Theorem eflt 13336

Description: The exponential function on the reals is strictly increasing. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Jim Kingdon, 21-May-2024.)

Assertion

Ref	Expression
eflt	|- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> ( exp ` A ) < ( exp ` B ) ) )

Proof of Theorem eflt

Dummy variables d x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		efltim 11639	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> ( exp ` A ) < ( exp ` B ) ) )
2		efcn 13329	. . . . 5 |- exp e. ( CC -cn-> CC )
3		simplr 520	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) -> B e. RR )
4	3	recnd 7927	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) -> B e. CC )
5		simpr 109	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) -> ( exp ` A ) < ( exp ` B ) )
6		simpll 519	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) -> A e. RR )
7	6	reefcld 11610	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) -> ( exp ` A ) e. RR )
8	3	reefcld 11610	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) -> ( exp ` B ) e. RR )
9		difrp 9628	. . . . . . 7 |- ( ( ( exp ` A ) e. RR /\ ( exp ` B ) e. RR ) -> ( ( exp ` A ) < ( exp ` B ) <-> ( ( exp ` B ) - ( exp ` A ) ) e. RR+ ) )
10	7, 8, 9	syl2anc 409	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) -> ( ( exp ` A ) < ( exp ` B ) <-> ( ( exp ` B ) - ( exp ` A ) ) e. RR+ ) )
11	5, 10	mpbid 146	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) -> ( ( exp ` B ) - ( exp ` A ) ) e. RR+ )
12		cncfi 13205	. . . . 5 |- ( ( exp e. ( CC -cn-> CC ) /\ B e. CC /\ ( ( exp ` B ) - ( exp ` A ) ) e. RR+ ) -> E. d e. RR+ A. x e. CC ( ( abs ` ( x - B ) ) < d -> ( abs ` ( ( exp ` x ) - ( exp ` B ) ) ) < ( ( exp ` B ) - ( exp ` A ) ) ) )
13	2, 4, 11, 12	mp3an2i 1332	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) -> E. d e. RR+ A. x e. CC ( ( abs ` ( x - B ) ) < d -> ( abs ` ( ( exp ` x ) - ( exp ` B ) ) ) < ( ( exp ` B ) - ( exp ` A ) ) ) )
14	6	adantr 274	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) /\ ( d e. RR+ /\ A. x e. CC ( ( abs ` ( x - B ) ) < d -> ( abs ` ( ( exp ` x ) - ( exp ` B ) ) ) < ( ( exp ` B ) - ( exp ` A ) ) ) ) ) -> A e. RR )
15	3	adantr 274	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) /\ ( d e. RR+ /\ A. x e. CC ( ( abs ` ( x - B ) ) < d -> ( abs ` ( ( exp ` x ) - ( exp ` B ) ) ) < ( ( exp ` B ) - ( exp ` A ) ) ) ) ) -> B e. RR )
16		simplr 520	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) /\ ( d e. RR+ /\ A. x e. CC ( ( abs ` ( x - B ) ) < d -> ( abs ` ( ( exp ` x ) - ( exp ` B ) ) ) < ( ( exp ` B ) - ( exp ` A ) ) ) ) ) -> ( exp ` A ) < ( exp ` B ) )
17		simprl 521	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) /\ ( d e. RR+ /\ A. x e. CC ( ( abs ` ( x - B ) ) < d -> ( abs ` ( ( exp ` x ) - ( exp ` B ) ) ) < ( ( exp ` B ) - ( exp ` A ) ) ) ) ) -> d e. RR+ )
18		fvoveq1 5865	. . . . . . . 8 |- ( x = A -> ( abs ` ( x - B ) ) = ( abs ` ( A - B ) ) )
19	18	breq1d 3992	. . . . . . 7 |- ( x = A -> ( ( abs ` ( x - B ) ) < d <-> ( abs ` ( A - B ) ) < d ) )
20	19	imbrov2fvoveq 5867	. . . . . 6 |- ( x = A -> ( ( ( abs ` ( x - B ) ) < d -> ( abs ` ( ( exp ` x ) - ( exp ` B ) ) ) < ( ( exp ` B ) - ( exp ` A ) ) ) <-> ( ( abs ` ( A - B ) ) < d -> ( abs ` ( ( exp ` A ) - ( exp ` B ) ) ) < ( ( exp ` B ) - ( exp ` A ) ) ) ) )
21		simprr 522	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) /\ ( d e. RR+ /\ A. x e. CC ( ( abs ` ( x - B ) ) < d -> ( abs ` ( ( exp ` x ) - ( exp ` B ) ) ) < ( ( exp ` B ) - ( exp ` A ) ) ) ) ) -> A. x e. CC ( ( abs ` ( x - B ) ) < d -> ( abs ` ( ( exp ` x ) - ( exp ` B ) ) ) < ( ( exp ` B ) - ( exp ` A ) ) ) )
22	14	recnd 7927	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) /\ ( d e. RR+ /\ A. x e. CC ( ( abs ` ( x - B ) ) < d -> ( abs ` ( ( exp ` x ) - ( exp ` B ) ) ) < ( ( exp ` B ) - ( exp ` A ) ) ) ) ) -> A e. CC )
23	20, 21, 22	rspcdva 2835	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) /\ ( d e. RR+ /\ A. x e. CC ( ( abs ` ( x - B ) ) < d -> ( abs ` ( ( exp ` x ) - ( exp ` B ) ) ) < ( ( exp ` B ) - ( exp ` A ) ) ) ) ) -> ( ( abs ` ( A - B ) ) < d -> ( abs ` ( ( exp ` A ) - ( exp ` B ) ) ) < ( ( exp ` B ) - ( exp ` A ) ) ) )
24	14, 15, 16, 17, 23	efltlemlt 13335	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) /\ ( d e. RR+ /\ A. x e. CC ( ( abs ` ( x - B ) ) < d -> ( abs ` ( ( exp ` x ) - ( exp ` B ) ) ) < ( ( exp ` B ) - ( exp ` A ) ) ) ) ) -> A < B )
25	13, 24	rexlimddv 2588	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) -> A < B )
26	25	ex 114	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( exp ` A ) < ( exp ` B ) -> A < B ) )
27	1, 26	impbid 128	1 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> ( exp ` A ) < ( exp ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   A.wral 2444   E.wrex 2445   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   CCcc 7751   RRcr 7752   < clt 7933   - cmin 8069   RR+crp 9589   abscabs 10939   expce 11583   -cn->ccncf 13197

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873  ax-addf 7875  ax-mulf 7876

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-disj 3960  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-of 6050  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-frec 6359  df-1o 6384  df-oadd 6388  df-er 6501  df-map 6616  df-pm 6617  df-en 6707  df-dom 6708  df-fin 6709  df-sup 6949  df-inf 6950  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-xneg 9708  df-xadd 9709  df-ico 9830  df-fz 9945  df-fzo 10078  df-seqfrec 10381  df-exp 10455  df-fac 10639  df-bc 10661  df-ihash 10689  df-shft 10757  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-clim 11220  df-sumdc 11295  df-ef 11589  df-rest 12558  df-topgen 12577  df-psmet 12627  df-xmet 12628  df-met 12629  df-bl 12630  df-mopn 12631  df-top 12636  df-topon 12649  df-bases 12681  df-ntr 12736  df-cn 12828  df-cnp 12829  df-tx 12893  df-cncf 13198  df-limced 13265  df-dvap 13266

This theorem is referenced by:  efle  13337  reefiso  13338  reapef  13339  logdivlti  13442  cxplt  13476  rpcxplt2  13479