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Theorem subsq 11257
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 443 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  A  e.  CC )
2 simpr 447 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  B  e.  CC )
3 subcl 9096 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
41, 2, 3adddird 8905 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  x.  ( A  -  B )
)  =  ( ( A  x.  ( A  -  B ) )  +  ( B  x.  ( A  -  B
) ) ) )
5 subdi 9258 . . . . 5  |-  ( ( A  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  ( A  -  B ) )  =  ( ( A  x.  A )  -  ( A  x.  B )
) )
653anidm12 1239 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  ( A  -  B )
)  =  ( ( A  x.  A )  -  ( A  x.  B ) ) )
7 sqval 11210 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 2 )  =  ( A  x.  A
) )
87adantr 451 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A ^ 2 )  =  ( A  x.  A ) )
98oveq1d 5915 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
2 )  -  ( A  x.  B )
)  =  ( ( A  x.  A )  -  ( A  x.  B ) ) )
106, 9eqtr4d 2351 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  ( A  -  B )
)  =  ( ( A ^ 2 )  -  ( A  x.  B ) ) )
112, 1, 2subdid 9280 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  x.  ( A  -  B )
)  =  ( ( B  x.  A )  -  ( B  x.  B ) ) )
12 mulcom 8868 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
13 sqval 11210 . . . . . 6  |-  ( B  e.  CC  ->  ( B ^ 2 )  =  ( B  x.  B
) )
1413adantl 452 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B ^ 2 )  =  ( B  x.  B ) )
1512, 14oveq12d 5918 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  -  ( B ^ 2 ) )  =  ( ( B  x.  A )  -  ( B  x.  B
) ) )
1611, 15eqtr4d 2351 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  x.  ( A  -  B )
)  =  ( ( A  x.  B )  -  ( B ^
2 ) ) )
1710, 16oveq12d 5918 . 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 11213 . . . 4  |-  ( A  e.  CC  ->  ( A ^ 2 )  e.  CC )
1918adantr 451 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A ^ 2 )  e.  CC )
20 mulcl 8866 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
21 sqcl 11213 . . . 4  |-  ( B  e.  CC  ->  ( B ^ 2 )  e.  CC )
2221adantl 452 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B ^ 2 )  e.  CC )
2319, 20, 22npncand 9226 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A ^ 2 )  -  ( A  x.  B
) )  +  ( ( A  x.  B
)  -  ( B ^ 2 ) ) )  =  ( ( A ^ 2 )  -  ( B ^
2 ) ) )
244, 17, 233eqtrrd 2353 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
2 )  -  ( B ^ 2 ) )  =  ( ( A  +  B )  x.  ( A  -  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701  (class class class)co 5900   CCcc 8780    + caddc 8785    x. cmul 8787    - cmin 9082   2c2 9840   ^cexp 11151
This theorem is referenced by:  subsq2  11258  subsqi  11261  pythagtriplem4  12919  pythagtriplem6  12921  pythagtriplem7  12922  pythagtriplem12  12926  pythagtriplem14  12928  pythagtriplem16  12930  4sqlem8  13039  4sqlem10  13041  4sqlem11  13049  chordthmlem4  20185  dcubic2  20193  cubic  20198  dquart  20202  asinlem2  20218  asinsin  20241  efiatan2  20266  atans2  20280  dvatan  20284  wilthlem1  20359  lgslem1  20588  lgsqrlem2  20634  2sqlem4  20659  2sqblem  20669  rplogsumlem1  20686  pellexlem2  26063  pell1234qrne0  26086  pell1234qrreccl  26087  pell1234qrmulcl  26088  pell14qrdich  26102  rmxyneg  26153  stoweidlem1  26898
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-n0 10013  df-z 10072  df-uz 10278  df-seq 11094  df-exp 11152
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