Theorem eflt 13490

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 11661	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> ( exp ` A ) < ( exp ` B ) ) )
2		efcn 13483	. . . . 5 |- exp e. ( CC -cn-> CC )
3		simplr 525	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) -> B e. RR )
4	3	recnd 7948	. . . . 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 524	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) -> A e. RR )
7	6	reefcld 11632	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) -> ( exp ` A ) e. RR )
8	3	reefcld 11632	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( exp ` A ) < ( exp ` B ) ) -> ( exp ` B ) e. RR )
9		difrp 9649	. . . . . . 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 13359	. . . . 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 1337	. . . 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 525	. . . . 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 526	. . . . 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 5876	. . . . . . . 8 |- ( x = A -> ( abs ` ( x - B ) ) = ( abs ` ( A - B ) ) )
19	18	breq1d 3999	. . . . . . 7 |- ( x = A -> ( ( abs ` ( x - B ) ) < d <-> ( abs ` ( A - B ) ) < d ) )
20	19	imbrov2fvoveq 5878	. . . . . 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 527	. . . . . 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 7948	. . . . . 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 2839	. . . . 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 13489	. . . 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 2592	. . 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 1348   e. wcel 2141   A.wral 2448   E.wrex 2449   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   CCcc 7772   RRcr 7773   < clt 7954   - cmin 8090   RR+crp 9610   abscabs 10961   expce 11605   -cn->ccncf 13351

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894  ax-addf 7896  ax-mulf 7897

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-disj 3967  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-of 6061  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-oadd 6399  df-er 6513  df-map 6628  df-pm 6629  df-en 6719  df-dom 6720  df-fin 6721  df-sup 6961  df-inf 6962  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-xneg 9729  df-xadd 9730  df-ico 9851  df-fz 9966  df-fzo 10099  df-seqfrec 10402  df-exp 10476  df-fac 10660  df-bc 10682  df-ihash 10710  df-shft 10779  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-sumdc 11317  df-ef 11611  df-rest 12581  df-topgen 12600  df-psmet 12781  df-xmet 12782  df-met 12783  df-bl 12784  df-mopn 12785  df-top 12790  df-topon 12803  df-bases 12835  df-ntr 12890  df-cn 12982  df-cnp 12983  df-tx 13047  df-cncf 13352  df-limced 13419  df-dvap 13420

This theorem is referenced by:  efle  13491  reefiso  13492  reapef  13493  logdivlti  13596  cxplt  13630  rpcxplt2  13633