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Theorem binom2i 11971
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 9386 . . . 4  |-  ( A  +  B )  e.  CC
43, 1, 2adddii 9392 . . 3  |-  ( ( A  +  B )  x.  ( A  +  B ) )  =  ( ( ( A  +  B )  x.  A )  +  ( ( A  +  B
)  x.  B ) )
51, 2, 1adddiri 9393 . . . . . 6  |-  ( ( A  +  B )  x.  A )  =  ( ( A  x.  A )  +  ( B  x.  A ) )
62, 1mulcomi 9388 . . . . . . 7  |-  ( B  x.  A )  =  ( A  x.  B
)
76oveq2i 6101 . . . . . 6  |-  ( ( A  x.  A )  +  ( B  x.  A ) )  =  ( ( A  x.  A )  +  ( A  x.  B ) )
85, 7eqtri 2461 . . . . 5  |-  ( ( A  +  B )  x.  A )  =  ( ( A  x.  A )  +  ( A  x.  B ) )
91, 2, 2adddiri 9393 . . . . 5  |-  ( ( A  +  B )  x.  B )  =  ( ( A  x.  B )  +  ( B  x.  B ) )
108, 9oveq12i 6102 . . . 4  |-  ( ( ( A  +  B
)  x.  A )  +  ( ( A  +  B )  x.  B ) )  =  ( ( ( A  x.  A )  +  ( A  x.  B
) )  +  ( ( A  x.  B
)  +  ( B  x.  B ) ) )
111, 1mulcli 9387 . . . . . 6  |-  ( A  x.  A )  e.  CC
121, 2mulcli 9387 . . . . . 6  |-  ( A  x.  B )  e.  CC
1311, 12addcli 9386 . . . . 5  |-  ( ( A  x.  A )  +  ( A  x.  B ) )  e.  CC
142, 2mulcli 9387 . . . . 5  |-  ( B  x.  B )  e.  CC
1513, 12, 14addassi 9390 . . . 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 9390 . . . . 5  |-  ( ( ( A  x.  A
)  +  ( A  x.  B ) )  +  ( A  x.  B ) )  =  ( ( A  x.  A )  +  ( ( A  x.  B
)  +  ( A  x.  B ) ) )
1716oveq1i 6100 . . . 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 2467 . . 3  |-  ( ( ( A  +  B
)  x.  A )  +  ( ( A  +  B )  x.  B ) )  =  ( ( ( A  x.  A )  +  ( ( A  x.  B )  +  ( A  x.  B ) ) )  +  ( B  x.  B ) )
194, 18eqtri 2461 . 2  |-  ( ( A  +  B )  x.  ( A  +  B ) )  =  ( ( ( A  x.  A )  +  ( ( A  x.  B )  +  ( A  x.  B ) ) )  +  ( B  x.  B ) )
203sqvali 11941 . 2  |-  ( ( A  +  B ) ^ 2 )  =  ( ( A  +  B )  x.  ( A  +  B )
)
211sqvali 11941 . . . 4  |-  ( A ^ 2 )  =  ( A  x.  A
)
22122timesi 10438 . . . 4  |-  ( 2  x.  ( A  x.  B ) )  =  ( ( A  x.  B )  +  ( A  x.  B ) )
2321, 22oveq12i 6102 . . 3  |-  ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  B
) ) )  =  ( ( A  x.  A )  +  ( ( A  x.  B
)  +  ( A  x.  B ) ) )
242sqvali 11941 . . 3  |-  ( B ^ 2 )  =  ( B  x.  B
)
2523, 24oveq12i 6102 . 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 2471 1  |-  ( ( A  +  B ) ^ 2 )  =  ( ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  B )
) )  +  ( B ^ 2 ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1364    e. wcel 1761  (class class class)co 6090   CCcc 9276    + caddc 9281    x. cmul 9283   2c2 10367   ^cexp 11861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-n0 10576  df-z 10643  df-uz 10858  df-seq 11803  df-exp 11862
This theorem is referenced by:  binom2  11977  nn0opthlem1  12042  ax5seglem7  23116  norm-ii-i  24474
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