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Mirrors > Home > ILE Home > Th. List > expmul | Unicode version |
Description: Product of exponents law for nonnegative 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 5844 | . . . . . . 7 | |
2 | 1 | oveq2d 5852 | . . . . . 6 |
3 | oveq2 5844 | . . . . . 6 | |
4 | 2, 3 | eqeq12d 2179 | . . . . 5 |
5 | 4 | imbi2d 229 | . . . 4 |
6 | oveq2 5844 | . . . . . . 7 | |
7 | 6 | oveq2d 5852 | . . . . . 6 |
8 | oveq2 5844 | . . . . . 6 | |
9 | 7, 8 | eqeq12d 2179 | . . . . 5 |
10 | 9 | imbi2d 229 | . . . 4 |
11 | oveq2 5844 | . . . . . . 7 | |
12 | 11 | oveq2d 5852 | . . . . . 6 |
13 | oveq2 5844 | . . . . . 6 | |
14 | 12, 13 | eqeq12d 2179 | . . . . 5 |
15 | 14 | imbi2d 229 | . . . 4 |
16 | oveq2 5844 | . . . . . . 7 | |
17 | 16 | oveq2d 5852 | . . . . . 6 |
18 | oveq2 5844 | . . . . . 6 | |
19 | 17, 18 | eqeq12d 2179 | . . . . 5 |
20 | 19 | imbi2d 229 | . . . 4 |
21 | nn0cn 9115 | . . . . . . . 8 | |
22 | 21 | mul01d 8282 | . . . . . . 7 |
23 | 22 | oveq2d 5852 | . . . . . 6 |
24 | exp0 10449 | . . . . . 6 | |
25 | 23, 24 | sylan9eqr 2219 | . . . . 5 |
26 | expcl 10463 | . . . . . 6 | |
27 | exp0 10449 | . . . . . 6 | |
28 | 26, 27 | syl 14 | . . . . 5 |
29 | 25, 28 | eqtr4d 2200 | . . . 4 |
30 | oveq1 5843 | . . . . . . 7 | |
31 | nn0cn 9115 | . . . . . . . . . . . 12 | |
32 | ax-1cn 7837 | . . . . . . . . . . . . . 14 | |
33 | adddi 7876 | . . . . . . . . . . . . . 14 | |
34 | 32, 33 | mp3an3 1315 | . . . . . . . . . . . . 13 |
35 | mulid1 7887 | . . . . . . . . . . . . . . 15 | |
36 | 35 | adantr 274 | . . . . . . . . . . . . . 14 |
37 | 36 | oveq2d 5852 | . . . . . . . . . . . . 13 |
38 | 34, 37 | eqtrd 2197 | . . . . . . . . . . . 12 |
39 | 21, 31, 38 | syl2an 287 | . . . . . . . . . . 11 |
40 | 39 | adantll 468 | . . . . . . . . . 10 |
41 | 40 | oveq2d 5852 | . . . . . . . . 9 |
42 | simpll 519 | . . . . . . . . . 10 | |
43 | nn0mulcl 9141 | . . . . . . . . . . 11 | |
44 | 43 | adantll 468 | . . . . . . . . . 10 |
45 | simplr 520 | . . . . . . . . . 10 | |
46 | expadd 10487 | . . . . . . . . . 10 | |
47 | 42, 44, 45, 46 | syl3anc 1227 | . . . . . . . . 9 |
48 | 41, 47 | eqtrd 2197 | . . . . . . . 8 |
49 | expp1 10452 | . . . . . . . . 9 | |
50 | 26, 49 | sylan 281 | . . . . . . . 8 |
51 | 48, 50 | eqeq12d 2179 | . . . . . . 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 9296 | . . 3 |
56 | 55 | expdcom 1429 | . 2 |
57 | 56 | 3imp 1182 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 967 wceq 1342 wcel 2135 (class class class)co 5836 cc 7742 cc0 7744 c1 7745 caddc 7747 cmul 7749 cn0 9105 cexp 10444 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-frec 6350 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-n0 9106 df-z 9183 df-uz 9458 df-seqfrec 10371 df-exp 10445 |
This theorem is referenced by: expmulzap 10491 expnass 10550 expmuld 10580 |
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