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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 5861 | . . . . . . 7 | |
2 | 1 | oveq2d 5869 | . . . . . 6 |
3 | oveq2 5861 | . . . . . . 7 | |
4 | 3 | oveq2d 5869 | . . . . . 6 |
5 | 2, 4 | eqeq12d 2185 | . . . . 5 |
6 | 5 | imbi2d 229 | . . . 4 |
7 | oveq2 5861 | . . . . . . 7 | |
8 | 7 | oveq2d 5869 | . . . . . 6 |
9 | oveq2 5861 | . . . . . . 7 | |
10 | 9 | oveq2d 5869 | . . . . . 6 |
11 | 8, 10 | eqeq12d 2185 | . . . . 5 |
12 | 11 | imbi2d 229 | . . . 4 |
13 | oveq2 5861 | . . . . . . 7 | |
14 | 13 | oveq2d 5869 | . . . . . 6 |
15 | oveq2 5861 | . . . . . . 7 | |
16 | 15 | oveq2d 5869 | . . . . . 6 |
17 | 14, 16 | eqeq12d 2185 | . . . . 5 |
18 | 17 | imbi2d 229 | . . . 4 |
19 | oveq2 5861 | . . . . . . 7 | |
20 | 19 | oveq2d 5869 | . . . . . 6 |
21 | oveq2 5861 | . . . . . . 7 | |
22 | 21 | oveq2d 5869 | . . . . . 6 |
23 | 20, 22 | eqeq12d 2185 | . . . . 5 |
24 | 23 | imbi2d 229 | . . . 4 |
25 | nn0cn 9145 | . . . . . . . . 9 | |
26 | 25 | addid1d 8068 | . . . . . . . 8 |
27 | 26 | adantl 275 | . . . . . . 7 |
28 | 27 | oveq2d 5869 | . . . . . 6 |
29 | expcl 10494 | . . . . . . 7 | |
30 | 29 | mulid1d 7937 | . . . . . 6 |
31 | 28, 30 | eqtr4d 2206 | . . . . 5 |
32 | exp0 10480 | . . . . . . 7 | |
33 | 32 | adantr 274 | . . . . . 6 |
34 | 33 | oveq2d 5869 | . . . . 5 |
35 | 31, 34 | eqtr4d 2206 | . . . 4 |
36 | oveq1 5860 | . . . . . . 7 | |
37 | nn0cn 9145 | . . . . . . . . . . . 12 | |
38 | ax-1cn 7867 | . . . . . . . . . . . . 13 | |
39 | addass 7904 | . . . . . . . . . . . . 13 | |
40 | 38, 39 | mp3an3 1321 | . . . . . . . . . . . 12 |
41 | 25, 37, 40 | syl2an 287 | . . . . . . . . . . 11 |
42 | 41 | adantll 473 | . . . . . . . . . 10 |
43 | 42 | oveq2d 5869 | . . . . . . . . 9 |
44 | simpll 524 | . . . . . . . . . 10 | |
45 | nn0addcl 9170 | . . . . . . . . . . 11 | |
46 | 45 | adantll 473 | . . . . . . . . . 10 |
47 | expp1 10483 | . . . . . . . . . 10 | |
48 | 44, 46, 47 | syl2anc 409 | . . . . . . . . 9 |
49 | 43, 48 | eqtr3d 2205 | . . . . . . . 8 |
50 | expp1 10483 | . . . . . . . . . . 11 | |
51 | 50 | adantlr 474 | . . . . . . . . . 10 |
52 | 51 | oveq2d 5869 | . . . . . . . . 9 |
53 | 29 | adantr 274 | . . . . . . . . . 10 |
54 | expcl 10494 | . . . . . . . . . . 11 | |
55 | 54 | adantlr 474 | . . . . . . . . . 10 |
56 | 53, 55, 44 | mulassd 7943 | . . . . . . . . 9 |
57 | 52, 56 | eqtr4d 2206 | . . . . . . . 8 |
58 | 49, 57 | eqeq12d 2185 | . . . . . . 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 9326 | . . 3 |
63 | 62 | expdcom 1435 | . 2 |
64 | 63 | 3imp 1188 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 973 wceq 1348 wcel 2141 (class class class)co 5853 cc 7772 cc0 7774 c1 7775 caddc 7777 cmul 7779 cn0 9135 cexp 10475 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-n0 9136 df-z 9213 df-uz 9488 df-seqfrec 10402 df-exp 10476 |
This theorem is referenced by: expaddzaplem 10519 expaddzap 10520 expmul 10521 i4 10578 expaddd 10611 ef01bndlem 11719 |
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