MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  subsq2 Structured version   Unicode version

Theorem subsq2 11989
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 10407 . . . . . . . 8  |-  2  e.  CC
2 mulcl 9381 . . . . . . . 8  |-  ( ( 2  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  B
)  e.  CC )
31, 2mpan 670 . . . . . . 7  |-  ( B  e.  CC  ->  (
2  x.  B )  e.  CC )
43adantl 466 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  B
)  e.  CC )
5 subadd23 9637 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  (
2  x.  B )  e.  CC )  -> 
( ( A  -  B )  +  ( 2  x.  B ) )  =  ( A  +  ( ( 2  x.  B )  -  B ) ) )
64, 5mpd3an3 1315 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B )  +  ( 2  x.  B ) )  =  ( A  +  ( ( 2  x.  B )  -  B ) ) )
7 2times 10455 . . . . . . . . 9  |-  ( B  e.  CC  ->  (
2  x.  B )  =  ( B  +  B ) )
87oveq1d 6121 . . . . . . . 8  |-  ( B  e.  CC  ->  (
( 2  x.  B
)  -  B )  =  ( ( B  +  B )  -  B ) )
9 pncan 9631 . . . . . . . . 9  |-  ( ( B  e.  CC  /\  B  e.  CC )  ->  ( ( B  +  B )  -  B
)  =  B )
109anidms 645 . . . . . . . 8  |-  ( B  e.  CC  ->  (
( B  +  B
)  -  B )  =  B )
118, 10eqtrd 2475 . . . . . . 7  |-  ( B  e.  CC  ->  (
( 2  x.  B
)  -  B )  =  B )
1211adantl 466 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 2  x.  B )  -  B
)  =  B )
1312oveq2d 6122 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  ( ( 2  x.  B
)  -  B ) )  =  ( A  +  B ) )
146, 13eqtrd 2475 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B )  +  ( 2  x.  B ) )  =  ( A  +  B ) )
1514oveq1d 6121 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  -  B )  +  ( 2  x.  B
) )  x.  ( A  -  B )
)  =  ( ( A  +  B )  x.  ( A  -  B ) ) )
16 subcl 9624 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
1716, 4, 16adddird 9426 . . 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 2477 . 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 11988 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
2 )  -  ( B ^ 2 ) )  =  ( ( A  +  B )  x.  ( A  -  B
) ) )
20 sqval 11940 . . . 4  |-  ( ( A  -  B )  e.  CC  ->  (
( A  -  B
) ^ 2 )  =  ( ( A  -  B )  x.  ( A  -  B
) ) )
2116, 20syl 16 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B ) ^ 2 )  =  ( ( A  -  B )  x.  ( A  -  B ) ) )
2221oveq1d 6121 . 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 2485 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
2 )  -  ( B ^ 2 ) )  =  ( ( ( A  -  B ) ^ 2 )  +  ( ( 2  x.  B )  x.  ( A  -  B )
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756  (class class class)co 6106   CCcc 9295    + caddc 9300    x. cmul 9302    - cmin 9610   2c2 10386   ^cexp 11880
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 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374
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 2735  df-rex 2736  df-reu 2737  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-om 6492  df-2nd 6593  df-recs 6847  df-rdg 6881  df-er 7116  df-en 7326  df-dom 7327  df-sdom 7328  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-nn 10338  df-2 10395  df-n0 10595  df-z 10662  df-uz 10877  df-seq 11822  df-exp 11881
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator