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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 8144 | . . 3 | |
2 | prid1g 3496 | . . 3 | |
3 | 1, 2 | ax-mp 7 | . 2 |
4 | neg1ap0 8148 | . 2 # | |
5 | ax-1cn 7069 | . . . 4 | |
6 | prssi 3543 | . . . 4 | |
7 | 1, 5, 6 | mp2an 416 | . . 3 |
8 | elpri 3421 | . . . . 5 | |
9 | 7 | sseli 2995 | . . . . . . . . 9 |
10 | 9 | mulm1d 7514 | . . . . . . . 8 |
11 | elpri 3421 | . . . . . . . . 9 | |
12 | negeq 7301 | . . . . . . . . . . 11 | |
13 | negneg1e1 8149 | . . . . . . . . . . . 12 | |
14 | 1ex 7114 | . . . . . . . . . . . . 13 | |
15 | 14 | prid2 3499 | . . . . . . . . . . . 12 |
16 | 13, 15 | eqeltri 2151 | . . . . . . . . . . 11 |
17 | 12, 16 | syl6eqel 2169 | . . . . . . . . . 10 |
18 | negeq 7301 | . . . . . . . . . . 11 | |
19 | 18, 3 | syl6eqel 2169 | . . . . . . . . . 10 |
20 | 17, 19 | jaoi 668 | . . . . . . . . 9 |
21 | 11, 20 | syl 14 | . . . . . . . 8 |
22 | 10, 21 | eqeltrd 2155 | . . . . . . 7 |
23 | oveq1 5539 | . . . . . . . 8 | |
24 | 23 | eleq1d 2147 | . . . . . . 7 |
25 | 22, 24 | syl5ibr 154 | . . . . . 6 |
26 | 9 | mulid2d 7137 | . . . . . . . 8 |
27 | id 19 | . . . . . . . 8 | |
28 | 26, 27 | eqeltrd 2155 | . . . . . . 7 |
29 | oveq1 5539 | . . . . . . . 8 | |
30 | 29 | eleq1d 2147 | . . . . . . 7 |
31 | 28, 30 | syl5ibr 154 | . . . . . 6 |
32 | 25, 31 | jaoi 668 | . . . . 5 |
33 | 8, 32 | syl 14 | . . . 4 |
34 | 33 | imp 122 | . . 3 |
35 | oveq2 5540 | . . . . . . 7 | |
36 | 1ap0 7690 | . . . . . . . . . 10 # | |
37 | divneg2ap 7824 | . . . . . . . . . 10 # | |
38 | 5, 5, 36, 37 | mp3an 1268 | . . . . . . . . 9 |
39 | 1div1e1 7792 | . . . . . . . . . 10 | |
40 | 39 | negeqi 7302 | . . . . . . . . 9 |
41 | 38, 40 | eqtr3i 2103 | . . . . . . . 8 |
42 | 41, 3 | eqeltri 2151 | . . . . . . 7 |
43 | 35, 42 | syl6eqel 2169 | . . . . . 6 |
44 | oveq2 5540 | . . . . . . 7 | |
45 | 39, 15 | eqeltri 2151 | . . . . . . 7 |
46 | 44, 45 | syl6eqel 2169 | . . . . . 6 |
47 | 43, 46 | jaoi 668 | . . . . 5 |
48 | 8, 47 | syl 14 | . . . 4 |
49 | 48 | adantr 270 | . . 3 # |
50 | 7, 34, 15, 49 | expcl2lemap 9488 | . 2 # |
51 | 3, 4, 50 | mp3an12 1258 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wo 661 wceq 1284 wcel 1433 wss 2973 cpr 3399 class class class wbr 3785 (class class class)co 5532 cc 6979 cc0 6981 c1 6982 cmul 6986 cneg 7280 # cap 7681 cdiv 7760 cz 8351 cexp 9475 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-if 3352 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-frec 6001 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-n0 8289 df-z 8352 df-uz 8620 df-iseq 9432 df-iexp 9476 |
This theorem is referenced by: m1expcl 9499 m1expeven 9523 |
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