Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > efexp | Unicode version |
Description: The exponential of an integer power. Corollary 15-4.4 of [Gleason] p. 309, restricted to integers. (Contributed by NM, 13-Jan-2006.) (Revised by Mario Carneiro, 5-Jun-2014.) |
Ref | Expression |
---|---|
efexp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 9217 | . . . 4 | |
2 | mulcom 7903 | . . . 4 | |
3 | 1, 2 | sylan2 284 | . . 3 |
4 | 3 | fveq2d 5500 | . 2 |
5 | oveq2 5861 | . . . . . 6 | |
6 | 5 | fveq2d 5500 | . . . . 5 |
7 | oveq2 5861 | . . . . 5 | |
8 | 6, 7 | eqeq12d 2185 | . . . 4 |
9 | oveq2 5861 | . . . . . 6 | |
10 | 9 | fveq2d 5500 | . . . . 5 |
11 | oveq2 5861 | . . . . 5 | |
12 | 10, 11 | eqeq12d 2185 | . . . 4 |
13 | oveq2 5861 | . . . . . 6 | |
14 | 13 | fveq2d 5500 | . . . . 5 |
15 | oveq2 5861 | . . . . 5 | |
16 | 14, 15 | eqeq12d 2185 | . . . 4 |
17 | oveq2 5861 | . . . . . 6 | |
18 | 17 | fveq2d 5500 | . . . . 5 |
19 | oveq2 5861 | . . . . 5 | |
20 | 18, 19 | eqeq12d 2185 | . . . 4 |
21 | oveq2 5861 | . . . . . 6 | |
22 | 21 | fveq2d 5500 | . . . . 5 |
23 | oveq2 5861 | . . . . 5 | |
24 | 22, 23 | eqeq12d 2185 | . . . 4 |
25 | ef0 11635 | . . . . 5 | |
26 | mul01 8308 | . . . . . 6 | |
27 | 26 | fveq2d 5500 | . . . . 5 |
28 | efcl 11627 | . . . . . 6 | |
29 | 28 | exp0d 10603 | . . . . 5 |
30 | 25, 27, 29 | 3eqtr4a 2229 | . . . 4 |
31 | oveq1 5860 | . . . . . . 7 | |
32 | 31 | adantl 275 | . . . . . 6 |
33 | nn0cn 9145 | . . . . . . . . . 10 | |
34 | ax-1cn 7867 | . . . . . . . . . . . 12 | |
35 | adddi 7906 | . . . . . . . . . . . 12 | |
36 | 34, 35 | mp3an3 1321 | . . . . . . . . . . 11 |
37 | mulid1 7917 | . . . . . . . . . . . . 13 | |
38 | 37 | adantr 274 | . . . . . . . . . . . 12 |
39 | 38 | oveq2d 5869 | . . . . . . . . . . 11 |
40 | 36, 39 | eqtrd 2203 | . . . . . . . . . 10 |
41 | 33, 40 | sylan2 284 | . . . . . . . . 9 |
42 | 41 | fveq2d 5500 | . . . . . . . 8 |
43 | mulcl 7901 | . . . . . . . . . 10 | |
44 | 33, 43 | sylan2 284 | . . . . . . . . 9 |
45 | simpl 108 | . . . . . . . . 9 | |
46 | efadd 11638 | . . . . . . . . 9 | |
47 | 44, 45, 46 | syl2anc 409 | . . . . . . . 8 |
48 | 42, 47 | eqtrd 2203 | . . . . . . 7 |
49 | 48 | adantr 274 | . . . . . 6 |
50 | expp1 10483 | . . . . . . . 8 | |
51 | 28, 50 | sylan 281 | . . . . . . 7 |
52 | 51 | adantr 274 | . . . . . 6 |
53 | 32, 49, 52 | 3eqtr4d 2213 | . . . . 5 |
54 | 53 | exp31 362 | . . . 4 |
55 | oveq2 5861 | . . . . . 6 | |
56 | nncn 8886 | . . . . . . . . . 10 | |
57 | mulneg2 8315 | . . . . . . . . . 10 | |
58 | 56, 57 | sylan2 284 | . . . . . . . . 9 |
59 | 58 | fveq2d 5500 | . . . . . . . 8 |
60 | 56, 43 | sylan2 284 | . . . . . . . . 9 |
61 | efneg 11642 | . . . . . . . . 9 | |
62 | 60, 61 | syl 14 | . . . . . . . 8 |
63 | 59, 62 | eqtrd 2203 | . . . . . . 7 |
64 | efap0 11640 | . . . . . . . 8 # | |
65 | nnnn0 9142 | . . . . . . . 8 | |
66 | expnegap0 10484 | . . . . . . . 8 # | |
67 | 28, 64, 65, 66 | syl2an3an 1293 | . . . . . . 7 |
68 | 63, 67 | eqeq12d 2185 | . . . . . 6 |
69 | 55, 68 | syl5ibr 155 | . . . . 5 |
70 | 69 | ex 114 | . . . 4 |
71 | 8, 12, 16, 20, 24, 30, 54, 70 | zindd 9330 | . . 3 |
72 | 71 | imp 123 | . 2 |
73 | 4, 72 | eqtr3d 2205 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wcel 2141 class class class wbr 3989 cfv 5198 (class class class)co 5853 cc 7772 cc0 7774 c1 7775 caddc 7777 cmul 7779 cneg 8091 # cap 8500 cdiv 8589 cn 8878 cn0 9135 cz 9212 cexp 10475 ce 11605 |
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 |
This theorem depends on definitions: df-bi 116 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-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-frec 6370 df-1o 6395 df-oadd 6399 df-er 6513 df-en 6719 df-dom 6720 df-fin 6721 df-sup 6961 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-ico 9851 df-fz 9966 df-fzo 10099 df-seqfrec 10402 df-exp 10476 df-fac 10660 df-bc 10682 df-ihash 10710 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-clim 11242 df-sumdc 11317 df-ef 11611 |
This theorem is referenced by: efzval 11646 efgt0 11647 tanval3ap 11677 demoivre 11735 ef2kpi 13521 reexplog 13586 relogexp 13587 |
Copyright terms: Public domain | W3C validator |