Theorem efltim 11404

Description: The exponential function on the reals is strictly increasing. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Jim Kingdon, 20-Dec-2022.)

Assertion

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

Proof of Theorem efltim

Step	Hyp	Ref	Expression
1		simp2 982	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> B e. RR )
2		simp1 981	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> A e. RR )
3	1, 2	resubcld 8143	. . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( B - A ) e. RR )
4		posdif 8217	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> 0 < ( B - A ) ) )
5	4	biimp3a 1323	. . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> 0 < ( B - A ) )
6	3, 5	elrpd 9481	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( B - A ) e. RR+ )
7		efgt1 11403	. . . . 5 |- ( ( B - A ) e. RR+ -> 1 < ( exp ` ( B - A ) ) )
8	6, 7	syl 14	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> 1 < ( exp ` ( B - A ) ) )
9	2	reefcld 11375	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( exp ` A ) e. RR )
10	3	reefcld 11375	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( exp ` ( B - A ) ) e. RR )
11		efgt0 11390	. . . . . 6 |- ( A e. RR -> 0 < ( exp ` A ) )
12	2, 11	syl 14	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> 0 < ( exp ` A ) )
13		ltmulgt11 8622	. . . . 5 |- ( ( ( exp ` A ) e. RR /\ ( exp ` ( B - A ) ) e. RR /\ 0 < ( exp ` A ) ) -> ( 1 < ( exp ` ( B - A ) ) <-> ( exp ` A ) < ( ( exp ` A ) x. ( exp ` ( B - A ) ) ) ) )
14	9, 10, 12, 13	syl3anc 1216	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( 1 < ( exp ` ( B - A ) ) <-> ( exp ` A ) < ( ( exp ` A ) x. ( exp ` ( B - A ) ) ) ) )
15	8, 14	mpbid 146	. . 3 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( exp ` A ) < ( ( exp ` A ) x. ( exp ` ( B - A ) ) ) )
16	2	recnd 7794	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> A e. CC )
17	3	recnd 7794	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( B - A ) e. CC )
18		efadd 11381	. . . . 5 |- ( ( A e. CC /\ ( B - A ) e. CC ) -> ( exp ` ( A + ( B - A ) ) ) = ( ( exp ` A ) x. ( exp ` ( B - A ) ) ) )
19	16, 17, 18	syl2anc 408	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( exp ` ( A + ( B - A ) ) ) = ( ( exp ` A ) x. ( exp ` ( B - A ) ) ) )
20	1	recnd 7794	. . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> B e. CC )
21	16, 20	pncan3d 8076	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( A + ( B - A ) ) = B )
22	21	fveq2d 5425	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( exp ` ( A + ( B - A ) ) ) = ( exp ` B ) )
23	19, 22	eqtr3d 2174	. . 3 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( ( exp ` A ) x. ( exp ` ( B - A ) ) ) = ( exp ` B ) )
24	15, 23	breqtrd 3954	. 2 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( exp ` A ) < ( exp ` B ) )
25	24	3expia 1183	1 |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> ( exp ` A ) < ( exp ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   CCcc 7618   RRcr 7619   0cc0 7620   1c1 7621   + caddc 7623   x. cmul 7625   < clt 7800   - cmin 7933   RR+crp 9441   expce 11348

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-disj 3907  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-frec 6288  df-1o 6313  df-oadd 6317  df-er 6429  df-en 6635  df-dom 6636  df-fin 6637  df-sup 6871  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-ico 9677  df-fz 9791  df-fzo 9920  df-seqfrec 10219  df-exp 10293  df-fac 10472  df-bc 10494  df-ihash 10522  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-clim 11048  df-sumdc 11123  df-ef 11354

This theorem is referenced by:  efler  11405  reef11  11406