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Mirrors > Home > ILE Home > Th. List > exp3val | Unicode version |
Description: Value of exponentiation to integer powers. (Contributed by Jim Kingdon, 7-Jun-2020.) |
Ref | Expression |
---|---|
exp3val | # |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1cnd 7750 | . . 3 # | |
2 | simp1 966 | . . . . . . 7 # | |
3 | nnuz 9317 | . . . . . . . 8 | |
4 | 1zzd 9039 | . . . . . . . 8 | |
5 | fvconst2g 5602 | . . . . . . . . 9 | |
6 | simpl 108 | . . . . . . . . 9 | |
7 | 5, 6 | eqeltrd 2194 | . . . . . . . 8 |
8 | mulcl 7715 | . . . . . . . . 9 | |
9 | 8 | adantl 275 | . . . . . . . 8 |
10 | 3, 4, 7, 9 | seqf 10189 | . . . . . . 7 |
11 | 2, 10 | syl 14 | . . . . . 6 # |
12 | 11 | ad2antrr 479 | . . . . 5 # |
13 | simp2 967 | . . . . . . 7 # | |
14 | 13 | ad2antrr 479 | . . . . . 6 # |
15 | simpr 109 | . . . . . 6 # | |
16 | elnnz 9022 | . . . . . 6 | |
17 | 14, 15, 16 | sylanbrc 413 | . . . . 5 # |
18 | 12, 17 | ffvelrnd 5524 | . . . 4 # |
19 | 11 | ad2antrr 479 | . . . . . 6 # |
20 | 13 | ad2antrr 479 | . . . . . . . 8 # |
21 | 20 | znegcld 9133 | . . . . . . 7 # |
22 | simpr 109 | . . . . . . . . . . 11 # | |
23 | simplr 504 | . . . . . . . . . . . 12 # | |
24 | eqcom 2119 | . . . . . . . . . . . 12 | |
25 | 23, 24 | sylnib 650 | . . . . . . . . . . 11 # |
26 | ioran 726 | . . . . . . . . . . 11 | |
27 | 22, 25, 26 | sylanbrc 413 | . . . . . . . . . 10 # |
28 | 0zd 9024 | . . . . . . . . . . 11 # | |
29 | zleloe 9059 | . . . . . . . . . . 11 | |
30 | 28, 20, 29 | syl2anc 408 | . . . . . . . . . 10 # |
31 | 27, 30 | mtbird 647 | . . . . . . . . 9 # |
32 | zltnle 9058 | . . . . . . . . . 10 | |
33 | 20, 28, 32 | syl2anc 408 | . . . . . . . . 9 # |
34 | 31, 33 | mpbird 166 | . . . . . . . 8 # |
35 | 20 | zred 9131 | . . . . . . . . 9 # |
36 | 35 | lt0neg1d 8245 | . . . . . . . 8 # |
37 | 34, 36 | mpbid 146 | . . . . . . 7 # |
38 | elnnz 9022 | . . . . . . 7 | |
39 | 21, 37, 38 | sylanbrc 413 | . . . . . 6 # |
40 | 19, 39 | ffvelrnd 5524 | . . . . 5 # |
41 | 2 | ad2antrr 479 | . . . . . 6 # |
42 | simpll3 1007 | . . . . . . 7 # # | |
43 | 31, 42 | ecased 1312 | . . . . . 6 # # |
44 | 41, 43, 39 | exp3vallem 10249 | . . . . 5 # # |
45 | 40, 44 | recclapd 8508 | . . . 4 # |
46 | 0zd 9024 | . . . . 5 # | |
47 | simpl2 970 | . . . . 5 # | |
48 | zdclt 9086 | . . . . 5 DECID | |
49 | 46, 47, 48 | syl2anc 408 | . . . 4 # DECID |
50 | 18, 45, 49 | ifcldadc 3471 | . . 3 # |
51 | 0zd 9024 | . . . 4 # | |
52 | zdceq 9084 | . . . 4 DECID | |
53 | 13, 51, 52 | syl2anc 408 | . . 3 # DECID |
54 | 1, 50, 53 | ifcldadc 3471 | . 2 # |
55 | sneq 3508 | . . . . . . . 8 | |
56 | 55 | xpeq2d 4533 | . . . . . . 7 |
57 | 56 | seqeq3d 10181 | . . . . . 6 |
58 | 57 | fveq1d 5391 | . . . . 5 |
59 | 57 | fveq1d 5391 | . . . . . 6 |
60 | 59 | oveq2d 5758 | . . . . 5 |
61 | 58, 60 | ifeq12d 3461 | . . . 4 |
62 | 61 | ifeq2d 3460 | . . 3 |
63 | eqeq1 2124 | . . . 4 | |
64 | breq2 3903 | . . . . 5 | |
65 | fveq2 5389 | . . . . 5 | |
66 | negeq 7923 | . . . . . . 7 | |
67 | 66 | fveq2d 5393 | . . . . . 6 |
68 | 67 | oveq2d 5758 | . . . . 5 |
69 | 64, 65, 68 | ifbieq12d 3468 | . . . 4 |
70 | 63, 69 | ifbieq2d 3466 | . . 3 |
71 | df-exp 10248 | . . 3 | |
72 | 62, 70, 71 | ovmpog 5873 | . 2 |
73 | 54, 72 | syld3an3 1246 | 1 # |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 682 DECID wdc 804 w3a 947 wceq 1316 wcel 1465 cif 3444 csn 3497 class class class wbr 3899 cxp 4507 wf 5089 cfv 5093 (class class class)co 5742 cc 7586 cc0 7588 c1 7589 cmul 7593 clt 7768 cle 7769 cneg 7902 # cap 8310 cdiv 8399 cn 8684 cz 9012 cseq 10173 cexp 10247 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-ilim 4261 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-frec 6256 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8304 df-ap 8311 df-div 8400 df-inn 8685 df-n0 8936 df-z 9013 df-uz 9283 df-seqfrec 10174 df-exp 10248 |
This theorem is referenced by: expnnval 10251 exp0 10252 expnegap0 10256 |
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