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Theorem subsq 11210
Description: Factor the difference of two squares. (Contributed by NM, 21-Feb-2008.)
Assertion
Ref Expression
subsq  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
2 )  -  ( B ^ 2 ) )  =  ( ( A  +  B )  x.  ( A  -  B
) ) )

Proof of Theorem subsq
StepHypRef Expression
1 simpl 443 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  A  e.  CC )
2 simpr 447 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  B  e.  CC )
3 subcl 9051 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
41, 2, 3adddird 8860 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  x.  ( A  -  B )
)  =  ( ( A  x.  ( A  -  B ) )  +  ( B  x.  ( A  -  B
) ) ) )
5 subdi 9213 . . . . 5  |-  ( ( A  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  ( A  -  B ) )  =  ( ( A  x.  A )  -  ( A  x.  B )
) )
653anidm12 1239 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  ( A  -  B )
)  =  ( ( A  x.  A )  -  ( A  x.  B ) ) )
7 sqval 11163 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 2 )  =  ( A  x.  A
) )
87adantr 451 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A ^ 2 )  =  ( A  x.  A ) )
98oveq1d 5873 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
2 )  -  ( A  x.  B )
)  =  ( ( A  x.  A )  -  ( A  x.  B ) ) )
106, 9eqtr4d 2318 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  ( A  -  B )
)  =  ( ( A ^ 2 )  -  ( A  x.  B ) ) )
112, 1, 2subdid 9235 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  x.  ( A  -  B )
)  =  ( ( B  x.  A )  -  ( B  x.  B ) ) )
12 mulcom 8823 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
13 sqval 11163 . . . . . 6  |-  ( B  e.  CC  ->  ( B ^ 2 )  =  ( B  x.  B
) )
1413adantl 452 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B ^ 2 )  =  ( B  x.  B ) )
1512, 14oveq12d 5876 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  -  ( B ^ 2 ) )  =  ( ( B  x.  A )  -  ( B  x.  B
) ) )
1611, 15eqtr4d 2318 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  x.  ( A  -  B )
)  =  ( ( A  x.  B )  -  ( B ^
2 ) ) )
1710, 16oveq12d 5876 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  ( A  -  B
) )  +  ( B  x.  ( A  -  B ) ) )  =  ( ( ( A ^ 2 )  -  ( A  x.  B ) )  +  ( ( A  x.  B )  -  ( B ^ 2 ) ) ) )
18 sqcl 11166 . . . 4  |-  ( A  e.  CC  ->  ( A ^ 2 )  e.  CC )
1918adantr 451 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A ^ 2 )  e.  CC )
20 mulcl 8821 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
21 sqcl 11166 . . . 4  |-  ( B  e.  CC  ->  ( B ^ 2 )  e.  CC )
2221adantl 452 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B ^ 2 )  e.  CC )
2319, 20, 22npncand 9181 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A ^ 2 )  -  ( A  x.  B
) )  +  ( ( A  x.  B
)  -  ( B ^ 2 ) ) )  =  ( ( A ^ 2 )  -  ( B ^
2 ) ) )
244, 17, 233eqtrrd 2320 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
2 )  -  ( B ^ 2 ) )  =  ( ( A  +  B )  x.  ( A  -  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684  (class class class)co 5858   CCcc 8735    + caddc 8740    x. cmul 8742    - cmin 9037   2c2 9795   ^cexp 11104
This theorem is referenced by:  subsq2  11211  subsqi  11214  pythagtriplem4  12872  pythagtriplem6  12874  pythagtriplem7  12875  pythagtriplem12  12879  pythagtriplem14  12881  pythagtriplem16  12883  4sqlem8  12992  4sqlem10  12994  4sqlem11  13002  chordthmlem4  20132  dcubic2  20140  cubic  20145  dquart  20149  asinlem2  20165  asinsin  20188  efiatan2  20213  atans2  20227  dvatan  20231  wilthlem1  20306  lgslem1  20535  lgsqrlem2  20581  2sqlem4  20606  2sqblem  20616  rplogsumlem1  20633  pellexlem2  26915  pell1234qrne0  26938  pell1234qrreccl  26939  pell1234qrmulcl  26940  pell14qrdich  26954  rmxyneg  27005  stoweidlem1  27750
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-seq 11047  df-exp 11105
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