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Mirrors > Home > MPE Home > Th. List > modxai | Structured version Visualization version Unicode version |
Description: Add exponents in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.) (Revised by Mario Carneiro, 5-Feb-2015.) |
Ref | Expression |
---|---|
modxai.1 | |
modxai.2 | |
modxai.3 | |
modxai.4 | |
modxai.5 | |
modxai.6 | |
modxai.7 | |
modxai.8 | |
modxai.11 | |
modxai.12 | |
modxai.9 | |
modxai.10 |
Ref | Expression |
---|---|
modxai |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | modxai.9 | . . . . 5 | |
2 | 1 | oveq2i 6661 | . . . 4 |
3 | modxai.2 | . . . . . 6 | |
4 | 3 | nncni 11030 | . . . . 5 |
5 | modxai.3 | . . . . 5 | |
6 | modxai.7 | . . . . 5 | |
7 | expadd 12902 | . . . . 5 | |
8 | 4, 5, 6, 7 | mp3an 1424 | . . . 4 |
9 | 2, 8 | eqtr3i 2646 | . . 3 |
10 | 9 | oveq1i 6660 | . 2 |
11 | nnexpcl 12873 | . . . . . . . . 9 | |
12 | 3, 5, 11 | mp2an 708 | . . . . . . . 8 |
13 | 12 | nnzi 11401 | . . . . . . 7 |
14 | 13 | a1i 11 | . . . . . 6 |
15 | modxai.5 | . . . . . . . 8 | |
16 | 15 | nn0zi 11402 | . . . . . . 7 |
17 | 16 | a1i 11 | . . . . . 6 |
18 | nnexpcl 12873 | . . . . . . . . 9 | |
19 | 3, 6, 18 | mp2an 708 | . . . . . . . 8 |
20 | 19 | nnzi 11401 | . . . . . . 7 |
21 | 20 | a1i 11 | . . . . . 6 |
22 | modxai.8 | . . . . . . . 8 | |
23 | 22 | nn0zi 11402 | . . . . . . 7 |
24 | 23 | a1i 11 | . . . . . 6 |
25 | modxai.1 | . . . . . . . 8 | |
26 | nnrp 11842 | . . . . . . . 8 | |
27 | 25, 26 | ax-mp 5 | . . . . . . 7 |
28 | 27 | a1i 11 | . . . . . 6 |
29 | modxai.11 | . . . . . . 7 | |
30 | 29 | a1i 11 | . . . . . 6 |
31 | modxai.12 | . . . . . . 7 | |
32 | 31 | a1i 11 | . . . . . 6 |
33 | 14, 17, 21, 24, 28, 30, 32 | modmul12d 12724 | . . . . 5 |
34 | 33 | trud 1493 | . . . 4 |
35 | modxai.10 | . . . . . 6 | |
36 | modxai.4 | . . . . . . . . 9 | |
37 | zcn 11382 | . . . . . . . . 9 | |
38 | 36, 37 | ax-mp 5 | . . . . . . . 8 |
39 | 25 | nncni 11030 | . . . . . . . 8 |
40 | 38, 39 | mulcli 10045 | . . . . . . 7 |
41 | modxai.6 | . . . . . . . 8 | |
42 | 41 | nn0cni 11304 | . . . . . . 7 |
43 | 40, 42 | addcomi 10227 | . . . . . 6 |
44 | 35, 43 | eqtr3i 2646 | . . . . 5 |
45 | 44 | oveq1i 6660 | . . . 4 |
46 | 34, 45 | eqtri 2644 | . . 3 |
47 | 41 | nn0rei 11303 | . . . 4 |
48 | modcyc 12705 | . . . 4 | |
49 | 47, 27, 36, 48 | mp3an 1424 | . . 3 |
50 | 46, 49 | eqtri 2644 | . 2 |
51 | 10, 50 | eqtri 2644 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wceq 1483 wtru 1484 wcel 1990 (class class class)co 6650 cc 9934 cr 9935 caddc 9939 cmul 9941 cn 11020 cn0 11292 cz 11377 crp 11832 cmo 12668 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 ax-pre-sup 10014 |
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-rmo 2920 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-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 |
This theorem is referenced by: mod2xi 15773 modxp1i 15774 1259lem3 15840 1259lem4 15841 2503lem2 15845 4001lem3 15850 |
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