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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 5750 | . . . . . . 7 | |
2 | 1 | oveq2d 5758 | . . . . . 6 |
3 | oveq2 5750 | . . . . . . 7 | |
4 | 3 | oveq2d 5758 | . . . . . 6 |
5 | 2, 4 | eqeq12d 2132 | . . . . 5 |
6 | 5 | imbi2d 229 | . . . 4 |
7 | oveq2 5750 | . . . . . . 7 | |
8 | 7 | oveq2d 5758 | . . . . . 6 |
9 | oveq2 5750 | . . . . . . 7 | |
10 | 9 | oveq2d 5758 | . . . . . 6 |
11 | 8, 10 | eqeq12d 2132 | . . . . 5 |
12 | 11 | imbi2d 229 | . . . 4 |
13 | oveq2 5750 | . . . . . . 7 | |
14 | 13 | oveq2d 5758 | . . . . . 6 |
15 | oveq2 5750 | . . . . . . 7 | |
16 | 15 | oveq2d 5758 | . . . . . 6 |
17 | 14, 16 | eqeq12d 2132 | . . . . 5 |
18 | 17 | imbi2d 229 | . . . 4 |
19 | oveq2 5750 | . . . . . . 7 | |
20 | 19 | oveq2d 5758 | . . . . . 6 |
21 | oveq2 5750 | . . . . . . 7 | |
22 | 21 | oveq2d 5758 | . . . . . 6 |
23 | 20, 22 | eqeq12d 2132 | . . . . 5 |
24 | 23 | imbi2d 229 | . . . 4 |
25 | nn0cn 8955 | . . . . . . . . 9 | |
26 | 25 | addid1d 7879 | . . . . . . . 8 |
27 | 26 | adantl 275 | . . . . . . 7 |
28 | 27 | oveq2d 5758 | . . . . . 6 |
29 | expcl 10279 | . . . . . . 7 | |
30 | 29 | mulid1d 7751 | . . . . . 6 |
31 | 28, 30 | eqtr4d 2153 | . . . . 5 |
32 | exp0 10265 | . . . . . . 7 | |
33 | 32 | adantr 274 | . . . . . 6 |
34 | 33 | oveq2d 5758 | . . . . 5 |
35 | 31, 34 | eqtr4d 2153 | . . . 4 |
36 | oveq1 5749 | . . . . . . 7 | |
37 | nn0cn 8955 | . . . . . . . . . . . 12 | |
38 | ax-1cn 7681 | . . . . . . . . . . . . 13 | |
39 | addass 7718 | . . . . . . . . . . . . 13 | |
40 | 38, 39 | mp3an3 1289 | . . . . . . . . . . . 12 |
41 | 25, 37, 40 | syl2an 287 | . . . . . . . . . . 11 |
42 | 41 | adantll 467 | . . . . . . . . . 10 |
43 | 42 | oveq2d 5758 | . . . . . . . . 9 |
44 | simpll 503 | . . . . . . . . . 10 | |
45 | nn0addcl 8980 | . . . . . . . . . . 11 | |
46 | 45 | adantll 467 | . . . . . . . . . 10 |
47 | expp1 10268 | . . . . . . . . . 10 | |
48 | 44, 46, 47 | syl2anc 408 | . . . . . . . . 9 |
49 | 43, 48 | eqtr3d 2152 | . . . . . . . 8 |
50 | expp1 10268 | . . . . . . . . . . 11 | |
51 | 50 | adantlr 468 | . . . . . . . . . 10 |
52 | 51 | oveq2d 5758 | . . . . . . . . 9 |
53 | 29 | adantr 274 | . . . . . . . . . 10 |
54 | expcl 10279 | . . . . . . . . . . 11 | |
55 | 54 | adantlr 468 | . . . . . . . . . 10 |
56 | 53, 55, 44 | mulassd 7757 | . . . . . . . . 9 |
57 | 52, 56 | eqtr4d 2153 | . . . . . . . 8 |
58 | 49, 57 | eqeq12d 2132 | . . . . . . 7 |
59 | 36, 58 | syl5ibr 155 | . . . . . 6 |
60 | 59 | expcom 115 | . . . . 5 |
61 | 60 | a2d 26 | . . . 4 |
62 | 6, 12, 18, 24, 35, 61 | nn0ind 9133 | . . 3 |
63 | 62 | expdcom 1403 | . 2 |
64 | 63 | 3imp 1160 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 947 wceq 1316 wcel 1465 (class class class)co 5742 cc 7586 cc0 7588 c1 7589 caddc 7591 cmul 7593 cn0 8945 cexp 10260 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-ilim 4261 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-frec 6256 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 df-div 8401 df-inn 8689 df-n0 8946 df-z 9023 df-uz 9295 df-seqfrec 10187 df-exp 10261 |
This theorem is referenced by: expaddzaplem 10304 expaddzap 10305 expmul 10306 i4 10363 expaddd 10394 ef01bndlem 11390 |
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