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Mirrors > Home > ILE Home > Th. List > expmul | Unicode version |
Description: Product of exponents law for positive integer exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by NM, 4-Jan-2006.) |
Ref | Expression |
---|---|
expmul |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 5782 | . . . . . . 7 | |
2 | 1 | oveq2d 5790 | . . . . . 6 |
3 | oveq2 5782 | . . . . . 6 | |
4 | 2, 3 | eqeq12d 2154 | . . . . 5 |
5 | 4 | imbi2d 229 | . . . 4 |
6 | oveq2 5782 | . . . . . . 7 | |
7 | 6 | oveq2d 5790 | . . . . . 6 |
8 | oveq2 5782 | . . . . . 6 | |
9 | 7, 8 | eqeq12d 2154 | . . . . 5 |
10 | 9 | imbi2d 229 | . . . 4 |
11 | oveq2 5782 | . . . . . . 7 | |
12 | 11 | oveq2d 5790 | . . . . . 6 |
13 | oveq2 5782 | . . . . . 6 | |
14 | 12, 13 | eqeq12d 2154 | . . . . 5 |
15 | 14 | imbi2d 229 | . . . 4 |
16 | oveq2 5782 | . . . . . . 7 | |
17 | 16 | oveq2d 5790 | . . . . . 6 |
18 | oveq2 5782 | . . . . . 6 | |
19 | 17, 18 | eqeq12d 2154 | . . . . 5 |
20 | 19 | imbi2d 229 | . . . 4 |
21 | nn0cn 8987 | . . . . . . . 8 | |
22 | 21 | mul01d 8155 | . . . . . . 7 |
23 | 22 | oveq2d 5790 | . . . . . 6 |
24 | exp0 10297 | . . . . . 6 | |
25 | 23, 24 | sylan9eqr 2194 | . . . . 5 |
26 | expcl 10311 | . . . . . 6 | |
27 | exp0 10297 | . . . . . 6 | |
28 | 26, 27 | syl 14 | . . . . 5 |
29 | 25, 28 | eqtr4d 2175 | . . . 4 |
30 | oveq1 5781 | . . . . . . 7 | |
31 | nn0cn 8987 | . . . . . . . . . . . 12 | |
32 | ax-1cn 7713 | . . . . . . . . . . . . . 14 | |
33 | adddi 7752 | . . . . . . . . . . . . . 14 | |
34 | 32, 33 | mp3an3 1304 | . . . . . . . . . . . . 13 |
35 | mulid1 7763 | . . . . . . . . . . . . . . 15 | |
36 | 35 | adantr 274 | . . . . . . . . . . . . . 14 |
37 | 36 | oveq2d 5790 | . . . . . . . . . . . . 13 |
38 | 34, 37 | eqtrd 2172 | . . . . . . . . . . . 12 |
39 | 21, 31, 38 | syl2an 287 | . . . . . . . . . . 11 |
40 | 39 | adantll 467 | . . . . . . . . . 10 |
41 | 40 | oveq2d 5790 | . . . . . . . . 9 |
42 | simpll 518 | . . . . . . . . . 10 | |
43 | nn0mulcl 9013 | . . . . . . . . . . 11 | |
44 | 43 | adantll 467 | . . . . . . . . . 10 |
45 | simplr 519 | . . . . . . . . . 10 | |
46 | expadd 10335 | . . . . . . . . . 10 | |
47 | 42, 44, 45, 46 | syl3anc 1216 | . . . . . . . . 9 |
48 | 41, 47 | eqtrd 2172 | . . . . . . . 8 |
49 | expp1 10300 | . . . . . . . . 9 | |
50 | 26, 49 | sylan 281 | . . . . . . . 8 |
51 | 48, 50 | eqeq12d 2154 | . . . . . . 7 |
52 | 30, 51 | syl5ibr 155 | . . . . . 6 |
53 | 52 | expcom 115 | . . . . 5 |
54 | 53 | a2d 26 | . . . 4 |
55 | 5, 10, 15, 20, 29, 54 | nn0ind 9165 | . . 3 |
56 | 55 | expdcom 1418 | . 2 |
57 | 56 | 3imp 1175 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wcel 1480 (class class class)co 5774 cc 7618 cc0 7620 c1 7621 caddc 7623 cmul 7625 cn0 8977 cexp 10292 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-n0 8978 df-z 9055 df-uz 9327 df-seqfrec 10219 df-exp 10293 |
This theorem is referenced by: expmulzap 10339 expnass 10398 expmuld 10427 |
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