pre-lttrn 7825  ax-pre-apti 7826  ax-pre-ltadd 7827  ax-pre-mulgt0 7828  ax-pre-mulext 7829  ax-arch 7830  ax-caucvg 7831

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-nel 2420  df-ral 2437  df-rex 2438  df-reu 2439  df-rmo 2440  df-rab 2441  df-v 2711  df-sbc 2934  df-csb 3028  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-nul 3391  df-if 3502  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-int 3804  df-iun 3847  df-disj 3939  df-br 3962  df-opab 4022  df-mpt 4023  df-tr 4059  df-id 4248  df-po 4251  df-iso 4252  df-iord 4321  df-on 4323  df-ilim 4324  df-suc 4326  df-iom 4544  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592  df-iota 5128  df-fun 5165  df-fn 5166  df-f 5167  df-f1 5168  df-fo 5169  df-f1o 5170  df-fv 5171  df-isom 5172  df-riota 5770  df-ov 5817  df-oprab 5818  df-mpo 5819  df-1st 6078  df-2nd 6079  df-recs 6242  df-irdg 6307  df-frec 6328  df-1o 6353  df-oadd 6357  df-er 6469  df-en 6675  df-dom 6676  df-fin 6677  df-sup 6916  df-pnf 7893  df-mnf 7894  df-xr 7895  df-ltxr 7896  df-le 7897  df-sub 8027  df-neg 8028  df-reap 8429  df-ap 8436  df-div 8525  df-inn 8813  df-2 8871  df-3 8872  df-4 8873  df-n0 9070  df-z 9147  df-uz 9419  df-q 9507  df-rp 9539  df-ico 9776  df-fz 9891  df-fzo 10020  df-seqfrec 10323  df-exp 10397  df-fac 10577  df-bc 10599  df-ihash 10627  df-cj 10719  df-re 10720  df-im 10721  df-rsqrt 10875  df-abs 10876  df-clim 11153  df-sumdc 11228  df-ef 11522

This theorem is referenced by:  eflt  13035