ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  binom2sub Unicode version

Theorem binom2sub 10651
Description: Expand the square of a subtraction. (Contributed by Scott Fenton, 10-Jun-2013.)
Assertion
Ref Expression
binom2sub  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B ) ^ 2 )  =  ( ( ( A ^ 2 )  -  ( 2  x.  ( A  x.  B ) ) )  +  ( B ^
2 ) ) )

Proof of Theorem binom2sub
StepHypRef Expression
1 negcl 8174 . . . 4  |-  ( B  e.  CC  ->  -u B  e.  CC )
2 binom2 10649 . . . 4  |-  ( ( A  e.  CC  /\  -u B  e.  CC )  ->  ( ( A  +  -u B ) ^
2 )  =  ( ( ( A ^
2 )  +  ( 2  x.  ( A  x.  -u B ) ) )  +  ( -u B ^ 2 ) ) )
31, 2sylan2 286 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  -u B ) ^ 2 )  =  ( ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  -u B ) ) )  +  ( -u B ^ 2 ) ) )
4 negsub 8222 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  -u B )  =  ( A  -  B ) )
54oveq1d 5905 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  -u B ) ^ 2 )  =  ( ( A  -  B ) ^ 2 ) )
63, 5eqtr3d 2223 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  -u B ) ) )  +  (
-u B ^ 2 ) )  =  ( ( A  -  B
) ^ 2 ) )
7 mulneg2 8370 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  -u B
)  =  -u ( A  x.  B )
)
87oveq2d 5906 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  ( A  x.  -u B ) )  =  ( 2  x.  -u ( A  x.  B ) ) )
9 2cn 9007 . . . . . . 7  |-  2  e.  CC
10 mulcl 7955 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
11 mulneg2 8370 . . . . . . 7  |-  ( ( 2  e.  CC  /\  ( A  x.  B
)  e.  CC )  ->  ( 2  x.  -u ( A  x.  B
) )  =  -u ( 2  x.  ( A  x.  B )
) )
129, 10, 11sylancr 414 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  -u ( A  x.  B )
)  =  -u (
2  x.  ( A  x.  B ) ) )
138, 12eqtr2d 2222 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> 
-u ( 2  x.  ( A  x.  B
) )  =  ( 2  x.  ( A  x.  -u B ) ) )
1413oveq2d 5906 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
2 )  +  -u ( 2  x.  ( A  x.  B )
) )  =  ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  -u B ) ) ) )
15 sqcl 10598 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 2 )  e.  CC )
1615adantr 276 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A ^ 2 )  e.  CC )
17 mulcl 7955 . . . . . 6  |-  ( ( 2  e.  CC  /\  ( A  x.  B
)  e.  CC )  ->  ( 2  x.  ( A  x.  B
) )  e.  CC )
189, 10, 17sylancr 414 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  ( A  x.  B )
)  e.  CC )
1916, 18negsubd 8291 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
2 )  +  -u ( 2  x.  ( A  x.  B )
) )  =  ( ( A ^ 2 )  -  ( 2  x.  ( A  x.  B ) ) ) )
2014, 19eqtr3d 2223 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
2 )  +  ( 2  x.  ( A  x.  -u B ) ) )  =  ( ( A ^ 2 )  -  ( 2  x.  ( A  x.  B
) ) ) )
21 sqneg 10596 . . . 4  |-  ( B  e.  CC  ->  ( -u B ^ 2 )  =  ( B ^
2 ) )
2221adantl 277 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( -u B ^
2 )  =  ( B ^ 2 ) )
2320, 22oveq12d 5908 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  -u B ) ) )  +  (
-u B ^ 2 ) )  =  ( ( ( A ^
2 )  -  (
2  x.  ( A  x.  B ) ) )  +  ( B ^ 2 ) ) )
246, 23eqtr3d 2223 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B ) ^ 2 )  =  ( ( ( A ^ 2 )  -  ( 2  x.  ( A  x.  B ) ) )  +  ( B ^
2 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1363    e. wcel 2159  (class class class)co 5890   CCcc 7826    + caddc 7831    x. cmul 7833    - cmin 8145   -ucneg 8146   2c2 8987   ^cexp 10536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-coll 4132  ax-sep 4135  ax-nul 4143  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550  ax-iinf 4601  ax-cnex 7919  ax-resscn 7920  ax-1cn 7921  ax-1re 7922  ax-icn 7923  ax-addcl 7924  ax-addrcl 7925  ax-mulcl 7926  ax-mulrcl 7927  ax-addcom 7928  ax-mulcom 7929  ax-addass 7930  ax-mulass 7931  ax-distr 7932  ax-i2m1 7933  ax-0lt1 7934  ax-1rid 7935  ax-0id 7936  ax-rnegex 7937  ax-precex 7938  ax-cnre 7939  ax-pre-ltirr 7940  ax-pre-ltwlin 7941  ax-pre-lttrn 7942  ax-pre-apti 7943  ax-pre-ltadd 7944  ax-pre-mulgt0 7945  ax-pre-mulext 7946
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-nel 2455  df-ral 2472  df-rex 2473  df-reu 2474  df-rmo 2475  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-nul 3437  df-if 3549  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-iun 3902  df-br 4018  df-opab 4079  df-mpt 4080  df-tr 4116  df-id 4307  df-po 4310  df-iso 4311  df-iord 4380  df-on 4382  df-ilim 4383  df-suc 4385  df-iom 4604  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-riota 5846  df-ov 5893  df-oprab 5894  df-mpo 5895  df-1st 6158  df-2nd 6159  df-recs 6323  df-frec 6409  df-pnf 8011  df-mnf 8012  df-xr 8013  df-ltxr 8014  df-le 8015  df-sub 8147  df-neg 8148  df-reap 8549  df-ap 8556  df-div 8647  df-inn 8937  df-2 8995  df-n0 9194  df-z 9271  df-uz 9546  df-seqfrec 10463  df-exp 10537
This theorem is referenced by:  binom2sub1  10652  binom2subi  10653  resqrexlemover  11036  resqrexlemcalc1  11040  amgm2  11144  bdtrilem  11264  pythagtriplem1  12282  pythagtriplem14  12294  tangtx  14642
  Copyright terms: Public domain W3C validator