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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 8893 | . . . 4 | |
2 | 1 | oveq2i 5835 | . . 3 |
3 | addcl 7857 | . . . 4 | |
4 | 2nn0 9107 | . . . 4 | |
5 | expp1 10426 | . . . 4 | |
6 | 3, 4, 5 | sylancl 410 | . . 3 |
7 | 2, 6 | syl5eq 2202 | . 2 |
8 | sqcl 10480 | . . . . 5 | |
9 | 3, 8 | syl 14 | . . . 4 |
10 | simpl 108 | . . . 4 | |
11 | simpr 109 | . . . 4 | |
12 | 9, 10, 11 | adddid 7902 | . . 3 |
13 | binom2 10529 | . . . . . 6 | |
14 | 13 | oveq1d 5839 | . . . . 5 |
15 | sqcl 10480 | . . . . . . . 8 | |
16 | 10, 15 | syl 14 | . . . . . . 7 |
17 | 2cn 8904 | . . . . . . . 8 | |
18 | mulcl 7859 | . . . . . . . 8 | |
19 | mulcl 7859 | . . . . . . . 8 | |
20 | 17, 18, 19 | sylancr 411 | . . . . . . 7 |
21 | 16, 20 | addcld 7897 | . . . . . 6 |
22 | sqcl 10480 | . . . . . . 7 | |
23 | 11, 22 | syl 14 | . . . . . 6 |
24 | 21, 23, 10 | adddird 7903 | . . . . 5 |
25 | 16, 20, 10 | adddird 7903 | . . . . . . 7 |
26 | 1 | oveq2i 5835 | . . . . . . . . 9 |
27 | expp1 10426 | . . . . . . . . . 10 | |
28 | 10, 4, 27 | sylancl 410 | . . . . . . . . 9 |
29 | 26, 28 | syl5eq 2202 | . . . . . . . 8 |
30 | sqval 10477 | . . . . . . . . . . . . 13 | |
31 | 10, 30 | syl 14 | . . . . . . . . . . . 12 |
32 | 31 | oveq1d 5839 | . . . . . . . . . . 11 |
33 | 10, 10, 11 | mul32d 8028 | . . . . . . . . . . 11 |
34 | 32, 33 | eqtrd 2190 | . . . . . . . . . 10 |
35 | 34 | oveq2d 5840 | . . . . . . . . 9 |
36 | 2cnd 8906 | . . . . . . . . . 10 | |
37 | 36, 18, 10 | mulassd 7901 | . . . . . . . . 9 |
38 | 35, 37 | eqtr4d 2193 | . . . . . . . 8 |
39 | 29, 38 | oveq12d 5842 | . . . . . . 7 |
40 | 25, 39 | eqtr4d 2193 | . . . . . 6 |
41 | 23, 10 | mulcomd 7899 | . . . . . 6 |
42 | 40, 41 | oveq12d 5842 | . . . . 5 |
43 | 14, 24, 42 | 3eqtrd 2194 | . . . 4 |
44 | 13 | oveq1d 5839 | . . . . 5 |
45 | 21, 23, 11 | adddird 7903 | . . . . 5 |
46 | sqval 10477 | . . . . . . . . . . . . . 14 | |
47 | 11, 46 | syl 14 | . . . . . . . . . . . . 13 |
48 | 47 | oveq2d 5840 | . . . . . . . . . . . 12 |
49 | 10, 11, 11 | mulassd 7901 | . . . . . . . . . . . 12 |
50 | 48, 49 | eqtr4d 2193 | . . . . . . . . . . 11 |
51 | 50 | oveq2d 5840 | . . . . . . . . . 10 |
52 | 36, 18, 11 | mulassd 7901 | . . . . . . . . . 10 |
53 | 51, 52 | eqtr4d 2193 | . . . . . . . . 9 |
54 | 53 | oveq2d 5840 | . . . . . . . 8 |
55 | 16, 20, 11 | adddird 7903 | . . . . . . . 8 |
56 | 54, 55 | eqtr4d 2193 | . . . . . . 7 |
57 | 1 | oveq2i 5835 | . . . . . . . 8 |
58 | expp1 10426 | . . . . . . . . 9 | |
59 | 11, 4, 58 | sylancl 410 | . . . . . . . 8 |
60 | 57, 59 | syl5eq 2202 | . . . . . . 7 |
61 | 56, 60 | oveq12d 5842 | . . . . . 6 |
