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Theorem subsq2 11258
Description: Express the difference of the squares of two numbers as a polynomial in the difference of the numbers. (Contributed by NM, 21-Feb-2008.)
Assertion
Ref Expression
subsq2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
2 )  -  ( B ^ 2 ) )  =  ( ( ( A  -  B ) ^ 2 )  +  ( ( 2  x.  B )  x.  ( A  -  B )
) ) )

Proof of Theorem subsq2
StepHypRef Expression
1 2cn 9861 . . . . . . . 8  |-  2  e.  CC
2 mulcl 8866 . . . . . . . 8  |-  ( ( 2  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  B
)  e.  CC )
31, 2mpan 651 . . . . . . 7  |-  ( B  e.  CC  ->  (
2  x.  B )  e.  CC )
43adantl 452 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  B
)  e.  CC )
5 subadd23 9108 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  (
2  x.  B )  e.  CC )  -> 
( ( A  -  B )  +  ( 2  x.  B ) )  =  ( A  +  ( ( 2  x.  B )  -  B ) ) )
64, 5mpd3an3 1278 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B )  +  ( 2  x.  B ) )  =  ( A  +  ( ( 2  x.  B )  -  B ) ) )
7 2times 9890 . . . . . . . . 9  |-  ( B  e.  CC  ->  (
2  x.  B )  =  ( B  +  B ) )
87oveq1d 5915 . . . . . . . 8  |-  ( B  e.  CC  ->  (
( 2  x.  B
)  -  B )  =  ( ( B  +  B )  -  B ) )
9 pncan 9102 . . . . . . . . 9  |-  ( ( B  e.  CC  /\  B  e.  CC )  ->  ( ( B  +  B )  -  B
)  =  B )
109anidms 626 . . . . . . . 8  |-  ( B  e.  CC  ->  (
( B  +  B
)  -  B )  =  B )
118, 10eqtrd 2348 . . . . . . 7  |-  ( B  e.  CC  ->  (
( 2  x.  B
)  -  B )  =  B )
1211adantl 452 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 2  x.  B )  -  B
)  =  B )
1312oveq2d 5916 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  ( ( 2  x.  B
)  -  B ) )  =  ( A  +  B ) )
146, 13eqtrd 2348 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B )  +  ( 2  x.  B ) )  =  ( A  +  B ) )
1514oveq1d 5915 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  -  B )  +  ( 2  x.  B
) )  x.  ( A  -  B )
)  =  ( ( A  +  B )  x.  ( A  -  B ) ) )
16 subcl 9096 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
1716, 4, 16adddird 8905 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  -  B )  +  ( 2  x.  B
) )  x.  ( A  -  B )
)  =  ( ( ( A  -  B
)  x.  ( A  -  B ) )  +  ( ( 2  x.  B )  x.  ( A  -  B
) ) ) )
1815, 17eqtr3d 2350 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  x.  ( A  -  B )
)  =  ( ( ( A  -  B
)  x.  ( A  -  B ) )  +  ( ( 2  x.  B )  x.  ( A  -  B
) ) ) )
19 subsq 11257 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
2 )  -  ( B ^ 2 ) )  =  ( ( A  +  B )  x.  ( A  -  B
) ) )
20 sqval 11210 . . . 4  |-  ( ( A  -  B )  e.  CC  ->  (
( A  -  B
) ^ 2 )  =  ( ( A  -  B )  x.  ( A  -  B
) ) )
2116, 20syl 15 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B ) ^ 2 )  =  ( ( A  -  B )  x.  ( A  -  B ) ) )
2221oveq1d 5915 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  -  B ) ^
2 )  +  ( ( 2  x.  B
)  x.  ( A  -  B ) ) )  =  ( ( ( A  -  B
)  x.  ( A  -  B ) )  +  ( ( 2  x.  B )  x.  ( A  -  B
) ) ) )
2318, 19, 223eqtr4d 2358 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
2 )  -  ( B ^ 2 ) )  =  ( ( ( A  -  B ) ^ 2 )  +  ( ( 2  x.  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 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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