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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 9240 | . . . . . . 7 | |
2 | divides 11751 | . . . . . . 7 | |
3 | 1, 2 | mpan 422 | . . . . . 6 |
4 | oveq2 5861 | . . . . . . . . 9 | |
5 | 4 | eqcoms 2173 | . . . . . . . 8 |
6 | zcn 9217 | . . . . . . . . . . 11 | |
7 | 2cnd 8951 | . . . . . . . . . . 11 | |
8 | 6, 7 | mulcomd 7941 | . . . . . . . . . 10 |
9 | 8 | oveq2d 5869 | . . . . . . . . 9 |
10 | m1expeven 10523 | . . . . . . . . 9 | |
11 | 9, 10 | eqtrd 2203 | . . . . . . . 8 |
12 | 5, 11 | sylan9eqr 2225 | . . . . . . 7 |
13 | 12 | rexlimiva 2582 | . . . . . 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 8946 | . . . . . 6 | |
20 | eqcom 2172 | . . . . . . 7 | |
21 | ax-1cn 7867 | . . . . . . . 8 | |
22 | 21 | eqnegi 8658 | . . . . . . 7 |
23 | 20, 22 | bitri 183 | . . . . . 6 |
24 | 19, 23 | nemtbir 2429 | . . . . 5 |
25 | odd2np1 11832 | . . . . . . . 8 | |
26 | oveq2 5861 | . . . . . . . . . . 11 | |
27 | 26 | eqcoms 2173 | . . . . . . . . . 10 |
28 | neg1cn 8983 | . . . . . . . . . . . . 13 | |
29 | 28 | a1i 9 | . . . . . . . . . . . 12 |
30 | neg1ap0 8987 | . . . . . . . . . . . . 13 # | |
31 | 30 | a1i 9 | . . . . . . . . . . . 12 # |
32 | 1 | a1i 9 | . . . . . . . . . . . . 13 |
33 | id 19 | . . . . . . . . . . . . 13 | |
34 | 32, 33 | zmulcld 9340 | . . . . . . . . . . . 12 |
35 | 29, 31, 34 | expp1zapd 10618 | . . . . . . . . . . 11 |
36 | 10 | oveq1d 5868 | . . . . . . . . . . . 12 |
37 | 28 | mulid2i 7923 | . . . . . . . . . . . 12 |
38 | 36, 37 | eqtrdi 2219 | . . . . . . . . . . 11 |
39 | 35, 38 | eqtrd 2203 | . . . . . . . . . 10 |
40 | 27, 39 | sylan9eqr 2225 | . . . . . . . . 9 |
41 | 40 | rexlimiva 2582 | . . . . . . . 8 |
42 | 25, 41 | syl6bi 162 | . . . . . . 7 |
43 | 42 | impcom 124 | . . . . . 6 |
44 | 43 | eqeq1d 2179 | . . . . 5 |
45 | 24, 44 | mtbiri 670 | . . . 4 |
46 | simpl 108 | . . . 4 | |
47 | 45, 46 | 2falsed 697 | . . 3 |
48 | 47 | expcom 115 | . 2 |
49 | zeo3 11827 | . 2 | |
50 | 18, 48, 49 | mpjaod 713 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1348 wcel 2141 wrex 2449 class class class wbr 3989 (class class class)co 5853 cc 7772 cc0 7774 c1 7775 caddc 7777 cmul 7779 cneg 8091 # cap 8500 c2 8929 cz 9212 cexp 10475 cdvds 11749 |
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 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-xor 1371 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-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-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 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-n0 9136 df-z 9213 df-uz 9488 df-seqfrec 10402 df-exp 10476 df-dvds 11750 |
This theorem is referenced by: (None) |
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