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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 9036 | . . 3 | |
2 | simpl 108 | . . . . . 6 # | |
3 | simpl 108 | . . . . . 6 # | |
4 | 2, 3 | anim12i 336 | . . . . 5 # # |
5 | mulexp 10300 | . . . . . 6 | |
6 | 5 | 3expa 1166 | . . . . 5 |
7 | 4, 6 | sylan 281 | . . . 4 # # |
8 | simplll 507 | . . . . . . 7 # # | |
9 | simplrl 509 | . . . . . . 7 # # | |
10 | 8, 9 | mulcld 7754 | . . . . . 6 # # |
11 | simpllr 508 | . . . . . . 7 # # # | |
12 | simplrr 510 | . . . . . . 7 # # # | |
13 | 8, 9, 11, 12 | mulap0d 8387 | . . . . . 6 # # # |
14 | recn 7721 | . . . . . . 7 | |
15 | 14 | ad2antrl 481 | . . . . . 6 # # |
16 | nnnn0 8952 | . . . . . . 7 | |
17 | 16 | ad2antll 482 | . . . . . 6 # # |
18 | expineg2 10270 | . . . . . 6 # | |
19 | 10, 13, 15, 17, 18 | syl22anc 1202 | . . . . 5 # # |
20 | expineg2 10270 | . . . . . . . 8 # | |
21 | 8, 11, 15, 17, 20 | syl22anc 1202 | . . . . . . 7 # # |
22 | expineg2 10270 | . . . . . . . 8 # | |
23 | 9, 12, 15, 17, 22 | syl22anc 1202 | . . . . . . 7 # # |
24 | 21, 23 | oveq12d 5760 | . . . . . 6 # # |
25 | mulexp 10300 | . . . . . . . . . 10 | |
26 | 8, 9, 17, 25 | syl3anc 1201 | . . . . . . . . 9 # # |
27 | 26 | oveq2d 5758 | . . . . . . . 8 # # |
28 | 1t1e1 8840 | . . . . . . . . 9 | |
29 | 28 | oveq1i 5752 | . . . . . . . 8 |
30 | 27, 29 | syl6eqr 2168 | . . . . . . 7 # # |
31 | expcl 10279 | . . . . . . . . 9 | |
32 | 8, 17, 31 | syl2anc 408 | . . . . . . . 8 # # |
33 | nnz 9041 | . . . . . . . . . 10 | |
34 | 33 | ad2antll 482 | . . . . . . . . 9 # # |
35 | expap0i 10293 | . . . . . . . . 9 # # | |
36 | 8, 11, 34, 35 | syl3anc 1201 | . . . . . . . 8 # # # |
37 | expcl 10279 | . . . . . . . . 9 | |
38 | 9, 17, 37 | syl2anc 408 | . . . . . . . 8 # # |
39 | expap0i 10293 | . . . . . . . . 9 # # | |
40 | 9, 12, 34, 39 | syl3anc 1201 | . . . . . . . 8 # # # |
41 | ax-1cn 7681 | . . . . . . . . 9 | |
42 | divmuldivap 8440 | . . . . . . . . 9 # # | |
43 | 41, 41, 42 | mpanl12 432 | . . . . . . . 8 # # |
44 | 32, 36, 38, 40, 43 | syl22anc 1202 | . . . . . . 7 # # |
45 | 30, 44 | eqtr4d 2153 | . . . . . 6 # # |
46 | 24, 45 | eqtr4d 2153 | . . . . 5 # # |
47 | 19, 46 | eqtr4d 2153 | . . . 4 # # |
48 | 7, 47 | jaodan 771 | . . 3 # # |
49 | 1, 48 | sylan2b 285 | . 2 # # |
50 | 49 | 3impa 1161 | 1 # # |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 682 w3a 947 wceq 1316 wcel 1465 class class class wbr 3899 (class class class)co 5742 cc 7586 cr 7587 cc0 7588 c1 7589 cmul 7593 cneg 7902 # cap 8311 cdiv 8400 cn 8688 cn0 8945 cz 9022 cexp 10260 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-ilim 4261 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-frec 6256 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 df-div 8401 df-inn 8689 df-n0 8946 df-z 9023 df-uz 9295 df-seqfrec 10187 df-exp 10261 |
This theorem is referenced by: exprecap 10302 |
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