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Theorem binom2sub 12265
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 9832 . . . 4  |-  ( B  e.  CC  ->  -u B  e.  CC )
2 binom2 12263 . . . 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 474 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  -u B ) ^ 2 )  =  ( ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  -u B ) ) )  +  ( -u B ^ 2 ) ) )
4 negsub 9879 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  -u B )  =  ( A  -  B ) )
54oveq1d 6310 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  -u B ) ^ 2 )  =  ( ( A  -  B ) ^ 2 ) )
63, 5eqtr3d 2510 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  -u B ) ) )  +  (
-u B ^ 2 ) )  =  ( ( A  -  B
) ^ 2 ) )
7 mulneg2 10006 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  -u B
)  =  -u ( A  x.  B )
)
87oveq2d 6311 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  ( A  x.  -u B ) )  =  ( 2  x.  -u ( A  x.  B ) ) )
9 2cn 10618 . . . . . . 7  |-  2  e.  CC
10 mulcl 9588 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
11 mulneg2 10006 . . . . . . 7  |-  ( ( 2  e.  CC  /\  ( A  x.  B
)  e.  CC )  ->  ( 2  x.  -u ( A  x.  B
) )  =  -u ( 2  x.  ( A  x.  B )
) )
129, 10, 11sylancr 663 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  -u ( A  x.  B )
)  =  -u (
2  x.  ( A  x.  B ) ) )
138, 12eqtr2d 2509 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> 
-u ( 2  x.  ( A  x.  B
) )  =  ( 2  x.  ( A  x.  -u B ) ) )
1413oveq2d 6311 . . . 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 12210 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 2 )  e.  CC )
1615adantr 465 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A ^ 2 )  e.  CC )
17 mulcl 9588 . . . . . 6  |-  ( ( 2  e.  CC  /\  ( A  x.  B
)  e.  CC )  ->  ( 2  x.  ( A  x.  B
) )  e.  CC )
189, 10, 17sylancr 663 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  ( A  x.  B )
)  e.  CC )
1916, 18negsubd 9948 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
2 )  +  -u ( 2  x.  ( A  x.  B )
) )  =  ( ( A ^ 2 )  -  ( 2  x.  ( A  x.  B ) ) ) )
2014, 19eqtr3d 2510 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
2 )  +  ( 2  x.  ( A  x.  -u B ) ) )  =  ( ( A ^ 2 )  -  ( 2  x.  ( A  x.  B
) ) ) )
21 sqneg 12208 . . . 4  |-  ( B  e.  CC  ->  ( -u B ^ 2 )  =  ( B ^
2 ) )
2221adantl 466 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( -u B ^
2 )  =  ( B ^ 2 ) )
2320, 22oveq12d 6313 . 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 2510 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B ) ^ 2 )  =  ( ( ( A ^ 2 )  -  ( 2  x.  ( A  x.  B ) ) )  +  ( B ^
2 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767  (class class class)co 6295   CCcc 9502    + caddc 9507    x. cmul 9509    - cmin 9817   -ucneg 9818   2c2 10597   ^cexp 12146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-seq 12088  df-exp 12147
This theorem is referenced by:  binom2subi  12266  amgm2  13182  arisum2  13652  pythagtriplem1  14216  pythagtriplem14  14228  tangtx  22764  loglesqrt  22998  heron  23035  dcubic1  23042  dquart  23050  asinsin  23089  rmspecsqrtnq  30770
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