Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > expcllem | Unicode version |
Description: Lemma for proving nonnegative integer exponentiation closure laws. (Contributed by NM, 14-Dec-2005.) |
Ref | Expression |
---|---|
expcllem.1 | |
expcllem.2 | |
expcllem.3 |
Ref | Expression |
---|---|
expcllem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 8290 | . 2 | |
2 | oveq2 5540 | . . . . . . 7 | |
3 | 2 | eleq1d 2147 | . . . . . 6 |
4 | 3 | imbi2d 228 | . . . . 5 |
5 | oveq2 5540 | . . . . . . 7 | |
6 | 5 | eleq1d 2147 | . . . . . 6 |
7 | 6 | imbi2d 228 | . . . . 5 |
8 | oveq2 5540 | . . . . . . 7 | |
9 | 8 | eleq1d 2147 | . . . . . 6 |
10 | 9 | imbi2d 228 | . . . . 5 |
11 | oveq2 5540 | . . . . . . 7 | |
12 | 11 | eleq1d 2147 | . . . . . 6 |
13 | 12 | imbi2d 228 | . . . . 5 |
14 | expcllem.1 | . . . . . . . . 9 | |
15 | 14 | sseli 2995 | . . . . . . . 8 |
16 | exp1 9482 | . . . . . . . 8 | |
17 | 15, 16 | syl 14 | . . . . . . 7 |
18 | 17 | eleq1d 2147 | . . . . . 6 |
19 | 18 | ibir 175 | . . . . 5 |
20 | expcllem.2 | . . . . . . . . . . . 12 | |
21 | 20 | caovcl 5675 | . . . . . . . . . . 11 |
22 | 21 | ancoms 264 | . . . . . . . . . 10 |
23 | 22 | adantlr 460 | . . . . . . . . 9 |
24 | nnnn0 8295 | . . . . . . . . . . . 12 | |
25 | expp1 9483 | . . . . . . . . . . . 12 | |
26 | 15, 24, 25 | syl2an 283 | . . . . . . . . . . 11 |
27 | 26 | eleq1d 2147 | . . . . . . . . . 10 |
28 | 27 | adantr 270 | . . . . . . . . 9 |
29 | 23, 28 | mpbird 165 | . . . . . . . 8 |
30 | 29 | exp31 356 | . . . . . . 7 |
31 | 30 | com12 30 | . . . . . 6 |
32 | 31 | a2d 26 | . . . . 5 |
33 | 4, 7, 10, 13, 19, 32 | nnind 8055 | . . . 4 |
34 | 33 | impcom 123 | . . 3 |
35 | oveq2 5540 | . . . . 5 | |
36 | exp0 9480 | . . . . . 6 | |
37 | 15, 36 | syl 14 | . . . . 5 |
38 | 35, 37 | sylan9eqr 2135 | . . . 4 |
39 | expcllem.3 | . . . 4 | |
40 | 38, 39 | syl6eqel 2169 | . . 3 |
41 | 34, 40 | jaodan 743 | . 2 |
42 | 1, 41 | sylan2b 281 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wo 661 wceq 1284 wcel 1433 wss 2973 (class class class)co 5532 cc 6979 cc0 6981 c1 6982 caddc 6984 cmul 6986 cn 8039 cn0 8288 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: expcl2lemap 9488 nnexpcl 9489 nn0expcl 9490 zexpcl 9491 qexpcl 9492 reexpcl 9493 expcl 9494 expge0 9512 expge1 9513 |
Copyright terms: Public domain | W3C validator |