Theorem eflt 14067

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 11699	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> ( exp ` A ) < ( exp ` B ) ) )
2		efcn 14060	. . . . 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 7982	. . . . 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 11670	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) -> ( exp ` A ) e. RR )
8	3	reefcld 11670	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) -> ( exp ` B ) e. RR )
9		difrp 9688	. . . . . . 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 13936	. . . . 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 5895	. . . . . . . 8 |- ( x = A -> ( abs ` ( x - B ) ) = ( abs ` ( A - B ) ) )
19	18	breq1d 4012	. . . . . . 7 |- ( x = A -> ( ( abs ` ( x - B ) ) < d <-> ( abs ` ( A - B ) ) < d ) )
20	19	imbrov2fvoveq 5897	. . . . . 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 7982	. . . . . 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 2846	. . . . 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 14066	. . . 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 4002   ` cfv 5215  (class class class)co 5872   CCcc 7806   RRcr 7807   < clt 7988   - cmin 8124   RR+crp 9649   abscabs 10999   expce 11643   -cn->ccncf 13928

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 4117  ax-sep 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-iinf 4586  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-mulrcl 7907  ax-addcom 7908  ax-mulcom 7909  ax-addass 7910  ax-mulass 7911  ax-distr 7912  ax-i2m1 7913  ax-0lt1 7914  ax-1rid 7915  ax-0id 7916  ax-rnegex 7917  ax-precex 7918  ax-cnre 7919  ax-pre-ltirr 7920  ax-pre-ltwlin 7921  ax-pre-lttrn 7922  ax-pre-apti 7923  ax-pre-ltadd 7924  ax-pre-mulgt0 7925  ax-pre-mulext 7926  ax-arch 7927  ax-caucvg 7928  ax-addf 7930  ax-mulf 7931

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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-disj 3980  df-br 4003  df-opab 4064  df-mpt 4065  df-tr 4101  df-id 4292  df-po 4295  df-iso 4296  df-iord 4365  df-on 4367  df-ilim 4368  df-suc 4370  df-iom 4589  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-isom 5224  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-of 6080  df-1st 6138  df-2nd 6139  df-recs 6303  df-irdg 6368  df-frec 6389  df-1o 6414  df-oadd 6418  df-er 6532  df-map 6647  df-pm 6648  df-en 6738  df-dom 6739  df-fin 6740  df-sup 6980  df-inf 6981  df-pnf 7990  df-mnf 7991  df-xr 7992  df-ltxr 7993  df-le 7994  df-sub 8126  df-neg 8127  df-reap 8528  df-ap 8535  df-div 8626  df-inn 8916  df-2 8974  df-3 8975  df-4 8976  df-n0 9173  df-z 9250  df-uz 9525  df-q 9616  df-rp 9650  df-xneg 9768  df-xadd 9769  df-ico 9890  df-fz 10005  df-fzo 10138  df-seqfrec 10441  df-exp 10515  df-fac 10699  df-bc 10721  df-ihash 10749  df-shft 10817  df-cj 10844  df-re 10845  df-im 10846  df-rsqrt 11000  df-abs 11001  df-clim 11280  df-sumdc 11355  df-ef 11649  df-rest 12678  df-topgen 12697  df-psmet 13316  df-xmet 13317  df-met 13318  df-bl 13319  df-mopn 13320  df-top 13367  df-topon 13380  df-bases 13412  df-ntr 13467  df-cn 13559  df-cnp 13560  df-tx 13624  df-cncf 13929  df-limced 13996  df-dvap 13997

This theorem is referenced by:  efle  14068  reefiso  14069  reapef  14070  logdivlti  14173  cxplt  14207  rpcxplt2  14210