62 | 16, 11 | mulcld 7898 | . . . . . . 7 |
63 | 10, 23 | mulcld 7898 | . . . . . . . 8 |
64 | mulcl 7859 | . . . . . . . 8 | |
65 | 17, 63, 64 | sylancr 411 | . . . . . . 7 |
66 | 3nn0 9108 | . . . . . . . 8 | |
67 | expcl 10437 | . . . . . . . 8 | |
68 | 11, 66, 67 | sylancl 410 | . . . . . . 7 |
69 | 62, 65, 68 | addassd 7900 | . . . . . 6 |
70 | 61, 69 | eqtr3d 2192 | . . . . 5 |
71 | 44, 45, 70 | 3eqtrd 2194 | . . . 4 |
72 | 43, 71 | oveq12d 5842 | . . 3 |
73 | expcl 10437 | . . . . . 6 | |
74 | 10, 66, 73 | sylancl 410 | . . . . 5 |
75 | mulcl 7859 | . . . . . 6 | |
76 | 17, 62, 75 | sylancr 411 | . . . . 5 |
77 | 74, 76 | addcld 7897 | . . . 4 |
78 | 65, 68 | addcld 7897 | . . . 4 |
79 | 77, 63, 62, 78 | add4d 8044 | . . 3 |
80 | 12, 72, 79 | 3eqtrd 2194 | . 2 |
81 | 74, 76, 62 | addassd 7900 | . . . 4 |
82 | 1 | oveq1i 5834 | . . . . . . 7 |
83 | 1cnd 7894 | . . . . . . . 8 | |
84 | 36, 83, 62 | adddird 7903 | . . . . . . 7 |
85 | 82, 84 | syl5eq 2202 | . . . . . 6 |
86 | 62 | mulid2d 7896 | . . . . . . 7 |
87 | 86 | oveq2d 5840 | . . . . . 6 |
88 | 85, 87 | eqtrd 2190 | . . . . 5 |
89 | 88 | oveq2d 5840 | . . . 4 |
90 | 81, 89 | eqtr4d 2193 | . . 3 |
91 | 1p2e3 8967 | . . . . . . . 8 | |
92 | 91 | oveq1i 5834 | . . . . . . 7 |
93 | 83, 36, 63 | adddird 7903 | . . . . . . 7 |
94 | 92, 93 | eqtr3id 2204 | . . . . . 6 |
95 | 63 | mulid2d 7896 | . . . . . . 7 |
96 | 95 | oveq1d 5839 | . . . . . 6 |
97 | 94, 96 | eqtrd 2190 | . . . . 5 |
98 | 97 | oveq1d 5839 | . . . 4 |
99 | 63, 65, 68 | addassd 7900 | . . . 4 |
100 | 98, 99 | eqtr2d 2191 | . . 3 |
101 | 90, 100 | oveq12d 5842 | . 2 |
102 | 7, 80, 101 | 3eqtrd 2194 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 wcel 2128 (class class class)co 5824 cc 7730 c1 7733 caddc 7735 cmul 7737 c2 8884 c3 8885 cn0 9090 cexp 10418 |
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 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-mulrcl 7831 ax-addcom 7832 ax-mulcom 7833 ax-addass 7834 ax-mulass 7835 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-1rid 7839 ax-0id 7840 ax-rnegex 7841 ax-precex 7842 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-apti 7847 ax-pre-ltadd 7848 ax-pre-mulgt0 7849 ax-pre-mulext 7850 |
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 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-ilim 4329 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-recs 6252 df-frec 6338 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-sub 8048 df-neg 8049 df-reap 8450 df-ap 8457 df-div 8546 df-inn 8834 df-2 8892 df-3 8893 df-n0 9091 df-z 9168 df-uz 9440 df-seqfrec 10345 df-exp 10419 |
This theorem is referenced by: binom4 13307 |
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