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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 8938 | . . . 4 | |
2 | 1 | oveq2i 5864 | . . 3 |
3 | addcl 7899 | . . . 4 | |
4 | 2nn0 9152 | . . . 4 | |
5 | expp1 10483 | . . . 4 | |
6 | 3, 4, 5 | sylancl 411 | . . 3 |
7 | 2, 6 | eqtrid 2215 | . 2 |
8 | sqcl 10537 | . . . . 5 | |
9 | 3, 8 | syl 14 | . . . 4 |
10 | simpl 108 | . . . 4 | |
11 | simpr 109 | . . . 4 | |
12 | 9, 10, 11 | adddid 7944 | . . 3 |
13 | binom2 10587 | . . . . . 6 | |
14 | 13 | oveq1d 5868 | . . . . 5 |
15 | sqcl 10537 | . . . . . . . 8 | |
16 | 10, 15 | syl 14 | . . . . . . 7 |
17 | 2cn 8949 | . . . . . . . 8 | |
18 | mulcl 7901 | . . . . . . . 8 | |
19 | mulcl 7901 | . . . . . . . 8 | |
20 | 17, 18, 19 | sylancr 412 | . . . . . . 7 |
21 | 16, 20 | addcld 7939 | . . . . . 6 |
22 | sqcl 10537 | . . . . . . 7 | |
23 | 11, 22 | syl 14 | . . . . . 6 |
24 | 21, 23, 10 | adddird 7945 | . . . . 5 |
25 | 16, 20, 10 | adddird 7945 | . . . . . . 7 |
26 | 1 | oveq2i 5864 | . . . . . . . . 9 |
27 | expp1 10483 | . . . . . . . . . 10 | |
28 | 10, 4, 27 | sylancl 411 | . . . . . . . . 9 |
29 | 26, 28 | eqtrid 2215 | . . . . . . . 8 |
30 | sqval 10534 | . . . . . . . . . . . . 13 | |
31 | 10, 30 | syl 14 | . . . . . . . . . . . 12 |
32 | 31 | oveq1d 5868 | . . . . . . . . . . 11 |
33 | 10, 10, 11 | mul32d 8072 | . . . . . . . . . . 11 |
34 | 32, 33 | eqtrd 2203 | . . . . . . . . . 10 |
35 | 34 | oveq2d 5869 | . . . . . . . . 9 |
36 | 2cnd 8951 | . . . . . . . . . 10 | |
37 | 36, 18, 10 | mulassd 7943 | . . . . . . . . 9 |
38 | 35, 37 | eqtr4d 2206 | . . . . . . . 8 |
39 | 29, 38 | oveq12d 5871 | . . . . . . 7 |
40 | 25, 39 | eqtr4d 2206 | . . . . . 6 |
41 | 23, 10 | mulcomd 7941 | . . . . . 6 |
42 | 40, 41 | oveq12d 5871 | . . . . 5 |
43 | 14, 24, 42 | 3eqtrd 2207 | . . . 4 |
44 | 13 | oveq1d 5868 | . . . . 5 |
45 | 21, 23, 11 | adddird 7945 | . . . . 5 |
46 | sqval 10534 | . . . . . . . . . . . . . 14 | |
47 | 11, 46 | syl 14 | . . . . . . . . . . . . 13 |
48 | 47 | oveq2d 5869 | . . . . . . . . . . . 12 |
49 | 10, 11, 11 | mulassd 7943 | . . . . . . . . . . . 12 |
50 | 48, 49 | eqtr4d 2206 | . . . . . . . . . . 11 |
51 | 50 | oveq2d 5869 | . . . . . . . . . 10 |
52 | 36, 18, 11 | mulassd 7943 | . . . . . . . . . 10 |
53 | 51, 52 | eqtr4d 2206 | . . . . . . . . 9 |
54 | 53 | oveq2d 5869 | . . . . . . . 8 |
55 | 16, 20, 11 | adddird 7945 | . . . . . . . 8 |
56 | 54, 55 | eqtr4d 2206 | . . . . . . 7 |
57 | 1 | oveq2i 5864 | . . . . . . . 8 |
58 | expp1 10483 | . . . . . . . . 9 | |
59 | 11, 4, 58 | sylancl 411 | . . . . . . . 8 |
60 | 57, 59 | eqtrid 2215 | . . . . . . 7 |
61 | 56, 60 | oveq12d 5871 | . . . . . 6 |
62 | 16, 11 | mulcld 7940 | . . . . . . 7 |
63 | 10, 23 | mulcld 7940 | . . . . . . . 8 |
64 | mulcl 7901 | . . . . . . . 8 | |
65 | 17, 63, 64 | sylancr 412 | . . . . . . 7 |
66 | 3nn0 9153 | . . . . . . . 8 | |
67 | expcl 10494 | . . . . . . . 8 | |
68 | 11, 66, 67 | sylancl 411 | . . . . . . 7 |
69 | 62, 65, 68 | addassd 7942 | . . . . . 6 |
70 | 61, 69 | eqtr3d 2205 | . . . . 5 |
71 | 44, 45, 70 | 3eqtrd 2207 | . . . 4 |
72 | 43, 71 | oveq12d 5871 | . . 3 |
73 | expcl 10494 | . . . . . 6 | |
74 | 10, 66, 73 | sylancl 411 | . . . . 5 |
75 | mulcl 7901 | . . . . . 6 | |
76 | 17, 62, 75 | sylancr 412 | . . . . 5 |
77 | 74, 76 | addcld 7939 | . . . 4 |
78 | 65, 68 | addcld 7939 | . . . 4 |
79 | 77, 63, 62, 78 | add4d 8088 | . . 3 |
80 | 12, 72, 79 | 3eqtrd 2207 | . 2 |
81 | 74, 76, 62 | addassd 7942 | . . . 4 |
82 | 1 | oveq1i 5863 | . . . . . . 7 |
83 | 1cnd 7936 | . . . . . . . 8 | |
84 | 36, 83, 62 | adddird 7945 | . . . . . . 7 |
85 | 82, 84 | eqtrid 2215 | . . . . . 6 |
86 | 62 | mulid2d 7938 | . . . . . . 7 |
87 | 86 | oveq2d 5869 | . . . . . 6 |
88 | 85, 87 | eqtrd 2203 | . . . . 5 |
89 | 88 | oveq2d 5869 | . . . 4 |
90 | 81, 89 | eqtr4d 2206 | . . 3 |
91 | 1p2e3 9012 | . . . . . . . 8 | |
92 | 91 | oveq1i 5863 | . . . . . . 7 |
93 | 83, 36, 63 | adddird 7945 | . . . . . . 7 |
94 | 92, 93 | eqtr3id 2217 | . . . . . 6 |
95 | 63 | mulid2d 7938 | . . . . . . 7 |
96 | 95 | oveq1d 5868 | . . . . . 6 |
97 | 94, 96 | eqtrd 2203 | . . . . 5 |
98 | 97 | oveq1d 5868 | . . . 4 |
99 | 63, 65, 68 | addassd 7942 | . . . 4 |
100 | 98, 99 | eqtr2d 2204 | . . 3 |
101 | 90, 100 | oveq12d 5871 | . 2 |
102 | 7, 80, 101 | 3eqtrd 2207 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wcel 2141 (class class class)co 5853 cc 7772 c1 7775 caddc 7777 cmul 7779 c2 8929 c3 8930 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-2 8937 df-3 8938 df-n0 9136 df-z 9213 df-uz 9488 df-seqfrec 10402 df-exp 10476 |
This theorem is referenced by: binom4 13691 |
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