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Mirrors > Home > MPE Home > Th. List > expmul | Structured version Visualization version 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 6658 | . . . . . . 7 | |
2 | 1 | oveq2d 6666 | . . . . . 6 |
3 | oveq2 6658 | . . . . . 6 | |
4 | 2, 3 | eqeq12d 2637 | . . . . 5 |
5 | 4 | imbi2d 330 | . . . 4 |
6 | oveq2 6658 | . . . . . . 7 | |
7 | 6 | oveq2d 6666 | . . . . . 6 |
8 | oveq2 6658 | . . . . . 6 | |
9 | 7, 8 | eqeq12d 2637 | . . . . 5 |
10 | 9 | imbi2d 330 | . . . 4 |
11 | oveq2 6658 | . . . . . . 7 | |
12 | 11 | oveq2d 6666 | . . . . . 6 |
13 | oveq2 6658 | . . . . . 6 | |
14 | 12, 13 | eqeq12d 2637 | . . . . 5 |
15 | 14 | imbi2d 330 | . . . 4 |
16 | oveq2 6658 | . . . . . . 7 | |
17 | 16 | oveq2d 6666 | . . . . . 6 |
18 | oveq2 6658 | . . . . . 6 | |
19 | 17, 18 | eqeq12d 2637 | . . . . 5 |
20 | 19 | imbi2d 330 | . . . 4 |
21 | nn0cn 11302 | . . . . . . . 8 | |
22 | 21 | mul01d 10235 | . . . . . . 7 |
23 | 22 | oveq2d 6666 | . . . . . 6 |
24 | exp0 12864 | . . . . . 6 | |
25 | 23, 24 | sylan9eqr 2678 | . . . . 5 |
26 | expcl 12878 | . . . . . 6 | |
27 | exp0 12864 | . . . . . 6 | |
28 | 26, 27 | syl 17 | . . . . 5 |
29 | 25, 28 | eqtr4d 2659 | . . . 4 |
30 | oveq1 6657 | . . . . . . 7 | |
31 | nn0cn 11302 | . . . . . . . . . . . 12 | |
32 | ax-1cn 9994 | . . . . . . . . . . . . . 14 | |
33 | adddi 10025 | . . . . . . . . . . . . . 14 | |
34 | 32, 33 | mp3an3 1413 | . . . . . . . . . . . . 13 |
35 | mulid1 10037 | . . . . . . . . . . . . . . 15 | |
36 | 35 | adantr 481 | . . . . . . . . . . . . . 14 |
37 | 36 | oveq2d 6666 | . . . . . . . . . . . . 13 |
38 | 34, 37 | eqtrd 2656 | . . . . . . . . . . . 12 |
39 | 21, 31, 38 | syl2an 494 | . . . . . . . . . . 11 |
40 | 39 | adantll 750 | . . . . . . . . . 10 |
41 | 40 | oveq2d 6666 | . . . . . . . . 9 |
42 | simpll 790 | . . . . . . . . . 10 | |
43 | nn0mulcl 11329 | . . . . . . . . . . 11 | |
44 | 43 | adantll 750 | . . . . . . . . . 10 |
45 | simplr 792 | . . . . . . . . . 10 | |
46 | expadd 12902 | . . . . . . . . . 10 | |
47 | 42, 44, 45, 46 | syl3anc 1326 | . . . . . . . . 9 |
48 | 41, 47 | eqtrd 2656 | . . . . . . . 8 |
49 | expp1 12867 | . . . . . . . . 9 | |
50 | 26, 49 | sylan 488 | . . . . . . . 8 |
51 | 48, 50 | eqeq12d 2637 | . . . . . . 7 |
52 | 30, 51 | syl5ibr 236 | . . . . . 6 |
53 | 52 | expcom 451 | . . . . 5 |
54 | 53 | a2d 29 | . . . 4 |
55 | 5, 10, 15, 20, 29, 54 | nn0ind 11472 | . . 3 |
56 | 55 | expdcom 455 | . 2 |
57 | 56 | 3imp 1256 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 (class class class)co 6650 cc 9934 cc0 9936 c1 9937 caddc 9939 cmul 9941 cn0 11292 cexp 12860 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-seq 12802 df-exp 12861 |
This theorem is referenced by: expmulz 12906 expnass 12970 expmuld 13011 mcubic 24574 quart1 24583 log2cnv 24671 log2ublem2 24674 log2ub 24676 basellem3 24809 bclbnd 25005 hgt750lemd 30726 hgt750lem 30729 fmtnoprmfac1lem 41476 41prothprmlem2 41535 |
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