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Theorem subsq2 10526
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 8904 . . . . . . . 8  |-  2  e.  CC
2 mulcl 7859 . . . . . . . 8  |-  ( ( 2  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  B
)  e.  CC )
31, 2mpan 421 . . . . . . 7  |-  ( B  e.  CC  ->  (
2  x.  B )  e.  CC )
43adantl 275 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  B
)  e.  CC )
5 subadd23 8087 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  (
2  x.  B )  e.  CC )  -> 
( ( A  -  B )  +  ( 2  x.  B ) )  =  ( A  +  ( ( 2  x.  B )  -  B ) ) )
64, 5mpd3an3 1320 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B )  +  ( 2  x.  B ) )  =  ( A  +  ( ( 2  x.  B )  -  B ) ) )
7 2times 8961 . . . . . . . . 9  |-  ( B  e.  CC  ->  (
2  x.  B )  =  ( B  +  B ) )
87oveq1d 5839 . . . . . . . 8  |-  ( B  e.  CC  ->  (
( 2  x.  B
)  -  B )  =  ( ( B  +  B )  -  B ) )
9 pncan 8081 . . . . . . . . 9  |-  ( ( B  e.  CC  /\  B  e.  CC )  ->  ( ( B  +  B )  -  B
)  =  B )
109anidms 395 . . . . . . . 8  |-  ( B  e.  CC  ->  (
( B  +  B
)  -  B )  =  B )
118, 10eqtrd 2190 . . . . . . 7  |-  ( B  e.  CC  ->  (
( 2  x.  B
)  -  B )  =  B )
1211adantl 275 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 2  x.  B )  -  B
)  =  B )
1312oveq2d 5840 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  ( ( 2  x.  B
)  -  B ) )  =  ( A  +  B ) )
146, 13eqtrd 2190 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B )  +  ( 2  x.  B ) )  =  ( A  +  B ) )
1514oveq1d 5839 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  -  B )  +  ( 2  x.  B
) )  x.  ( A  -  B )
)  =  ( ( A  +  B )  x.  ( A  -  B ) ) )
16 subcl 8074 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
1716, 4, 16adddird 7903 . . 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 2192 . 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 10525 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
2 )  -  ( B ^ 2 ) )  =  ( ( A  +  B )  x.  ( A  -  B
) ) )
20 sqval 10477 . . . 4  |-  ( ( A  -  B )  e.  CC  ->  (
( A  -  B
) ^ 2 )  =  ( ( A  -  B )  x.  ( A  -  B
) ) )
2116, 20syl 14 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B ) ^ 2 )  =  ( ( A  -  B )  x.  ( A  -  B ) ) )
2221oveq1d 5839 . 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 2200 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 103    = wceq 1335    e. wcel 2128  (class class class)co 5824   CCcc 7730    + caddc 7735    x. cmul 7737    - cmin 8046   2c2 8884   ^cexp 10418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849  ax-pre-mulext 7850
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-frec 6338  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457  df-div 8546  df-inn 8834  df-2 8892  df-n0 9091  df-z 9168  df-uz 9440  df-seqfrec 10345  df-exp 10419
This theorem is referenced by: (None)
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