MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  subsq Structured version   Unicode version

Theorem subsq 11965
Description: Factor the difference of two squares. (Contributed by NM, 21-Feb-2008.)
Assertion
Ref Expression
subsq  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
2 )  -  ( B ^ 2 ) )  =  ( ( A  +  B )  x.  ( A  -  B
) ) )

Proof of Theorem subsq
StepHypRef Expression
1 simpl 457 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  A  e.  CC )
2 simpr 461 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  B  e.  CC )
3 subcl 9601 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
41, 2, 3adddird 9403 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  x.  ( A  -  B )
)  =  ( ( A  x.  ( A  -  B ) )  +  ( B  x.  ( A  -  B
) ) ) )
5 subdi 9770 . . . . 5  |-  ( ( A  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  ( A  -  B ) )  =  ( ( A  x.  A )  -  ( A  x.  B )
) )
653anidm12 1275 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  ( A  -  B )
)  =  ( ( A  x.  A )  -  ( A  x.  B ) ) )
7 sqval 11917 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 2 )  =  ( A  x.  A
) )
87adantr 465 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A ^ 2 )  =  ( A  x.  A ) )
98oveq1d 6101 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
2 )  -  ( A  x.  B )
)  =  ( ( A  x.  A )  -  ( A  x.  B ) ) )
106, 9eqtr4d 2473 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  ( A  -  B )
)  =  ( ( A ^ 2 )  -  ( A  x.  B ) ) )
112, 1, 2subdid 9792 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  x.  ( A  -  B )
)  =  ( ( B  x.  A )  -  ( B  x.  B ) ) )
12 mulcom 9360 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
13 sqval 11917 . . . . . 6  |-  ( B  e.  CC  ->  ( B ^ 2 )  =  ( B  x.  B
) )
1413adantl 466 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B ^ 2 )  =  ( B  x.  B ) )
1512, 14oveq12d 6104 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  -  ( B ^ 2 ) )  =  ( ( B  x.  A )  -  ( B  x.  B
) ) )
1611, 15eqtr4d 2473 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  x.  ( A  -  B )
)  =  ( ( A  x.  B )  -  ( B ^
2 ) ) )
1710, 16oveq12d 6104 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  ( A  -  B
) )  +  ( B  x.  ( A  -  B ) ) )  =  ( ( ( A ^ 2 )  -  ( A  x.  B ) )  +  ( ( A  x.  B )  -  ( B ^ 2 ) ) ) )
18 sqcl 11920 . . . 4  |-  ( A  e.  CC  ->  ( A ^ 2 )  e.  CC )
1918adantr 465 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A ^ 2 )  e.  CC )
20 mulcl 9358 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
21 sqcl 11920 . . . 4  |-  ( B  e.  CC  ->  ( B ^ 2 )  e.  CC )
2221adantl 466 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B ^ 2 )  e.  CC )
2319, 20, 22npncand 9735 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A ^ 2 )  -  ( A  x.  B
) )  +  ( ( A  x.  B
)  -  ( B ^ 2 ) ) )  =  ( ( A ^ 2 )  -  ( B ^
2 ) ) )
244, 17, 233eqtrrd 2475 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
2 )  -  ( B ^ 2 ) )  =  ( ( A  +  B )  x.  ( A  -  B
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756  (class class class)co 6086   CCcc 9272    + caddc 9277    x. cmul 9279    - cmin 9587   2c2 10363   ^cexp 11857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-n0 10572  df-z 10639  df-uz 10854  df-seq 11799  df-exp 11858
This theorem is referenced by:  subsq2  11966  subsqi  11969  pythagtriplem4  13878  pythagtriplem6  13880  pythagtriplem7  13881  pythagtriplem12  13885  pythagtriplem14  13887  pythagtriplem16  13889  4sqlem8  13998  4sqlem10  14000  4sqlem11  14008  chordthmlem4  22205  heron  22208  dcubic2  22214  cubic  22219  dquart  22223  asinlem2  22239  asinsin  22262  efiatan2  22287  atans2  22301  dvatan  22305  wilthlem1  22381  lgslem1  22610  lgsqrlem2  22656  2sqlem4  22681  2sqblem  22691  rplogsumlem1  22708  pellexlem2  29124  pell1234qrne0  29147  pell1234qrreccl  29148  pell1234qrmulcl  29149  pell14qrdich  29163  rmxyneg  29214  stoweidlem1  29749
  Copyright terms: Public domain W3C validator