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Theorem subsq 11490
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 445 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  A  e.  CC )
2 simpr 449 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  B  e.  CC )
3 subcl 9307 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
41, 2, 3adddird 9115 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  x.  ( A  -  B )
)  =  ( ( A  x.  ( A  -  B ) )  +  ( B  x.  ( A  -  B
) ) ) )
5 subdi 9469 . . . . 5  |-  ( ( A  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  ( A  -  B ) )  =  ( ( A  x.  A )  -  ( A  x.  B )
) )
653anidm12 1242 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  ( A  -  B )
)  =  ( ( A  x.  A )  -  ( A  x.  B ) ) )
7 sqval 11443 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 2 )  =  ( A  x.  A
) )
87adantr 453 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A ^ 2 )  =  ( A  x.  A ) )
98oveq1d 6098 . . . 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 9491 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  x.  ( A  -  B )
)  =  ( ( B  x.  A )  -  ( B  x.  B ) ) )
12 mulcom 9078 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
13 sqval 11443 . . . . . 6  |-  ( B  e.  CC  ->  ( B ^ 2 )  =  ( B  x.  B
) )
1413adantl 454 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B ^ 2 )  =  ( B  x.  B ) )
1512, 14oveq12d 6101 . . . 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 6101 . 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 11446 . . . 4  |-  ( A  e.  CC  ->  ( A ^ 2 )  e.  CC )
1918adantr 453 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A ^ 2 )  e.  CC )
20 mulcl 9076 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
21 sqcl 11446 . . . 4  |-  ( B  e.  CC  ->  ( B ^ 2 )  e.  CC )
2221adantl 454 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B ^ 2 )  e.  CC )
2319, 20, 22npncand 9437 . 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 set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726  (class class class)co 6083   CCcc 8990    + caddc 8995    x. cmul 8997    - cmin 9293   2c2 10051   ^cexp 11384
This theorem is referenced by:  subsq2  11491  subsqi  11494  pythagtriplem4  13195  pythagtriplem6  13197  pythagtriplem7  13198  pythagtriplem12  13202  pythagtriplem14  13204  pythagtriplem16  13206  4sqlem8  13315  4sqlem10  13317  4sqlem11  13325  chordthmlem4  20678  dcubic2  20686  cubic  20691  dquart  20695  asinlem2  20711  asinsin  20734  efiatan2  20759  atans2  20773  dvatan  20777  wilthlem1  20853  lgslem1  21082  lgsqrlem2  21128  2sqlem4  21153  2sqblem  21163  rplogsumlem1  21180  pellexlem2  26895  pell1234qrne0  26918  pell1234qrreccl  26919  pell1234qrmulcl  26920  pell14qrdich  26934  rmxyneg  26985  stoweidlem1  27728
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-n0 10224  df-z 10285  df-uz 10491  df-seq 11326  df-exp 11385
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