Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > mulexpzap | Unicode version |
Description: Integer exponentiation of a product. (Contributed by Jim Kingdon, 10-Jun-2020.) |
Ref | Expression |
---|---|
mulexpzap | # # |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elznn0nn 8259 | . . 3 | |
2 | simpl 102 | . . . . . 6 # | |
3 | simpl 102 | . . . . . 6 # | |
4 | 2, 3 | anim12i 321 | . . . . 5 # # |
5 | mulexp 9294 | . . . . . 6 | |
6 | 5 | 3expa 1104 | . . . . 5 |
7 | 4, 6 | sylan 267 | . . . 4 # # |
8 | simplll 485 | . . . . . . 7 # # | |
9 | simplrl 487 | . . . . . . 7 # # | |
10 | 8, 9 | mulcld 7047 | . . . . . 6 # # |
11 | simpllr 486 | . . . . . . 7 # # # | |
12 | simplrr 488 | . . . . . . 7 # # # | |
13 | 8, 9, 11, 12 | mulap0d 7639 | . . . . . 6 # # # |
14 | recn 7014 | . . . . . . 7 | |
15 | 14 | ad2antrl 459 | . . . . . 6 # # |
16 | nnnn0 8188 | . . . . . . 7 | |
17 | 16 | ad2antll 460 | . . . . . 6 # # |
18 | expineg2 9264 | . . . . . 6 # | |
19 | 10, 13, 15, 17, 18 | syl22anc 1136 | . . . . 5 # # |
20 | expineg2 9264 | . . . . . . . 8 # | |
21 | 8, 11, 15, 17, 20 | syl22anc 1136 | . . . . . . 7 # # |
22 | expineg2 9264 | . . . . . . . 8 # | |
23 | 9, 12, 15, 17, 22 | syl22anc 1136 | . . . . . . 7 # # |
24 | 21, 23 | oveq12d 5530 | . . . . . 6 # # |
25 | mulexp 9294 | . . . . . . . . . 10 | |
26 | 8, 9, 17, 25 | syl3anc 1135 | . . . . . . . . 9 # # |
27 | 26 | oveq2d 5528 | . . . . . . . 8 # # |
28 | 1t1e1 8067 | . . . . . . . . 9 | |
29 | 28 | oveq1i 5522 | . . . . . . . 8 |
30 | 27, 29 | syl6eqr 2090 | . . . . . . 7 # # |
31 | expcl 9273 | . . . . . . . . 9 | |
32 | 8, 17, 31 | syl2anc 391 | . . . . . . . 8 # # |
33 | nnz 8264 | . . . . . . . . . 10 | |
34 | 33 | ad2antll 460 | . . . . . . . . 9 # # |
35 | expap0i 9287 | . . . . . . . . 9 # # | |
36 | 8, 11, 34, 35 | syl3anc 1135 | . . . . . . . 8 # # # |
37 | expcl 9273 | . . . . . . . . 9 | |
38 | 9, 17, 37 | syl2anc 391 | . . . . . . . 8 # # |
39 | expap0i 9287 | . . . . . . . . 9 # # | |
40 | 9, 12, 34, 39 | syl3anc 1135 | . . . . . . . 8 # # # |
41 | ax-1cn 6977 | . . . . . . . . 9 | |
42 | divmuldivap 7688 | . . . . . . . . 9 # # | |
43 | 41, 41, 42 | mpanl12 412 | . . . . . . . 8 # # |
44 | 32, 36, 38, 40, 43 | syl22anc 1136 | . . . . . . 7 # # |
45 | 30, 44 | eqtr4d 2075 | . . . . . 6 # # |
46 | 24, 45 | eqtr4d 2075 | . . . . 5 # # |
47 | 19, 46 | eqtr4d 2075 | . . . 4 # # |
48 | 7, 47 | jaodan 710 | . . 3 # # |
49 | 1, 48 | sylan2b 271 | . 2 # # |
50 | 49 | 3impa 1099 | 1 # # |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wo 629 w3a 885 wceq 1243 wcel 1393 class class class wbr 3764 (class class class)co 5512 cc 6887 cr 6888 cc0 6889 c1 6890 cmul 6894 cneg 7183 # cap 7572 cdiv 7651 cn 7914 cn0 8181 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: exprecap 9296 |
Copyright terms: Public domain | W3C validator |