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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 5834 | . . . . . . 7 | |
2 | 1 | oveq2d 5842 | . . . . . 6 |
3 | oveq2 5834 | . . . . . . 7 | |
4 | 3 | oveq2d 5842 | . . . . . 6 |
5 | 2, 4 | eqeq12d 2172 | . . . . 5 |
6 | 5 | imbi2d 229 | . . . 4 |
7 | oveq2 5834 | . . . . . . 7 | |
8 | 7 | oveq2d 5842 | . . . . . 6 |
9 | oveq2 5834 | . . . . . . 7 | |
10 | 9 | oveq2d 5842 | . . . . . 6 |
11 | 8, 10 | eqeq12d 2172 | . . . . 5 |
12 | 11 | imbi2d 229 | . . . 4 |
13 | oveq2 5834 | . . . . . . 7 | |
14 | 13 | oveq2d 5842 | . . . . . 6 |
15 | oveq2 5834 | . . . . . . 7 | |
16 | 15 | oveq2d 5842 | . . . . . 6 |
17 | 14, 16 | eqeq12d 2172 | . . . . 5 |
18 | 17 | imbi2d 229 | . . . 4 |
19 | oveq2 5834 | . . . . . . 7 | |
20 | 19 | oveq2d 5842 | . . . . . 6 |
21 | oveq2 5834 | . . . . . . 7 | |
22 | 21 | oveq2d 5842 | . . . . . 6 |
23 | 20, 22 | eqeq12d 2172 | . . . . 5 |
24 | 23 | imbi2d 229 | . . . 4 |
25 | nn0cn 9105 | . . . . . . . . 9 | |
26 | 25 | addid1d 8028 | . . . . . . . 8 |
27 | 26 | adantl 275 | . . . . . . 7 |
28 | 27 | oveq2d 5842 | . . . . . 6 |
29 | expcl 10446 | . . . . . . 7 | |
30 | 29 | mulid1d 7897 | . . . . . 6 |
31 | 28, 30 | eqtr4d 2193 | . . . . 5 |
32 | exp0 10432 | . . . . . . 7 | |
33 | 32 | adantr 274 | . . . . . 6 |
34 | 33 | oveq2d 5842 | . . . . 5 |
35 | 31, 34 | eqtr4d 2193 | . . . 4 |
36 | oveq1 5833 | . . . . . . 7 | |
37 | nn0cn 9105 | . . . . . . . . . . . 12 | |
38 | ax-1cn 7827 | . . . . . . . . . . . . 13 | |
39 | addass 7864 | . . . . . . . . . . . . 13 | |
40 | 38, 39 | mp3an3 1308 | . . . . . . . . . . . 12 |
41 | 25, 37, 40 | syl2an 287 | . . . . . . . . . . 11 |
42 | 41 | adantll 468 | . . . . . . . . . 10 |
43 | 42 | oveq2d 5842 | . . . . . . . . 9 |
44 | simpll 519 | . . . . . . . . . 10 | |
45 | nn0addcl 9130 | . . . . . . . . . . 11 | |
46 | 45 | adantll 468 | . . . . . . . . . 10 |
47 | expp1 10435 | . . . . . . . . . 10 | |
48 | 44, 46, 47 | syl2anc 409 | . . . . . . . . 9 |
49 | 43, 48 | eqtr3d 2192 | . . . . . . . 8 |
50 | expp1 10435 | . . . . . . . . . . 11 | |
51 | 50 | adantlr 469 | . . . . . . . . . 10 |
52 | 51 | oveq2d 5842 | . . . . . . . . 9 |
53 | 29 | adantr 274 | . . . . . . . . . 10 |
54 | expcl 10446 | . . . . . . . . . . 11 | |
55 | 54 | adantlr 469 | . . . . . . . . . 10 |
56 | 53, 55, 44 | mulassd 7903 | . . . . . . . . 9 |
57 | 52, 56 | eqtr4d 2193 | . . . . . . . 8 |
58 | 49, 57 | eqeq12d 2172 | . . . . . . 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 9283 | . . 3 |
63 | 62 | expdcom 1422 | . 2 |
64 | 63 | 3imp 1176 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 963 wceq 1335 wcel 2128 (class class class)co 5826 cc 7732 cc0 7734 c1 7735 caddc 7737 cmul 7739 cn0 9095 cexp 10427 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4081 ax-sep 4084 ax-nul 4092 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-setind 4498 ax-iinf 4549 ax-cnex 7825 ax-resscn 7826 ax-1cn 7827 ax-1re 7828 ax-icn 7829 ax-addcl 7830 ax-addrcl 7831 ax-mulcl 7832 ax-mulrcl 7833 ax-addcom 7834 ax-mulcom 7835 ax-addass 7836 ax-mulass 7837 ax-distr 7838 ax-i2m1 7839 ax-0lt1 7840 ax-1rid 7841 ax-0id 7842 ax-rnegex 7843 ax-precex 7844 ax-cnre 7845 ax-pre-ltirr 7846 ax-pre-ltwlin 7847 ax-pre-lttrn 7848 ax-pre-apti 7849 ax-pre-ltadd 7850 ax-pre-mulgt0 7851 ax-pre-mulext 7852 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-if 3507 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-iun 3853 df-br 3968 df-opab 4028 df-mpt 4029 df-tr 4065 df-id 4255 df-po 4258 df-iso 4259 df-iord 4328 df-on 4330 df-ilim 4331 df-suc 4333 df-iom 4552 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-iota 5137 df-fun 5174 df-fn 5175 df-f 5176 df-f1 5177 df-fo 5178 df-f1o 5179 df-fv 5180 df-riota 5782 df-ov 5829 df-oprab 5830 df-mpo 5831 df-1st 6090 df-2nd 6091 df-recs 6254 df-frec 6340 df-pnf 7916 df-mnf 7917 df-xr 7918 df-ltxr 7919 df-le 7920 df-sub 8052 df-neg 8053 df-reap 8454 df-ap 8461 df-div 8550 df-inn 8839 df-n0 9096 df-z 9173 df-uz 9445 df-seqfrec 10354 df-exp 10428 |
This theorem is referenced by: expaddzaplem 10471 expaddzap 10472 expmul 10473 i4 10530 expaddd 10562 ef01bndlem 11664 |
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