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Mirrors > Home > MPE Home > Th. List > mulsubdivbinom2 | Structured version Visualization version Unicode version |
Description: The square of a binomial with factor minus a number divided by a nonzero number. (Contributed by AV, 19-Jul-2021.) |
Ref | Expression |
---|---|
mulsubdivbinom2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1061 | . . . 4 | |
2 | 1 | adantr 481 | . . 3 |
3 | simpl2 1065 | . . 3 | |
4 | simpl 473 | . . . 4 | |
5 | 4 | adantl 482 | . . 3 |
6 | mulbinom2 12984 | . . . . 5 | |
7 | 6 | oveq1d 6665 | . . . 4 |
8 | 7 | oveq1d 6665 | . . 3 |
9 | 2, 3, 5, 8 | syl3anc 1326 | . 2 |
10 | 5, 2 | mulcld 10060 | . . . . . . 7 |
11 | 10 | sqcld 13006 | . . . . . 6 |
12 | 2cnd 11093 | . . . . . . . . . 10 | |
13 | id 22 | . . . . . . . . . 10 | |
14 | 12, 13 | mulcld 10060 | . . . . . . . . 9 |
15 | 14 | adantr 481 | . . . . . . . 8 |
16 | 15 | adantl 482 | . . . . . . 7 |
17 | mulcl 10020 | . . . . . . . . 9 | |
18 | 17 | 3adant3 1081 | . . . . . . . 8 |
19 | 18 | adantr 481 | . . . . . . 7 |
20 | 16, 19 | mulcld 10060 | . . . . . 6 |
21 | 11, 20 | addcld 10059 | . . . . 5 |
22 | sqcl 12925 | . . . . . . 7 | |
23 | 22 | 3ad2ant2 1083 | . . . . . 6 |
24 | 23 | adantr 481 | . . . . 5 |
25 | 21, 24 | addcld 10059 | . . . 4 |
26 | simpl3 1066 | . . . 4 | |
27 | simpr 477 | . . . 4 | |
28 | divsubdir 10721 | . . . 4 | |
29 | 25, 26, 27, 28 | syl3anc 1326 | . . 3 |
30 | divdir 10710 | . . . . . 6 | |
31 | 21, 24, 27, 30 | syl3anc 1326 | . . . . 5 |
32 | divdir 10710 | . . . . . . . 8 | |
33 | 11, 20, 27, 32 | syl3anc 1326 | . . . . . . 7 |
34 | sqmul 12926 | . . . . . . . . . . 11 | |
35 | 4, 1, 34 | syl2anr 495 | . . . . . . . . . 10 |
36 | 35 | oveq1d 6665 | . . . . . . . . 9 |
37 | sqcl 12925 | . . . . . . . . . . . 12 | |
38 | 37 | adantr 481 | . . . . . . . . . . 11 |
39 | 38 | adantl 482 | . . . . . . . . . 10 |
40 | sqcl 12925 | . . . . . . . . . . . 12 | |
41 | 40 | 3ad2ant1 1082 | . . . . . . . . . . 11 |
42 | 41 | adantr 481 | . . . . . . . . . 10 |
43 | div23 10704 | . . . . . . . . . 10 | |
44 | 39, 42, 27, 43 | syl3anc 1326 | . . . . . . . . 9 |
45 | sqdivid 12929 | . . . . . . . . . . 11 | |
46 | 45 | adantl 482 | . . . . . . . . . 10 |
47 | 46 | oveq1d 6665 | . . . . . . . . 9 |
48 | 36, 44, 47 | 3eqtrd 2660 | . . . . . . . 8 |
49 | div23 10704 | . . . . . . . . . 10 | |
50 | 16, 19, 27, 49 | syl3anc 1326 | . . . . . . . . 9 |
51 | 2cnd 11093 | . . . . . . . . . . . 12 | |
52 | simpr 477 | . . . . . . . . . . . 12 | |
53 | 51, 4, 52 | divcan4d 10807 | . . . . . . . . . . 11 |
54 | 53 | adantl 482 | . . . . . . . . . 10 |
55 | 54 | oveq1d 6665 | . . . . . . . . 9 |
56 | 50, 55 | eqtrd 2656 | . . . . . . . 8 |
57 | 48, 56 | oveq12d 6668 | . . . . . . 7 |
58 | 33, 57 | eqtrd 2656 | . . . . . 6 |
59 | 58 | oveq1d 6665 | . . . . 5 |
60 | 31, 59 | eqtrd 2656 | . . . 4 |
61 | 60 | oveq1d 6665 | . . 3 |
62 | 5, 42 | mulcld 10060 | . . . . 5 |
63 | 2cnd 11093 | . . . . . . . 8 | |
64 | 63, 17 | mulcld 10060 | . . . . . . 7 |
65 | 64 | 3adant3 1081 | . . . . . 6 |
66 | 65 | adantr 481 | . . . . 5 |
67 | 62, 66 | addcld 10059 | . . . 4 |
68 | 52 | adantl 482 | . . . . 5 |
69 | 24, 5, 68 | divcld 10801 | . . . 4 |
70 | 26, 5, 68 | divcld 10801 | . . . 4 |
71 | 67, 69, 70 | addsubassd 10412 | . . 3 |
72 | 29, 61, 71 | 3eqtrd 2660 | . 2 |
73 | divsubdir 10721 | . . . . 5 | |
74 | 24, 26, 27, 73 | syl3anc 1326 | . . . 4 |
75 | 74 | eqcomd 2628 | . . 3 |
76 | 75 | oveq2d 6666 | . 2 |
77 | 9, 72, 76 | 3eqtrd 2660 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 wne 2794 (class class class)co 6650 cc 9934 cc0 9936 caddc 9939 cmul 9941 cmin 10266 cdiv 10684 c2 11070 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-rmo 2920 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-div 10685 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-seq 12802 df-exp 12861 |
This theorem is referenced by: muldivbinom2 13047 2lgsoddprmlem1 25133 |
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