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Mirrors > Home > ILE Home > Th. List > binom2 | Unicode version |
Description: The square of a binomial. (Contributed by FL, 10-Dec-2006.) |
Ref | Expression |
---|---|
binom2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addcl 7713 | . . . 4 | |
2 | simpl 108 | . . . 4 | |
3 | simpr 109 | . . . 4 | |
4 | 1, 2, 3 | adddid 7758 | . . 3 |
5 | 2, 3, 2 | adddird 7759 | . . . . . 6 |
6 | 3, 2 | mulcomd 7755 | . . . . . . 7 |
7 | 6 | oveq2d 5758 | . . . . . 6 |
8 | 5, 7 | eqtrd 2150 | . . . . 5 |
9 | 2, 3, 3 | adddird 7759 | . . . . 5 |
10 | 8, 9 | oveq12d 5760 | . . . 4 |
11 | 2, 2 | mulcld 7754 | . . . . . 6 |
12 | 2, 3 | mulcld 7754 | . . . . . 6 |
13 | 11, 12 | addcld 7753 | . . . . 5 |
14 | 3, 3 | mulcld 7754 | . . . . 5 |
15 | 13, 12, 14 | addassd 7756 | . . . 4 |
16 | 11, 12, 12 | addassd 7756 | . . . . 5 |
17 | 16 | oveq1d 5757 | . . . 4 |
18 | 10, 15, 17 | 3eqtr2d 2156 | . . 3 |
19 | 4, 18 | eqtrd 2150 | . 2 |
20 | sqval 10319 | . . 3 | |
21 | 1, 20 | syl 14 | . 2 |
22 | sqval 10319 | . . . . 5 | |
23 | 2, 22 | syl 14 | . . . 4 |
24 | 12 | 2timesd 8930 | . . . 4 |
25 | 23, 24 | oveq12d 5760 | . . 3 |
26 | sqval 10319 | . . . 4 | |
27 | 3, 26 | syl 14 | . . 3 |
28 | 25, 27 | oveq12d 5760 | . 2 |
29 | 19, 21, 28 | 3eqtr4d 2160 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wcel 1465 (class class class)co 5742 cc 7586 caddc 7591 cmul 7593 c2 8739 cexp 10260 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-ilim 4261 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-frec 6256 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 df-div 8401 df-inn 8689 df-2 8747 df-n0 8946 df-z 9023 df-uz 9295 df-seqfrec 10187 df-exp 10261 |
This theorem is referenced by: binom21 10372 binom2sub 10373 mulbinom2 10376 binom3 10377 nn0opthlem1d 10434 resqrexlemover 10750 resqrexlemcalc1 10754 abstri 10844 amgm2 10858 bdtrilem 10978 |
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