ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  binom2 Unicode version

Theorem binom2 10371
Description: The square of a binomial. (Contributed by FL, 10-Dec-2006.)
Assertion
Ref Expression
binom2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B ) ^ 2 )  =  ( ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  B ) ) )  +  ( B ^
2 ) ) )

Proof of Theorem binom2
StepHypRef Expression
1 addcl 7713 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
2 simpl 108 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  A  e.  CC )
3 simpr 109 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  B  e.  CC )
41, 2, 3adddid 7758 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  x.  ( A  +  B )
)  =  ( ( ( A  +  B
)  x.  A )  +  ( ( A  +  B )  x.  B ) ) )
52, 3, 2adddird 7759 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  x.  A
)  =  ( ( A  x.  A )  +  ( B  x.  A ) ) )
63, 2mulcomd 7755 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  x.  A
)  =  ( A  x.  B ) )
76oveq2d 5758 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  A )  +  ( B  x.  A ) )  =  ( ( A  x.  A )  +  ( A  x.  B ) ) )
85, 7eqtrd 2150 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  x.  A
)  =  ( ( A  x.  A )  +  ( A  x.  B ) ) )
92, 3, 3adddird 7759 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  x.  B
)  =  ( ( A  x.  B )  +  ( B  x.  B ) ) )
108, 9oveq12d 5760 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  x.  A )  +  ( ( A  +  B
)  x.  B ) )  =  ( ( ( A  x.  A
)  +  ( A  x.  B ) )  +  ( ( A  x.  B )  +  ( B  x.  B
) ) ) )
112, 2mulcld 7754 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  A
)  e.  CC )
122, 3mulcld 7754 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
1311, 12addcld 7753 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  A )  +  ( A  x.  B ) )  e.  CC )
143, 3mulcld 7754 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  x.  B
)  e.  CC )
1513, 12, 14addassd 7756 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( ( A  x.  A )  +  ( A  x.  B ) )  +  ( A  x.  B
) )  +  ( B  x.  B ) )  =  ( ( ( A  x.  A
)  +  ( A  x.  B ) )  +  ( ( A  x.  B )  +  ( B  x.  B
) ) ) )
1611, 12, 12addassd 7756 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  x.  A )  +  ( A  x.  B
) )  +  ( A  x.  B ) )  =  ( ( A  x.  A )  +  ( ( A  x.  B )  +  ( A  x.  B
) ) ) )
1716oveq1d 5757 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( ( A  x.  A )  +  ( A  x.  B ) )  +  ( A  x.  B
) )  +  ( B  x.  B ) )  =  ( ( ( A  x.  A
)  +  ( ( A  x.  B )  +  ( A  x.  B ) ) )  +  ( B  x.  B ) ) )
1810, 15, 173eqtr2d 2156 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  x.  A )  +  ( ( A  +  B
)  x.  B ) )  =  ( ( ( A  x.  A
)  +  ( ( A  x.  B )  +  ( A  x.  B ) ) )  +  ( B  x.  B ) ) )
194, 18eqtrd 2150 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  x.  ( A  +  B )
)  =  ( ( ( A  x.  A
)  +  ( ( A  x.  B )  +  ( A  x.  B ) ) )  +  ( B  x.  B ) ) )
20 sqval 10319 . . 3  |-  ( ( A  +  B )  e.  CC  ->  (
( A  +  B
) ^ 2 )  =  ( ( A  +  B )  x.  ( A  +  B
) ) )
211, 20syl 14 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B ) ^ 2 )  =  ( ( A  +  B )  x.  ( A  +  B ) ) )
22 sqval 10319 . . . . 5  |-  ( A  e.  CC  ->  ( A ^ 2 )  =  ( A  x.  A
) )
232, 22syl 14 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A ^ 2 )  =  ( A  x.  A ) )
24122timesd 8930 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  ( A  x.  B )
)  =  ( ( A  x.  B )  +  ( A  x.  B ) ) )
2523, 24oveq12d 5760 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
2 )  +  ( 2  x.  ( A  x.  B ) ) )  =  ( ( A  x.  A )  +  ( ( A  x.  B )  +  ( A  x.  B
) ) ) )
26 sqval 10319 . . . 4  |-  ( B  e.  CC  ->  ( B ^ 2 )  =  ( B  x.  B
) )
273, 26syl 14 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B ^ 2 )  =  ( B  x.  B ) )
2825, 27oveq12d 5760 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  B )
) )  +  ( B ^ 2 ) )  =  ( ( ( A  x.  A
)  +  ( ( A  x.  B )  +  ( A  x.  B ) ) )  +  ( B  x.  B ) ) )
2919, 21, 283eqtr4d 2160 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B ) ^ 2 )  =  ( ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  B ) ) )  +  ( B ^
2 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465  (class class class)co 5742   CCcc 7586    + caddc 7591    x. cmul 7593   2c2 8739   ^cexp 10260
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312  df-div 8401  df-inn 8689  df-2 8747  df-n0 8946  df-z 9023  df-uz 9295  df-seqfrec 10187  df-exp 10261
This theorem is referenced by:  binom21  10372  binom2sub  10373  mulbinom2  10376  binom3  10377  nn0opthlem1d  10434  resqrexlemover  10750  resqrexlemcalc1  10754  abstri  10844  amgm2  10858  bdtrilem  10978
  Copyright terms: Public domain W3C validator