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Mirrors > Home > ILE Home > Th. List > m1expcl2 | Unicode version |
Description: Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.) |
Ref | Expression |
---|---|
m1expcl2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neg1cn 8022 | . . 3 | |
2 | prid1g 3474 | . . 3 | |
3 | 1, 2 | ax-mp 7 | . 2 |
4 | neg1ap0 8026 | . 2 # | |
5 | ax-1cn 6977 | . . . 4 | |
6 | prssi 3522 | . . . 4 | |
7 | 1, 5, 6 | mp2an 402 | . . 3 |
8 | elpri 3398 | . . . . 5 | |
9 | 7 | sseli 2941 | . . . . . . . . 9 |
10 | 9 | mulm1d 7407 | . . . . . . . 8 |
11 | elpri 3398 | . . . . . . . . 9 | |
12 | negeq 7204 | . . . . . . . . . . 11 | |
13 | negneg1e1 8027 | . . . . . . . . . . . 12 | |
14 | 1ex 7022 | . . . . . . . . . . . . 13 | |
15 | 14 | prid2 3477 | . . . . . . . . . . . 12 |
16 | 13, 15 | eqeltri 2110 | . . . . . . . . . . 11 |
17 | 12, 16 | syl6eqel 2128 | . . . . . . . . . 10 |
18 | negeq 7204 | . . . . . . . . . . 11 | |
19 | 18, 3 | syl6eqel 2128 | . . . . . . . . . 10 |
20 | 17, 19 | jaoi 636 | . . . . . . . . 9 |
21 | 11, 20 | syl 14 | . . . . . . . 8 |
22 | 10, 21 | eqeltrd 2114 | . . . . . . 7 |
23 | oveq1 5519 | . . . . . . . 8 | |
24 | 23 | eleq1d 2106 | . . . . . . 7 |
25 | 22, 24 | syl5ibr 145 | . . . . . 6 |
26 | 9 | mulid2d 7045 | . . . . . . . 8 |
27 | id 19 | . . . . . . . 8 | |
28 | 26, 27 | eqeltrd 2114 | . . . . . . 7 |
29 | oveq1 5519 | . . . . . . . 8 | |
30 | 29 | eleq1d 2106 | . . . . . . 7 |
31 | 28, 30 | syl5ibr 145 | . . . . . 6 |
32 | 25, 31 | jaoi 636 | . . . . 5 |
33 | 8, 32 | syl 14 | . . . 4 |
34 | 33 | imp 115 | . . 3 |
35 | oveq2 5520 | . . . . . . 7 | |
36 | 1ap0 7581 | . . . . . . . . . 10 # | |
37 | divneg2ap 7712 | . . . . . . . . . 10 # | |
38 | 5, 5, 36, 37 | mp3an 1232 | . . . . . . . . 9 |
39 | 1div1e1 7681 | . . . . . . . . . 10 | |
40 | 39 | negeqi 7205 | . . . . . . . . 9 |
41 | 38, 40 | eqtr3i 2062 | . . . . . . . 8 |
42 | 41, 3 | eqeltri 2110 | . . . . . . 7 |
43 | 35, 42 | syl6eqel 2128 | . . . . . 6 |
44 | oveq2 5520 | . . . . . . 7 | |
45 | 39, 15 | eqeltri 2110 | . . . . . . 7 |
46 | 44, 45 | syl6eqel 2128 | . . . . . 6 |
47 | 43, 46 | jaoi 636 | . . . . 5 |
48 | 8, 47 | syl 14 | . . . 4 |
49 | 48 | adantr 261 | . . 3 # |
50 | 7, 34, 15, 49 | expcl2lemap 9267 | . 2 # |
51 | 3, 4, 50 | mp3an12 1222 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wo 629 wceq 1243 wcel 1393 wss 2917 cpr 3376 class class class wbr 3764 (class class class)co 5512 cc 6887 cc0 6889 c1 6890 cmul 6894 cneg 7183 # cap 7572 cdiv 7651 cz 8245 cexp 9254 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-mulrcl 6983 ax-addcom 6984 ax-mulcom 6985 ax-addass 6986 ax-mulass 6987 ax-distr 6988 ax-i2m1 6989 ax-1rid 6991 ax-0id 6992 ax-rnegex 6993 ax-precex 6994 ax-cnre 6995 ax-pre-ltirr 6996 ax-pre-ltwlin 6997 ax-pre-lttrn 6998 ax-pre-apti 6999 ax-pre-ltadd 7000 ax-pre-mulgt0 7001 ax-pre-mulext 7002 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-reu 2313 df-rmo 2314 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-if 3332 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-frec 5978 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-iltp 6568 df-enr 6811 df-nr 6812 df-ltr 6815 df-0r 6816 df-1r 6817 df-0 6896 df-1 6897 df-r 6899 df-lt 6902 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 df-sub 7184 df-neg 7185 df-reap 7566 df-ap 7573 df-div 7652 df-inn 7915 df-n0 8182 df-z 8246 df-uz 8474 df-iseq 9212 df-iexp 9255 |
This theorem is referenced by: m1expcl 9278 |
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