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Theorem binom2sub 12002
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 9629 . . . 4  |-  ( B  e.  CC  ->  -u B  e.  CC )
2 binom2 12000 . . . 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 9676 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  -u B )  =  ( A  -  B ) )
54oveq1d 6125 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  -u B ) ^ 2 )  =  ( ( A  -  B ) ^ 2 ) )
63, 5eqtr3d 2477 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  -u B ) ) )  +  (
-u B ^ 2 ) )  =  ( ( A  -  B
) ^ 2 ) )
7 mulneg2 9801 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  -u B
)  =  -u ( A  x.  B )
)
87oveq2d 6126 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  ( A  x.  -u B ) )  =  ( 2  x.  -u ( A  x.  B ) ) )
9 2cn 10411 . . . . . . 7  |-  2  e.  CC
10 mulcl 9385 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
11 mulneg2 9801 . . . . . . 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 2476 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> 
-u ( 2  x.  ( A  x.  B
) )  =  ( 2  x.  ( A  x.  -u B ) ) )
1413oveq2d 6126 . . . 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 11947 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 2 )  e.  CC )
1615adantr 465 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A ^ 2 )  e.  CC )
17 mulcl 9385 . . . . . 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 9744 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
2 )  +  -u ( 2  x.  ( A  x.  B )
) )  =  ( ( A ^ 2 )  -  ( 2  x.  ( A  x.  B ) ) ) )
2014, 19eqtr3d 2477 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
2 )  +  ( 2  x.  ( A  x.  -u B ) ) )  =  ( ( A ^ 2 )  -  ( 2  x.  ( A  x.  B
) ) ) )
21 sqneg 11945 . . . 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 6128 . 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 2477 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 1369    e. wcel 1756  (class class class)co 6110   CCcc 9299    + caddc 9304    x. cmul 9306    - cmin 9614   -ucneg 9615   2c2 10390   ^cexp 11884
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 2423  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-reu 2741  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-om 6496  df-2nd 6597  df-recs 6851  df-rdg 6885  df-er 7120  df-en 7330  df-dom 7331  df-sdom 7332  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-nn 10342  df-2 10399  df-n0 10599  df-z 10666  df-uz 10881  df-seq 11826  df-exp 11885
This theorem is referenced by:  binom2subi  12003  amgm2  12876  arisum2  13342  pythagtriplem1  13902  pythagtriplem14  13914  tangtx  21986  loglesqr  22215  heron  22252  dcubic1  22259  dquart  22267  asinsin  22306  rmspecsqrnq  29270
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