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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 8933 | . . 3 | |
2 | prid1g 3663 | . . 3 | |
3 | 1, 2 | ax-mp 5 | . 2 |
4 | neg1ap0 8937 | . 2 # | |
5 | ax-1cn 7820 | . . . 4 | |
6 | prssi 3714 | . . . 4 | |
7 | 1, 5, 6 | mp2an 423 | . . 3 |
8 | elpri 3583 | . . . . 5 | |
9 | 7 | sseli 3124 | . . . . . . . . 9 |
10 | 9 | mulm1d 8280 | . . . . . . . 8 |
11 | elpri 3583 | . . . . . . . . 9 | |
12 | negeq 8063 | . . . . . . . . . . 11 | |
13 | negneg1e1 8938 | . . . . . . . . . . . 12 | |
14 | 1ex 7868 | . . . . . . . . . . . . 13 | |
15 | 14 | prid2 3666 | . . . . . . . . . . . 12 |
16 | 13, 15 | eqeltri 2230 | . . . . . . . . . . 11 |
17 | 12, 16 | eqeltrdi 2248 | . . . . . . . . . 10 |
18 | negeq 8063 | . . . . . . . . . . 11 | |
19 | 18, 3 | eqeltrdi 2248 | . . . . . . . . . 10 |
20 | 17, 19 | jaoi 706 | . . . . . . . . 9 |
21 | 11, 20 | syl 14 | . . . . . . . 8 |
22 | 10, 21 | eqeltrd 2234 | . . . . . . 7 |
23 | oveq1 5828 | . . . . . . . 8 | |
24 | 23 | eleq1d 2226 | . . . . . . 7 |
25 | 22, 24 | syl5ibr 155 | . . . . . 6 |
26 | 9 | mulid2d 7891 | . . . . . . . 8 |
27 | id 19 | . . . . . . . 8 | |
28 | 26, 27 | eqeltrd 2234 | . . . . . . 7 |
29 | oveq1 5828 | . . . . . . . 8 | |
30 | 29 | eleq1d 2226 | . . . . . . 7 |
31 | 28, 30 | syl5ibr 155 | . . . . . 6 |
32 | 25, 31 | jaoi 706 | . . . . 5 |
33 | 8, 32 | syl 14 | . . . 4 |
34 | 33 | imp 123 | . . 3 |
35 | oveq2 5829 | . . . . . . 7 | |
36 | 1ap0 8460 | . . . . . . . . . 10 # | |
37 | divneg2ap 8604 | . . . . . . . . . 10 # | |
38 | 5, 5, 36, 37 | mp3an 1319 | . . . . . . . . 9 |
39 | 1div1e1 8572 | . . . . . . . . . 10 | |
40 | 39 | negeqi 8064 | . . . . . . . . 9 |
41 | 38, 40 | eqtr3i 2180 | . . . . . . . 8 |
42 | 41, 3 | eqeltri 2230 | . . . . . . 7 |
43 | 35, 42 | eqeltrdi 2248 | . . . . . 6 |
44 | oveq2 5829 | . . . . . . 7 | |
45 | 39, 15 | eqeltri 2230 | . . . . . . 7 |
46 | 44, 45 | eqeltrdi 2248 | . . . . . 6 |
47 | 43, 46 | jaoi 706 | . . . . 5 |
48 | 8, 47 | syl 14 | . . . 4 |
49 | 48 | adantr 274 | . . 3 # |
50 | 7, 34, 15, 49 | expcl2lemap 10426 | . 2 # |
51 | 3, 4, 50 | mp3an12 1309 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wo 698 wceq 1335 wcel 2128 wss 3102 cpr 3561 class class class wbr 3965 (class class class)co 5821 cc 7725 cc0 7727 c1 7728 cmul 7732 cneg 8042 # cap 8451 cdiv 8540 cz 9162 cexp 10413 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 ax-iinf 4546 ax-cnex 7818 ax-resscn 7819 ax-1cn 7820 ax-1re 7821 ax-icn 7822 ax-addcl 7823 ax-addrcl 7824 ax-mulcl 7825 ax-mulrcl 7826 ax-addcom 7827 ax-mulcom 7828 ax-addass 7829 ax-mulass 7830 ax-distr 7831 ax-i2m1 7832 ax-0lt1 7833 ax-1rid 7834 ax-0id 7835 ax-rnegex 7836 ax-precex 7837 ax-cnre 7838 ax-pre-ltirr 7839 ax-pre-ltwlin 7840 ax-pre-lttrn 7841 ax-pre-apti 7842 ax-pre-ltadd 7843 ax-pre-mulgt0 7844 ax-pre-mulext 7845 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-ilim 4329 df-suc 4331 df-iom 4549 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-rn 4596 df-res 4597 df-ima 4598 df-iota 5134 df-fun 5171 df-fn 5172 df-f 5173 df-f1 5174 df-fo 5175 df-f1o 5176 df-fv 5177 df-riota 5777 df-ov 5824 df-oprab 5825 df-mpo 5826 df-1st 6085 df-2nd 6086 df-recs 6249 df-frec 6335 df-pnf 7909 df-mnf 7910 df-xr 7911 df-ltxr 7912 df-le 7913 df-sub 8043 df-neg 8044 df-reap 8445 df-ap 8452 df-div 8541 df-inn 8829 df-n0 9086 df-z 9163 df-uz 9435 df-seqfrec 10340 df-exp 10414 |
This theorem is referenced by: m1expcl 10437 m1expeven 10461 |
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