Theorem eflt 15640

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 12384	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> ( exp ` A ) < ( exp ` B ) ) )
2		efcn 15633	. . . . 5 |- exp e. ( CC -cn-> CC )
3		simplr 529	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) -> B e. RR )
4	3	recnd 8302	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) -> B e. CC )
5		simpr 110	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) -> ( exp ` A ) < ( exp ` B ) )
6		simpll 527	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) -> A e. RR )
7	6	reefcld 12355	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) -> ( exp ` A ) e. RR )
8	3	reefcld 12355	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) -> ( exp ` B ) e. RR )
9		difrp 10025	. . . . . . 7 |- ( ( ( exp ` A ) e. RR /\ ( exp ` B ) e. RR ) -> ( ( exp ` A ) < ( exp ` B ) <-> ( ( exp ` B ) - ( exp ` A ) ) e. RR+ ) )
10	7, 8, 9	syl2anc 411	. . . . . 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 147	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) -> ( ( exp ` B ) - ( exp ` A ) ) e. RR+ )
12		cncfi 15443	. . . . 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 1379	. . . 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 276	. . . . 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 276	. . . . 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 529	. . . . 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 531	. . . . 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 6073	. . . . . . . 8 |- ( x = A -> ( abs ` ( x - B ) ) = ( abs ` ( A - B ) ) )
19	18	breq1d 4119	. . . . . . 7 |- ( x = A -> ( ( abs ` ( x - B ) ) < d <-> ( abs ` ( A - B ) ) < d ) )
20	19	imbrov2fvoveq 6075	. . . . . 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 533	. . . . . 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 8302	. . . . . 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 2926	. . . . 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 15639	. . . 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 2665	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) -> A < B )
26	25	ex 115	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( exp ` A ) < ( exp ` B ) -> A < B ) )
27	1, 26	impbid 129	1 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> ( exp ` A ) < ( exp ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   A.wral 2520   E.wrex 2521   class class class wbr 4109   ` cfv 5352  (class class class)co 6050   CCcc 8125   RRcr 8126   < clt 8308   - cmin 8444   RR+crp 9986   abscabs 11682   expce 12328   -cn->ccncf 15435

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247  ax-addf 8249  ax-mulf 8250

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-disj 4086  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-isom 5361  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-of 6266  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-frec 6622  df-1o 6647  df-oadd 6651  df-er 6767  df-map 6884  df-pm 6885  df-en 6976  df-dom 6977  df-fin 6978  df-sup 7275  df-inf 7276  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-xneg 10105  df-xadd 10106  df-ico 10227  df-fz 10343  df-fzo 10477  df-seqfrec 10810  df-exp 10901  df-fac 11088  df-bc 11110  df-ihash 11139  df-shft 11500  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-clim 11964  df-sumdc 12039  df-ef 12334  df-rest 13454  df-topgen 13473  df-psmet 14691  df-xmet 14692  df-met 14693  df-bl 14694  df-mopn 14695  df-top 14863  df-topon 14876  df-bases 14908  df-ntr 14961  df-cn 15053  df-cnp 15054  df-tx 15118  df-cncf 15436  df-limced 15521  df-dvap 15522

This theorem is referenced by:  efle  15641  reefiso  15642  reapef  15643  logdivlti  15746  cxplt  15781  rpcxplt2  15784