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Mirrors > Home > ILE Home > Th. List > expadd | Unicode version |
Description: Sum of exponents law for nonnegative integer exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by NM, 30-Nov-2004.) |
Ref | Expression |
---|---|
expadd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 5540 | . . . . . . 7 | |
2 | 1 | oveq2d 5548 | . . . . . 6 |
3 | oveq2 5540 | . . . . . . 7 | |
4 | 3 | oveq2d 5548 | . . . . . 6 |
5 | 2, 4 | eqeq12d 2095 | . . . . 5 |
6 | 5 | imbi2d 228 | . . . 4 |
7 | oveq2 5540 | . . . . . . 7 | |
8 | 7 | oveq2d 5548 | . . . . . 6 |
9 | oveq2 5540 | . . . . . . 7 | |
10 | 9 | oveq2d 5548 | . . . . . 6 |
11 | 8, 10 | eqeq12d 2095 | . . . . 5 |
12 | 11 | imbi2d 228 | . . . 4 |
13 | oveq2 5540 | . . . . . . 7 | |
14 | 13 | oveq2d 5548 | . . . . . 6 |
15 | oveq2 5540 | . . . . . . 7 | |
16 | 15 | oveq2d 5548 | . . . . . 6 |
17 | 14, 16 | eqeq12d 2095 | . . . . 5 |
18 | 17 | imbi2d 228 | . . . 4 |
19 | oveq2 5540 | . . . . . . 7 | |
20 | 19 | oveq2d 5548 | . . . . . 6 |
21 | oveq2 5540 | . . . . . . 7 | |
22 | 21 | oveq2d 5548 | . . . . . 6 |
23 | 20, 22 | eqeq12d 2095 | . . . . 5 |
24 | 23 | imbi2d 228 | . . . 4 |
25 | nn0cn 8298 | . . . . . . . . 9 | |
26 | 25 | addid1d 7257 | . . . . . . . 8 |
27 | 26 | adantl 271 | . . . . . . 7 |
28 | 27 | oveq2d 5548 | . . . . . 6 |
29 | expcl 9494 | . . . . . . 7 | |
30 | 29 | mulid1d 7136 | . . . . . 6 |
31 | 28, 30 | eqtr4d 2116 | . . . . 5 |
32 | exp0 9480 | . . . . . . 7 | |
33 | 32 | adantr 270 | . . . . . 6 |
34 | 33 | oveq2d 5548 | . . . . 5 |
35 | 31, 34 | eqtr4d 2116 | . . . 4 |
36 | oveq1 5539 | . . . . . . 7 | |
37 | nn0cn 8298 | . . . . . . . . . . . 12 | |
38 | ax-1cn 7069 | . . . . . . . . . . . . 13 | |
39 | addass 7103 | . . . . . . . . . . . . 13 | |
40 | 38, 39 | mp3an3 1257 | . . . . . . . . . . . 12 |
41 | 25, 37, 40 | syl2an 283 | . . . . . . . . . . 11 |
42 | 41 | adantll 459 | . . . . . . . . . 10 |
43 | 42 | oveq2d 5548 | . . . . . . . . 9 |
44 | simpll 495 | . . . . . . . . . 10 | |
45 | nn0addcl 8323 | . . . . . . . . . . 11 | |
46 | 45 | adantll 459 | . . . . . . . . . 10 |
47 | expp1 9483 | . . . . . . . . . 10 | |
48 | 44, 46, 47 | syl2anc 403 | . . . . . . . . 9 |
49 | 43, 48 | eqtr3d 2115 | . . . . . . . 8 |
50 | expp1 9483 | . . . . . . . . . . 11 | |
51 | 50 | adantlr 460 | . . . . . . . . . 10 |
52 | 51 | oveq2d 5548 | . . . . . . . . 9 |
53 | 29 | adantr 270 | . . . . . . . . . 10 |
54 | expcl 9494 | . . . . . . . . . . 11 | |
55 | 54 | adantlr 460 | . . . . . . . . . 10 |
56 | 53, 55, 44 | mulassd 7142 | . . . . . . . . 9 |
57 | 52, 56 | eqtr4d 2116 | . . . . . . . 8 |
58 | 49, 57 | eqeq12d 2095 | . . . . . . 7 |
59 | 36, 58 | syl5ibr 154 | . . . . . 6 |
60 | 59 | expcom 114 | . . . . 5 |
61 | 60 | a2d 26 | . . . 4 |
62 | 6, 12, 18, 24, 35, 61 | nn0ind 8461 | . . 3 |
63 | 62 | expdcom 1371 | . 2 |
64 | 63 | 3imp 1132 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 w3a 919 wceq 1284 wcel 1433 (class class class)co 5532 cc 6979 cc0 6981 c1 6982 caddc 6984 cmul 6986 cn0 8288 cexp 9475 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-if 3352 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-frec 6001 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-n0 8289 df-z 8352 df-uz 8620 df-iseq 9432 df-iexp 9476 |
This theorem is referenced by: expaddzaplem 9519 expaddzap 9520 expmul 9521 i4 9577 expaddd 9607 |
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