Theorem efltim 11255

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 965	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> B e. RR )
2		simp1 964	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> A e. RR )
3	1, 2	resubcld 8062	. . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( B - A ) e. RR )
4		posdif 8136	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> 0 < ( B - A ) ) )
5	4	biimp3a 1306	. . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> 0 < ( B - A ) )
6	3, 5	elrpd 9380	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( B - A ) e. RR+ )
7		efgt1 11254	. . . . 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 11226	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( exp ` A ) e. RR )
10	3	reefcld 11226	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( exp ` ( B - A ) ) e. RR )
11		efgt0 11241	. . . . . 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 8532	. . . . 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 1199	. . . 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 7718	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> A e. CC )
17	3	recnd 7718	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( B - A ) e. CC )
18		efadd 11232	. . . . 5 |- ( ( A e. CC /\ ( B - A ) e. CC ) -> ( exp ` ( A + ( B - A ) ) ) = ( ( exp ` A ) x. ( exp ` ( B - A ) ) ) )
19	16, 17, 18	syl2anc 406	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( exp ` ( A + ( B - A ) ) ) = ( ( exp ` A ) x. ( exp ` ( B - A ) ) ) )
20	1	recnd 7718	. . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> B e. CC )
21	16, 20	pncan3d 7999	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( A + ( B - A ) ) = B )
22	21	fveq2d 5379	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( exp ` ( A + ( B - A ) ) ) = ( exp ` B ) )
23	19, 22	eqtr3d 2149	. . 3 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( ( exp ` A ) x. ( exp ` ( B - A ) ) ) = ( exp ` B ) )
24	15, 23	breqtrd 3919	. 2 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( exp ` A ) < ( exp ` B ) )
25	24	3expia 1166	1 |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> ( exp ` A ) < ( exp ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 945   = wceq 1314   e. wcel 1463   class class class wbr 3895   ` cfv 5081  (class class class)co 5728   CCcc 7545   RRcr 7546   0cc0 7547   1c1 7548   + caddc 7550   x. cmul 7552   < clt 7724   - cmin 7856   RR+crp 9343   expce 11199

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-mulrcl 7644  ax-addcom 7645  ax-mulcom 7646  ax-addass 7647  ax-mulass 7648  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-1rid 7652  ax-0id 7653  ax-rnegex 7654  ax-precex 7655  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-apti 7660  ax-pre-ltadd 7661  ax-pre-mulgt0 7662  ax-pre-mulext 7663  ax-arch 7664  ax-caucvg 7665

This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rmo 2398  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-disj 3873  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-id 4175  df-po 4178  df-iso 4179  df-iord 4248  df-on 4250  df-ilim 4251  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-isom 5090  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-irdg 6221  df-frec 6242  df-1o 6267  df-oadd 6271  df-er 6383  df-en 6589  df-dom 6590  df-fin 6591  df-sup 6823  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-reap 8255  df-ap 8262  df-div 8346  df-inn 8631  df-2 8689  df-3 8690  df-4 8691  df-n0 8882  df-z 8959  df-uz 9229  df-q 9314  df-rp 9344  df-ico 9570  df-fz 9684  df-fzo 9813  df-seqfrec 10112  df-exp 10186  df-fac 10365  df-bc 10387  df-ihash 10415  df-cj 10507  df-re 10508  df-im 10509  df-rsqrt 10662  df-abs 10663  df-clim 10940  df-sumdc 11015  df-ef 11205

This theorem is referenced by:  efler  11256  reef11  11257