Theorem eflt 14281

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 11708	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> ( exp ` A ) < ( exp ` B ) ) )
2		efcn 14274	. . . . 5 |- exp e. ( CC -cn-> CC )
3		simplr 528	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) -> B e. RR )
4	3	recnd 7988	. . . . 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 11679	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) -> ( exp ` A ) e. RR )
8	3	reefcld 11679	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) -> ( exp ` B ) e. RR )
9		difrp 9694	. . . . . . 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 14150	. . . . 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 1342	. . . 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 528	. . . . 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 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 ) ) ) ) ) -> d e. RR+ )
18		fvoveq1 5900	. . . . . . . 8 |- ( x = A -> ( abs ` ( x - B ) ) = ( abs ` ( A - B ) ) )
19	18	breq1d 4015	. . . . . . 7 |- ( x = A -> ( ( abs ` ( x - B ) ) < d <-> ( abs ` ( A - B ) ) < d ) )
20	19	imbrov2fvoveq 5902	. . . . . 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 531	. . . . . 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 7988	. . . . . 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 2848	. . . . 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 14280	. . . 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 2599	. . 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 1353   e. wcel 2148   A.wral 2455   E.wrex 2456   class class class wbr 4005   ` cfv 5218  (class class class)co 5877   CCcc 7811   RRcr 7812   < clt 7994   - cmin 8130   RR+crp 9655   abscabs 11008   expce 11652   -cn->ccncf 14142

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933  ax-addf 7935  ax-mulf 7936

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-disj 3983  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-of 6085  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-frec 6394  df-1o 6419  df-oadd 6423  df-er 6537  df-map 6652  df-pm 6653  df-en 6743  df-dom 6744  df-fin 6745  df-sup 6985  df-inf 6986  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-xneg 9774  df-xadd 9775  df-ico 9896  df-fz 10011  df-fzo 10145  df-seqfrec 10448  df-exp 10522  df-fac 10708  df-bc 10730  df-ihash 10758  df-shft 10826  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-clim 11289  df-sumdc 11364  df-ef 11658  df-rest 12695  df-topgen 12714  df-psmet 13532  df-xmet 13533  df-met 13534  df-bl 13535  df-mopn 13536  df-top 13583  df-topon 13596  df-bases 13628  df-ntr 13681  df-cn 13773  df-cnp 13774  df-tx 13838  df-cncf 14143  df-limced 14210  df-dvap 14211

This theorem is referenced by:  efle  14282  reefiso  14283  reapef  14284  logdivlti  14387  cxplt  14421  rpcxplt2  14424