Theorem efltlemlt 15801

Description: Lemma for eflt 15802. The converse of efltim 12446 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 492	. . . 4 |- ( ( ( ph /\ ( B - D ) < A ) /\ B < A ) -> ( exp ` A ) < ( exp ` B ) )
3		efltlemlt.b	. . . . . . 7 |- ( ph -> B e. RR )
4	3	ad2antrr 492	. . . . . 6 |- ( ( ( ph /\ ( B - D ) < A ) /\ B < A ) -> B e. RR )
5	4	reefcld 12417	. . . . 5 |- ( ( ( ph /\ ( B - D ) < A ) /\ B < A ) -> ( exp ` B ) e. RR )
6		efltlemlt.a	. . . . . . 7 |- ( ph -> A e. RR )
7	6	ad2antrr 492	. . . . . 6 |- ( ( ( ph /\ ( B - D ) < A ) /\ B < A ) -> A e. RR )
8	7	reefcld 12417	. . . . 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 12446	. . . . . . 7 |- ( ( B e. RR /\ A e. RR ) -> ( B < A -> ( exp ` B ) < ( exp ` A ) ) )
11	3, 9, 10	syl2an2r 603	. . . . . 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 8439	. . . 4 |- ( ( ( ph /\ ( B - D ) < A ) /\ B < A ) -> -. ( exp ` A ) < ( exp ` B ) )
14	2, 13	pm2.21dd 629	. . 3 |- ( ( ( ph /\ ( B - D ) < A ) /\ B < A ) -> A < B )
15	6	reefcld 12417	. . . . . . 7 |- ( ph -> ( exp ` A ) e. RR )
16	3	reefcld 12417	. . . . . . 7 |- ( ph -> ( exp ` B ) e. RR )
17	15, 16, 1	ltled 8438	. . . . . . 7 |- ( ph -> ( exp ` A ) <_ ( exp ` B ) )
18	15, 16, 17	abssuble0d 11924	. . . . . 6 |- ( ph -> ( abs ` ( ( exp ` A ) - ( exp ` B ) ) ) = ( ( exp ` B ) - ( exp ` A ) ) )
19	18	ad2antrr 492	. . . . 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 10079	. . . . . . . . 9 |- ( ph -> D e. RR )
22	6, 3, 21	absdifltd 11925	. . . . . . . 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 492	. . . . . 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 4152	. . . 4 |- ( ( ( ph /\ ( B - D ) < A ) /\ A < ( B + D ) ) -> ( ( exp ` B ) - ( exp ` A ) ) < ( ( exp ` B ) - ( exp ` A ) ) )
29	16, 15	resubcld 8701	. . . . . 6 |- ( ph -> ( ( exp ` B ) - ( exp ` A ) ) e. RR )
30	29	ad2antrr 492	. . . . 5 |- ( ( ( ph /\ ( B - D ) < A ) /\ A < ( B + D ) ) -> ( ( exp ` B ) - ( exp ` A ) ) e. RR )
31	30	ltnrd 8430	. . . 4 |- ( ( ( ph /\ ( B - D ) < A ) /\ A < ( B + D ) ) -> -. ( ( exp ` B ) - ( exp ` A ) ) < ( ( exp ` B ) - ( exp ` A ) ) )
32	28, 31	pm2.21dd 629	. . 3 |- ( ( ( ph /\ ( B - D ) < A ) /\ A < ( B + D ) ) -> A < B )
33	3, 20	ltaddrpd 10113	. . . . 5 |- ( ph -> B < ( B + D ) )
34	3, 21	readdcld 8348	. . . . . 6 |- ( ph -> ( B + D ) e. RR )
35		axltwlin 8386	. . . . . 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 1278	. . . . 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 810	. 2 |- ( ( ph /\ ( B - D ) < A ) -> A < B )
40		simpr 110	. 2 |- ( ( ph /\ A < B ) -> A < B )
41	3, 20	ltsubrpd 10112	. . 3 |- ( ph -> ( B - D ) < B )
42	3, 21	resubcld 8701	. . . 4 |- ( ph -> ( B - D ) e. RR )
43		axltwlin 8386	. . . 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 1278	. . 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 810	1 |- ( ph -> A < B )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 720   = wceq 1402   e. wcel 2209   class class class wbr 4128   ` cfv 5375  (class class class)co 6078   RRcr 8171   + caddc 8175   < clt 8353   - cmin 8490   RR+crp 10036   abscabs 11744   expce 12390

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8263  ax-resscn 8264  ax-1cn 8265  ax-1re 8266  ax-icn 8267  ax-addcl 8268  ax-addrcl 8269  ax-mulcl 8270  ax-mulrcl 8271  ax-addcom 8272  ax-mulcom 8273  ax-addass 8274  ax-mulass 8275  ax-distr 8276  ax-i2m1 8277  ax-0lt1 8278  ax-1rid 8279  ax-0id 8280  ax-rnegex 8281  ax-precex 8282  ax-cnre 8283  ax-pre-ltirr 8284  ax-pre-ltwlin 8285  ax-pre-lttrn 8286  ax-pre-apti 8287  ax-pre-ltadd 8288  ax-pre-mulgt0 8289  ax-pre-mulext 8290  ax-arch 8291  ax-caucvg 8292

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-disj 4105  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-isom 5384  df-riota 6031  df-ov 6081  df-oprab 6082  df-mpo 6083  df-1st 6367  df-2nd 6368  df-recs 6569  df-irdg 6634  df-frec 6655  df-1o 6680  df-oadd 6684  df-er 6800  df-en 7016  df-dom 7017  df-fin 7018  df-sup 7317  df-pnf 8355  df-mnf 8356  df-xr 8357  df-ltxr 8358  df-le 8359  df-sub 8492  df-neg 8493  df-reap 8896  df-ap 8903  df-div 8996  df-inn 9287  df-2 9345  df-3 9346  df-4 9347  df-n0 9546  df-z 9627  df-uz 9904  df-q 10002  df-rp 10037  df-ico 10278  df-fz 10394  df-fzo 10531  df-seqfrec 10866  df-exp 10957  df-fac 11145  df-bc 11167  df-ihash 11196  df-cj 11588  df-re 11589  df-im 11590  df-rsqrt 11745  df-abs 11746  df-clim 12026  df-sumdc 12101  df-ef 12396

This theorem is referenced by:  eflt  15802