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Mirrors > Home > ILE Home > Th. List > m1exp1 | Unicode version |
Description: Exponentiation of negative one is one iff the exponent is even. (Contributed by AV, 20-Jun-2021.) |
Ref | Expression |
---|---|
m1exp1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2z 9082 | . . . . . . 7 | |
2 | divides 11495 | . . . . . . 7 | |
3 | 1, 2 | mpan 420 | . . . . . 6 |
4 | oveq2 5782 | . . . . . . . . 9 | |
5 | 4 | eqcoms 2142 | . . . . . . . 8 |
6 | zcn 9059 | . . . . . . . . . . 11 | |
7 | 2cnd 8793 | . . . . . . . . . . 11 | |
8 | 6, 7 | mulcomd 7787 | . . . . . . . . . 10 |
9 | 8 | oveq2d 5790 | . . . . . . . . 9 |
10 | m1expeven 10340 | . . . . . . . . 9 | |
11 | 9, 10 | eqtrd 2172 | . . . . . . . 8 |
12 | 5, 11 | sylan9eqr 2194 | . . . . . . 7 |
13 | 12 | rexlimiva 2544 | . . . . . 6 |
14 | 3, 13 | syl6bi 162 | . . . . 5 |
15 | 14 | impcom 124 | . . . 4 |
16 | simpl 108 | . . . 4 | |
17 | 15, 16 | 2thd 174 | . . 3 |
18 | 17 | expcom 115 | . 2 |
19 | 1ne0 8788 | . . . . . 6 | |
20 | eqcom 2141 | . . . . . . 7 | |
21 | ax-1cn 7713 | . . . . . . . 8 | |
22 | 21 | eqnegi 8501 | . . . . . . 7 |
23 | 20, 22 | bitri 183 | . . . . . 6 |
24 | 19, 23 | nemtbir 2397 | . . . . 5 |
25 | odd2np1 11570 | . . . . . . . 8 | |
26 | oveq2 5782 | . . . . . . . . . . 11 | |
27 | 26 | eqcoms 2142 | . . . . . . . . . 10 |
28 | neg1cn 8825 | . . . . . . . . . . . . 13 | |
29 | 28 | a1i 9 | . . . . . . . . . . . 12 |
30 | neg1ap0 8829 | . . . . . . . . . . . . 13 # | |
31 | 30 | a1i 9 | . . . . . . . . . . . 12 # |
32 | 1 | a1i 9 | . . . . . . . . . . . . 13 |
33 | id 19 | . . . . . . . . . . . . 13 | |
34 | 32, 33 | zmulcld 9179 | . . . . . . . . . . . 12 |
35 | 29, 31, 34 | expp1zapd 10433 | . . . . . . . . . . 11 |
36 | 10 | oveq1d 5789 | . . . . . . . . . . . 12 |
37 | 28 | mulid2i 7769 | . . . . . . . . . . . 12 |
38 | 36, 37 | syl6eq 2188 | . . . . . . . . . . 11 |
39 | 35, 38 | eqtrd 2172 | . . . . . . . . . 10 |
40 | 27, 39 | sylan9eqr 2194 | . . . . . . . . 9 |
41 | 40 | rexlimiva 2544 | . . . . . . . 8 |
42 | 25, 41 | syl6bi 162 | . . . . . . 7 |
43 | 42 | impcom 124 | . . . . . 6 |
44 | 43 | eqeq1d 2148 | . . . . 5 |
45 | 24, 44 | mtbiri 664 | . . . 4 |
46 | simpl 108 | . . . 4 | |
47 | 45, 46 | 2falsed 691 | . . 3 |
48 | 47 | expcom 115 | . 2 |
49 | zeo3 11565 | . 2 | |
50 | 18, 48, 49 | mpjaod 707 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wrex 2417 class class class wbr 3929 (class class class)co 5774 cc 7618 cc0 7620 c1 7621 caddc 7623 cmul 7625 cneg 7934 # cap 8343 c2 8771 cz 9054 cexp 10292 cdvds 11493 |
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 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-xor 1354 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-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-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 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-n0 8978 df-z 9055 df-uz 9327 df-seqfrec 10219 df-exp 10293 df-dvds 11494 |
This theorem is referenced by: (None) |
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