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Mirrors > Home > ILE Home > Th. List > efltim | Unicode version |
Description: The exponential function on the reals is strictly increasing. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Jim Kingdon, 20-Dec-2022.) |
Ref | Expression |
---|---|
efltim |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 965 |
. . . . . . 7
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2 | simp1 964 |
. . . . . . 7
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3 | 1, 2 | resubcld 8062 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | posdif 8136 |
. . . . . . 7
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5 | 4 | biimp3a 1306 |
. . . . . 6
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6 | 3, 5 | elrpd 9380 |
. . . . 5
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7 | efgt1 11254 |
. . . . 5
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8 | 6, 7 | syl 14 |
. . . 4
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9 | 2 | reefcld 11226 |
. . . . 5
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10 | 3 | reefcld 11226 |
. . . . 5
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11 | efgt0 11241 |
. . . . . 6
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12 | 2, 11 | syl 14 |
. . . . 5
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13 | ltmulgt11 8532 |
. . . . 5
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14 | 9, 10, 12, 13 | syl3anc 1199 |
. . . 4
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15 | 8, 14 | mpbid 146 |
. . 3
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16 | 2 | recnd 7718 |
. . . . 5
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17 | 3 | recnd 7718 |
. . . . 5
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18 | efadd 11232 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | 16, 17, 18 | syl2anc 406 |
. . . 4
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20 | 1 | recnd 7718 |
. . . . . 6
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21 | 16, 20 | pncan3d 7999 |
. . . . 5
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22 | 21 | fveq2d 5379 |
. . . 4
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23 | 19, 22 | eqtr3d 2149 |
. . 3
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24 | 15, 23 | breqtrd 3919 |
. 2
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25 | 24 | 3expia 1166 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 |
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