Theorem efltlemlt 15279

Description: Lemma for eflt 15280. The converse of efltim 12042 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 12013	. . . . 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 12013	. . . . 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 12042	. . . . . . 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 8194	. . . 4 |- ( ( ( ph /\ ( B - D ) < A ) /\ B < A ) -> -. ( exp ` A ) < ( exp ` B ) )
14	2, 13	pm2.21dd 621	. . 3 |- ( ( ( ph /\ ( B - D ) < A ) /\ B < A ) -> A < B )
15	6	reefcld 12013	. . . . . . 7 |- ( ph -> ( exp ` A ) e. RR )
16	3	reefcld 12013	. . . . . . 7 |- ( ph -> ( exp ` B ) e. RR )
17	15, 16, 1	ltled 8193	. . . . . . 7 |- ( ph -> ( exp ` A ) <_ ( exp ` B ) )
18	15, 16, 17	abssuble0d 11521	. . . . . 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 9820	. . . . . . . . 9 |- ( ph -> D e. RR )
22	6, 3, 21	absdifltd 11522	. . . . . . . 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 4069	. . . 4 |- ( ( ( ph /\ ( B - D ) < A ) /\ A < ( B + D ) ) -> ( ( exp ` B ) - ( exp ` A ) ) < ( ( exp ` B ) - ( exp ` A ) ) )
29	16, 15	resubcld 8455	. . . . . 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 8186	. . . 4 |- ( ( ( ph /\ ( B - D ) < A ) /\ A < ( B + D ) ) -> -. ( ( exp ` B ) - ( exp ` A ) ) < ( ( exp ` B ) - ( exp ` A ) ) )
32	28, 31	pm2.21dd 621	. . 3 |- ( ( ( ph /\ ( B - D ) < A ) /\ A < ( B + D ) ) -> A < B )
33	3, 20	ltaddrpd 9854	. . . . 5 |- ( ph -> B < ( B + D ) )
34	3, 21	readdcld 8104	. . . . . 6 |- ( ph -> ( B + D ) e. RR )
35		axltwlin 8142	. . . . . 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 1250	. . . . 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 800	. 2 |- ( ( ph /\ ( B - D ) < A ) -> A < B )
40		simpr 110	. 2 |- ( ( ph /\ A < B ) -> A < B )
41	3, 20	ltsubrpd 9853	. . 3 |- ( ph -> ( B - D ) < B )
42	3, 21	resubcld 8455	. . . 4 |- ( ph -> ( B - D ) e. RR )
43		axltwlin 8142	. . . 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 1250	. . 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 800	1 |- ( ph -> A < B )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 710   = wceq 1373   e. wcel 2176   class class class wbr 4045   ` cfv 5272  (class class class)co 5946   RRcr 7926   + caddc 7930   < clt 8109   - cmin 8245   RR+crp 9777   abscabs 11341   expce 11986

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-mulrcl 8026  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-mulass 8030  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-1rid 8034  ax-0id 8035  ax-rnegex 8036  ax-precex 8037  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-apti 8042  ax-pre-ltadd 8043  ax-pre-mulgt0 8044  ax-pre-mulext 8045  ax-arch 8046  ax-caucvg 8047

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 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-disj 4022  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-po 4344  df-iso 4345  df-iord 4414  df-on 4416  df-ilim 4417  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-isom 5281  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-irdg 6458  df-frec 6479  df-1o 6504  df-oadd 6508  df-er 6622  df-en 6830  df-dom 6831  df-fin 6832  df-sup 7088  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-reap 8650  df-ap 8657  df-div 8748  df-inn 9039  df-2 9097  df-3 9098  df-4 9099  df-n0 9298  df-z 9375  df-uz 9651  df-q 9743  df-rp 9778  df-ico 10018  df-fz 10133  df-fzo 10267  df-seqfrec 10595  df-exp 10686  df-fac 10873  df-bc 10895  df-ihash 10923  df-cj 11186  df-re 11187  df-im 11188  df-rsqrt 11342  df-abs 11343  df-clim 11623  df-sumdc 11698  df-ef 11992

This theorem is referenced by:  eflt  15280