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Theorem binom2 11088
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 8304 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
2 simpl 109 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> A e. CC )
3 simpr 110 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> B e. CC )
41, 2, 3adddid 8350 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) x. ( A + B ) ) = ( ( ( A + B ) x. A ) + ( ( A + B ) x. B ) ) )
52, 3, 2adddird 8351 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) x. A ) = ( ( A x. A ) + ( B x. A ) ) )
63, 2mulcomd 8347 . . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> ( B x. A ) = ( A x. B ) )
76oveq2d 6101 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. A ) + ( B x. A ) ) = ( ( A x. A ) + ( A x. B ) ) )
85, 7eqtrd 2271 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) x. A ) = ( ( A x. A ) + ( A x. B ) ) )
92, 3, 3adddird 8351 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) x. B ) = ( ( A x. B ) + ( B x. B ) ) )
108, 9oveq12d 6103 . . . 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 8346 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A x. A ) e. CC )
122, 3mulcld 8346 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
1311, 12addcld 8345 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. A ) + ( A x. B ) ) e. CC )
143, 3mulcld 8346 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( B x. B ) e. CC )
1513, 12, 14addassd 8348 . . . 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 8348 . . . . 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 6100 . . . 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 2277 . . 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 2271 . 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 11034 . . 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 11034 . . . . 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 9548 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( 2 x. ( A x. B ) ) = ( ( A x. B ) + ( A x. B ) ) )
2523, 24oveq12d 6103 . . 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 11034 . . . 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 6103 . 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 2281 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
This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209  (class class class)co 6085   CCcc 8177   + caddc 8182   x. cmul 8184   2c2 9355   ^cexp 10975
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297
This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-n0 9564  df-z 9645  df-uz 9922  df-seqfrec 10885  df-exp 10976
This theorem is used by:  binom21  11089  binom2sub  11090  mulbinom2  11093  binom3  11094  nn0opthlem1d  11158  sq01  11660  resqrexlemover  11776  resqrexlemcalc1  11780  abstri  11870  amgm2  11884  bdtrilem  12005  pythagtriplem1  13044  pythagtriplem12  13054
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