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Mirrors > Home > ILE Home > Th. List > binom3 | Unicode version |
Description: The cube of a binomial. (Contributed by Mario Carneiro, 24-Apr-2015.) |
Ref | Expression |
---|---|
binom3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-3 8780 | . . . 4 | |
2 | 1 | oveq2i 5785 | . . 3 |
3 | addcl 7745 | . . . 4 | |
4 | 2nn0 8994 | . . . 4 | |
5 | expp1 10300 | . . . 4 | |
6 | 3, 4, 5 | sylancl 409 | . . 3 |
7 | 2, 6 | syl5eq 2184 | . 2 |
8 | sqcl 10354 | . . . . 5 | |
9 | 3, 8 | syl 14 | . . . 4 |
10 | simpl 108 | . . . 4 | |
11 | simpr 109 | . . . 4 | |
12 | 9, 10, 11 | adddid 7790 | . . 3 |
13 | binom2 10403 | . . . . . 6 | |
14 | 13 | oveq1d 5789 | . . . . 5 |
15 | sqcl 10354 | . . . . . . . 8 | |
16 | 10, 15 | syl 14 | . . . . . . 7 |
17 | 2cn 8791 | . . . . . . . 8 | |
18 | mulcl 7747 | . . . . . . . 8 | |
19 | mulcl 7747 | . . . . . . . 8 | |
20 | 17, 18, 19 | sylancr 410 | . . . . . . 7 |
21 | 16, 20 | addcld 7785 | . . . . . 6 |
22 | sqcl 10354 | . . . . . . 7 | |
23 | 11, 22 | syl 14 | . . . . . 6 |
24 | 21, 23, 10 | adddird 7791 | . . . . 5 |
25 | 16, 20, 10 | adddird 7791 | . . . . . . 7 |
26 | 1 | oveq2i 5785 | . . . . . . . . 9 |
27 | expp1 10300 | . . . . . . . . . 10 | |
28 | 10, 4, 27 | sylancl 409 | . . . . . . . . 9 |
29 | 26, 28 | syl5eq 2184 | . . . . . . . 8 |
30 | sqval 10351 | . . . . . . . . . . . . 13 | |
31 | 10, 30 | syl 14 | . . . . . . . . . . . 12 |
32 | 31 | oveq1d 5789 | . . . . . . . . . . 11 |
33 | 10, 10, 11 | mul32d 7915 | . . . . . . . . . . 11 |
34 | 32, 33 | eqtrd 2172 | . . . . . . . . . 10 |
35 | 34 | oveq2d 5790 | . . . . . . . . 9 |
36 | 2cnd 8793 | . . . . . . . . . 10 | |
37 | 36, 18, 10 | mulassd 7789 | . . . . . . . . 9 |
38 | 35, 37 | eqtr4d 2175 | . . . . . . . 8 |
39 | 29, 38 | oveq12d 5792 | . . . . . . 7 |
40 | 25, 39 | eqtr4d 2175 | . . . . . 6 |
41 | 23, 10 | mulcomd 7787 | . . . . . 6 |
42 | 40, 41 | oveq12d 5792 | . . . . 5 |
43 | 14, 24, 42 | 3eqtrd 2176 | . . . 4 |
44 | 13 | oveq1d 5789 | . . . . 5 |
45 | 21, 23, 11 | adddird 7791 | . . . . 5 |
46 | sqval 10351 | . . . . . . . . . . . . . 14 | |
47 | 11, 46 | syl 14 | . . . . . . . . . . . . 13 |
48 | 47 | oveq2d 5790 | . . . . . . . . . . . 12 |
49 | 10, 11, 11 | mulassd 7789 | . . . . . . . . . . . 12 |
50 | 48, 49 | eqtr4d 2175 | . . . . . . . . . . 11 |
51 | 50 | oveq2d 5790 | . . . . . . . . . 10 |
52 | 36, 18, 11 | mulassd 7789 | . . . . . . . . . 10 |
