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Theorem binom2i 11491
Description: The square of a binomial. (Contributed by NM, 11-Aug-1999.)
Hypotheses
Ref Expression
binom2.1  |-  A  e.  CC
binom2.2  |-  B  e.  CC
Assertion
Ref Expression
binom2i  |-  ( ( A  +  B ) ^ 2 )  =  ( ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  B )
) )  +  ( B ^ 2 ) )

Proof of Theorem binom2i
StepHypRef Expression
1 binom2.1 . . . . 5  |-  A  e.  CC
2 binom2.2 . . . . 5  |-  B  e.  CC
31, 2addcli 9095 . . . 4  |-  ( A  +  B )  e.  CC
43, 1, 2adddii 9101 . . 3  |-  ( ( A  +  B )  x.  ( A  +  B ) )  =  ( ( ( A  +  B )  x.  A )  +  ( ( A  +  B
)  x.  B ) )
51, 2, 1adddiri 9102 . . . . . 6  |-  ( ( A  +  B )  x.  A )  =  ( ( A  x.  A )  +  ( B  x.  A ) )
62, 1mulcomi 9097 . . . . . . 7  |-  ( B  x.  A )  =  ( A  x.  B
)
76oveq2i 6093 . . . . . 6  |-  ( ( A  x.  A )  +  ( B  x.  A ) )  =  ( ( A  x.  A )  +  ( A  x.  B ) )
85, 7eqtri 2457 . . . . 5  |-  ( ( A  +  B )  x.  A )  =  ( ( A  x.  A )  +  ( A  x.  B ) )
91, 2, 2adddiri 9102 . . . . 5  |-  ( ( A  +  B )  x.  B )  =  ( ( A  x.  B )  +  ( B  x.  B ) )
108, 9oveq12i 6094 . . . 4  |-  ( ( ( A  +  B
)  x.  A )  +  ( ( A  +  B )  x.  B ) )  =  ( ( ( A  x.  A )  +  ( A  x.  B
) )  +  ( ( A  x.  B
)  +  ( B  x.  B ) ) )
111, 1mulcli 9096 . . . . . 6  |-  ( A  x.  A )  e.  CC
121, 2mulcli 9096 . . . . . 6  |-  ( A  x.  B )  e.  CC
1311, 12addcli 9095 . . . . 5  |-  ( ( A  x.  A )  +  ( A  x.  B ) )  e.  CC
142, 2mulcli 9096 . . . . 5  |-  ( B  x.  B )  e.  CC
1513, 12, 14addassi 9099 . . . 4  |-  ( ( ( ( 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, 12addassi 9099 . . . . 5  |-  ( ( ( A  x.  A
)  +  ( A  x.  B ) )  +  ( A  x.  B ) )  =  ( ( A  x.  A )  +  ( ( A  x.  B
)  +  ( A  x.  B ) ) )
1716oveq1i 6092 . . . 4  |-  ( ( ( ( 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, 173eqtr2i 2463 . . 3  |-  ( ( ( A  +  B
)  x.  A )  +  ( ( A  +  B )  x.  B ) )  =  ( ( ( A  x.  A )  +  ( ( A  x.  B )  +  ( A  x.  B ) ) )  +  ( B  x.  B ) )
194, 18eqtri 2457 . 2  |-  ( ( A  +  B )  x.  ( A  +  B ) )  =  ( ( ( A  x.  A )  +  ( ( A  x.  B )  +  ( A  x.  B ) ) )  +  ( B  x.  B ) )
203sqvali 11462 . 2  |-  ( ( A  +  B ) ^ 2 )  =  ( ( A  +  B )  x.  ( A  +  B )
)
211sqvali 11462 . . . 4  |-  ( A ^ 2 )  =  ( A  x.  A
)
22122timesi 10102 . . . 4  |-  ( 2  x.  ( A  x.  B ) )  =  ( ( A  x.  B )  +  ( A  x.  B ) )
2321, 22oveq12i 6094 . . 3  |-  ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  B
) ) )  =  ( ( A  x.  A )  +  ( ( A  x.  B
)  +  ( A  x.  B ) ) )
242sqvali 11462 . . 3  |-  ( B ^ 2 )  =  ( B  x.  B
)
2523, 24oveq12i 6094 . 2  |-  ( ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  B ) ) )  +  ( B ^
2 ) )  =  ( ( ( A  x.  A )  +  ( ( A  x.  B )  +  ( A  x.  B ) ) )  +  ( B  x.  B ) )
2619, 20, 253eqtr4i 2467 1  |-  ( ( A  +  B ) ^ 2 )  =  ( ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  B )
) )  +  ( B ^ 2 ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726  (class class class)co 6082   CCcc 8989    + caddc 8994    x. cmul 8996   2c2 10050   ^cexp 11383
This theorem is referenced by:  binom2  11497  nn0opthlem1  11562  norm-ii-i  22640  ax5seglem7  25875
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-nn 10002  df-2 10059  df-n0 10223  df-z 10284  df-uz 10490  df-seq 11325  df-exp 11384
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