Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > binom3 | Structured version Visualization version 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 11080 | . . . 4 | |
2 | 1 | oveq2i 6661 | . . 3 |
3 | addcl 10018 | . . . 4 | |
4 | 2nn0 11309 | . . . 4 | |
5 | expp1 12867 | . . . 4 | |
6 | 3, 4, 5 | sylancl 694 | . . 3 |
7 | 2, 6 | syl5eq 2668 | . 2 |
8 | sqcl 12925 | . . . . 5 | |
9 | 3, 8 | syl 17 | . . . 4 |
10 | simpl 473 | . . . 4 | |
11 | simpr 477 | . . . 4 | |
12 | 9, 10, 11 | adddid 10064 | . . 3 |
13 | binom2 12979 | . . . . . 6 | |
14 | 13 | oveq1d 6665 | . . . . 5 |
15 | sqcl 12925 | . . . . . . . 8 | |
16 | 10, 15 | syl 17 | . . . . . . 7 |
17 | 2cn 11091 | . . . . . . . 8 | |
18 | mulcl 10020 | . . . . . . . 8 | |
19 | mulcl 10020 | . . . . . . . 8 | |
20 | 17, 18, 19 | sylancr 695 | . . . . . . 7 |
21 | 16, 20 | addcld 10059 | . . . . . 6 |
22 | sqcl 12925 | . . . . . . 7 | |
23 | 11, 22 | syl 17 | . . . . . 6 |
24 | 21, 23, 10 | adddird 10065 | . . . . 5 |
25 | 16, 20, 10 | adddird 10065 | . . . . . . 7 |
26 | 1 | oveq2i 6661 | . . . . . . . . 9 |
27 | expp1 12867 | . . . . . . . . . 10 | |
28 | 10, 4, 27 | sylancl 694 | . . . . . . . . 9 |
29 | 26, 28 | syl5eq 2668 | . . . . . . . 8 |
30 | sqval 12922 | . . . . . . . . . . . . 13 | |
31 | 10, 30 | syl 17 | . . . . . . . . . . . 12 |
32 | 31 | oveq1d 6665 | . . . . . . . . . . 11 |
33 | 10, 10, 11 | mul32d 10246 | . . . . . . . . . . 11 |
34 | 32, 33 | eqtrd 2656 | . . . . . . . . . 10 |
35 | 34 | oveq2d 6666 | . . . . . . . . 9 |
36 | 2cnd 11093 | . . . . . . . . . 10 | |
37 | 36, 18, 10 | mulassd 10063 | . . . . . . . . 9 |
38 | 35, 37 | eqtr4d 2659 | . . . . . . . 8 |
39 | 29, 38 | oveq12d 6668 | . . . . . . 7 |
40 | 25, 39 | eqtr4d 2659 | . . . . . 6 |
41 | 23, 10 | mulcomd 10061 | . . . . . 6 |
42 | 40, 41 | oveq12d 6668 | . . . . 5 |
43 | 14, 24, 42 | 3eqtrd 2660 | . . . 4 |
44 | 13 | oveq1d 6665 | . . . . 5 |
45 | 21, 23, 11 | adddird 10065 | . . . . 5 |
46 | sqval 12922 | . . . . . . . . . . . . . 14 | |
47 | 11, 46 | syl 17 | . . . . . . . . . . . . 13 |
48 | 47 | oveq2d 6666 | . . . . . . . . . . . 12 |
49 | 10, 11, 11 | mulassd 10063 | . . . . . . . . . . . 12 |
50 | 48, 49 | eqtr4d 2659 | . . . . . . . . . . 11 |
51 | 50 | oveq2d 6666 | . . . . . . . . . 10 |
52 | 36, 18, 11 | mulassd 10063 | . . . . . . . . . 10 |
53 | 51, 52 | eqtr4d 2659 | . . . . . . . . 9 |
54 | 53 | oveq2d 6666 | . . . . . . . 8 |
55 | 16, 20, 11 | adddird 10065 | . . . . . . . 8 |
56 | 54, 55 | eqtr4d 2659 | . . . . . . 7 |
57 | 1 | oveq2i 6661 | . . . . . . . 8 |
58 | expp1 12867 | . . . . . . . . 9 | |
59 | 11, 4, 58 | sylancl 694 | . . . . . . . 8 |
60 | 57, 59 | syl5eq 2668 | . . . . . . 7 |
61 | 56, 60 | oveq12d 6668 | . . . . . 6 |
62 | 16, 11 | mulcld 10060 | . . . . . . 7 |
63 | 10, 23 | mulcld 10060 | . . . . . . . 8 |
64 | mulcl 10020 | . . . . . . . 8 | |
65 | 17, 63, 64 | sylancr 695 | . . . . . . 7 |
66 | 3nn0 11310 | . . . . . . . 8 | |
67 | expcl 12878 | . . . . . . . 8 | |
68 | 11, 66, 67 | sylancl 694 | . . . . . . 7 |
69 | 62, 65, 68 | addassd 10062 | . . . . . 6 |
70 | 61, 69 | eqtr3d 2658 | . . . . 5 |
71 | 44, 45, 70 | 3eqtrd 2660 | . . . 4 |
72 | 43, 71 | oveq12d 6668 | . . 3 |
73 | expcl 12878 | . . . . . 6 | |
74 | 10, 66, 73 | sylancl 694 | . . . . 5 |
75 | mulcl 10020 | . . . . . 6 | |
76 | 17, 62, 75 | sylancr 695 | . . . . 5 |
77 | 74, 76 | addcld 10059 | . . . 4 |
78 | 65, 68 | addcld 10059 | . . . 4 |
79 | 77, 63, 62, 78 | add4d 10264 | . . 3 |
80 | 12, 72, 79 | 3eqtrd 2660 | . 2 |
81 | 74, 76, 62 | addassd 10062 | . . . 4 |
82 | 1 | oveq1i 6660 | . . . . . . 7 |
83 | 1cnd 10056 | . . . . . . . 8 | |
84 | 36, 83, 62 | adddird 10065 | . . . . . . 7 |
85 | 82, 84 | syl5eq 2668 | . . . . . 6 |
86 | 62 | mulid2d 10058 | . . . . . . 7 |
87 | 86 | oveq2d 6666 | . . . . . 6 |
88 | 85, 87 | eqtrd 2656 | . . . . 5 |
89 | 88 | oveq2d 6666 | . . . 4 |
90 | 81, 89 | eqtr4d 2659 | . . 3 |
91 | 1p2e3 11152 | . . . . . . . 8 | |
92 | 91 | oveq1i 6660 | . . . . . . 7 |
93 | 83, 36, 63 | adddird 10065 | . . . . . . 7 |
94 | 92, 93 | syl5eqr 2670 | . . . . . 6 |
95 | 63 | mulid2d 10058 | . . . . . . 7 |
96 | 95 | oveq1d 6665 | . . . . . 6 |
97 | 94, 96 | eqtrd 2656 | . . . . 5 |
98 | 97 | oveq1d 6665 | . . . 4 |
99 | 63, 65, 68 | addassd 10062 | . . . 4 |
100 | 98, 99 | eqtr2d 2657 | . . 3 |
101 | 90, 100 | oveq12d 6668 | . 2 |
102 | 7, 80, 101 | 3eqtrd 2660 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 (class class class)co 6650 cc 9934 c1 9937 caddc 9939 cmul 9941 c2 11070 c3 11071 cn0 11292 cexp 12860 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-seq 12802 df-exp 12861 |
This theorem is referenced by: dcubic1lem 24570 mcubic 24574 binom4 24577 |
Copyright terms: Public domain | W3C validator |