53 | 51, 52 | eqtr4d 2175 | . . . . . . . . 9 |
54 | 53 | oveq2d 5790 | . . . . . . . 8 |
55 | 16, 20, 11 | adddird 7791 | . . . . . . . 8 |
56 | 54, 55 | eqtr4d 2175 | . . . . . . 7 |
57 | 1 | oveq2i 5785 | . . . . . . . 8 |
58 | expp1 10300 | . . . . . . . . 9 | |
59 | 11, 4, 58 | sylancl 409 | . . . . . . . 8 |
60 | 57, 59 | syl5eq 2184 | . . . . . . 7 |
61 | 56, 60 | oveq12d 5792 | . . . . . 6 |
62 | 16, 11 | mulcld 7786 | . . . . . . 7 |
63 | 10, 23 | mulcld 7786 | . . . . . . . 8 |
64 | mulcl 7747 | . . . . . . . 8 | |
65 | 17, 63, 64 | sylancr 410 | . . . . . . 7 |
66 | 3nn0 8995 | . . . . . . . 8 | |
67 | expcl 10311 | . . . . . . . 8 | |
68 | 11, 66, 67 | sylancl 409 | . . . . . . 7 |
69 | 62, 65, 68 | addassd 7788 | . . . . . 6 |
70 | 61, 69 | eqtr3d 2174 | . . . . 5 |
71 | 44, 45, 70 | 3eqtrd 2176 | . . . 4 |
72 | 43, 71 | oveq12d 5792 | . . 3 |
73 | expcl 10311 | . . . . . 6 | |
74 | 10, 66, 73 | sylancl 409 | . . . . 5 |
75 | mulcl 7747 | . . . . . 6 | |
76 | 17, 62, 75 | sylancr 410 | . . . . 5 |
77 | 74, 76 | addcld 7785 | . . . 4 |
78 | 65, 68 | addcld 7785 | . . . 4 |
79 | 77, 63, 62, 78 | add4d 7931 | . . 3 |
80 | 12, 72, 79 | 3eqtrd 2176 | . 2 |
81 | 74, 76, 62 | addassd 7788 | . . . 4 |
82 | 1 | oveq1i 5784 | . . . . . . 7 |
83 | 1cnd 7782 | . . . . . . . 8 | |
84 | 36, 83, 62 | adddird 7791 | . . . . . . 7 |
85 | 82, 84 | syl5eq 2184 | . . . . . 6 |
86 | 62 | mulid2d 7784 | . . . . . . 7 |
87 | 86 | oveq2d 5790 | . . . . . 6 |
88 | 85, 87 | eqtrd 2172 | . . . . 5 |
89 | 88 | oveq2d 5790 | . . . 4 |
90 | 81, 89 | eqtr4d 2175 | . . 3 |
91 | 1p2e3 8854 | . . . . . . . 8 | |
92 | 91 | oveq1i 5784 | . . . . . . 7 |
93 | 83, 36, 63 | adddird 7791 | . . . . . . 7 |
94 | 92, 93 | syl5eqr 2186 | . . . . . 6 |
95 | 63 | mulid2d 7784 | . . . . . . 7 |
96 | 95 | oveq1d 5789 | . . . . . 6 |
97 | 94, 96 | eqtrd 2172 | . . . . 5 |
98 | 97 | oveq1d 5789 | . . . 4 |
99 | 63, 65, 68 | addassd 7788 | . . . 4 |
100 | 98, 99 | eqtr2d 2173 | . . 3 |
101 | 90, 100 | oveq12d 5792 | . 2 |
102 | 7, 80, 101 | 3eqtrd 2176 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 (class class class)co 5774 cc 7618 c1 7621 caddc 7623 cmul 7625 c2 8771 c3 8772 cn0 8977 cexp 10292 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-n0 8978 df-z 9055 df-uz 9327 df-seqfrec 10219 df-exp 10293 |
This theorem is referenced by: (None) |
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