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Theorem subsq 12083
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 9713 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
41, 2, 3adddird 9515 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  x.  ( A  -  B )
)  =  ( ( A  x.  ( A  -  B ) )  +  ( B  x.  ( A  -  B
) ) ) )
5 subdi 9882 . . . . 5  |-  ( ( A  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  ( A  -  B ) )  =  ( ( A  x.  A )  -  ( A  x.  B )
) )
653anidm12 1276 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  ( A  -  B )
)  =  ( ( A  x.  A )  -  ( A  x.  B ) ) )
7 sqval 12035 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 2 )  =  ( A  x.  A
) )
87adantr 465 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A ^ 2 )  =  ( A  x.  A ) )
98oveq1d 6208 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
2 )  -  ( A  x.  B )
)  =  ( ( A  x.  A )  -  ( A  x.  B ) ) )
106, 9eqtr4d 2495 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  ( A  -  B )
)  =  ( ( A ^ 2 )  -  ( A  x.  B ) ) )
112, 1, 2subdid 9904 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  x.  ( A  -  B )
)  =  ( ( B  x.  A )  -  ( B  x.  B ) ) )
12 mulcom 9472 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
13 sqval 12035 . . . . . 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 6211 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  -  ( B ^ 2 ) )  =  ( ( B  x.  A )  -  ( B  x.  B
) ) )
1611, 15eqtr4d 2495 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  x.  ( A  -  B )
)  =  ( ( A  x.  B )  -  ( B ^
2 ) ) )
1710, 16oveq12d 6211 . 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 12038 . . . 4  |-  ( A  e.  CC  ->  ( A ^ 2 )  e.  CC )
1918adantr 465 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A ^ 2 )  e.  CC )
20 mulcl 9470 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
21 sqcl 12038 . . . 4  |-  ( B  e.  CC  ->  ( B ^ 2 )  e.  CC )
2221adantl 466 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B ^ 2 )  e.  CC )
2319, 20, 22npncand 9847 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A ^ 2 )  -  ( A  x.  B
) )  +  ( ( A  x.  B
)  -  ( B ^ 2 ) ) )  =  ( ( A ^ 2 )  -  ( B ^
2 ) ) )
244, 17, 233eqtrrd 2497 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 1370    e. wcel 1758  (class class class)co 6193   CCcc 9384    + caddc 9389    x. cmul 9391    - cmin 9699   2c2 10475   ^cexp 11975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-2nd 6681  df-recs 6935  df-rdg 6969  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-2 10484  df-n0 10684  df-z 10751  df-uz 10966  df-seq 11917  df-exp 11976
This theorem is referenced by:  subsq2  12084  subsqi  12087  pythagtriplem4  13997  pythagtriplem6  13999  pythagtriplem7  14000  pythagtriplem12  14004  pythagtriplem14  14006  pythagtriplem16  14008  4sqlem8  14117  4sqlem10  14119  4sqlem11  14127  chordthmlem4  22356  heron  22359  dcubic2  22365  cubic  22370  dquart  22374  asinlem2  22390  asinsin  22413  efiatan2  22438  atans2  22452  dvatan  22456  wilthlem1  22532  lgslem1  22761  lgsqrlem2  22807  2sqlem4  22832  2sqblem  22842  rplogsumlem1  22859  pellexlem2  29312  pell1234qrne0  29335  pell1234qrreccl  29336  pell1234qrmulcl  29337  pell14qrdich  29351  rmxyneg  29402  stoweidlem1  29937
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