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Theorem binom2i 10425
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 7789 . . . 4  |-  ( A  +  B )  e.  CC
43, 1, 2adddii 7795 . . 3  |-  ( ( A  +  B )  x.  ( A  +  B ) )  =  ( ( ( A  +  B )  x.  A )  +  ( ( A  +  B
)  x.  B ) )
51, 2, 1adddiri 7796 . . . . . 6  |-  ( ( A  +  B )  x.  A )  =  ( ( A  x.  A )  +  ( B  x.  A ) )
62, 1mulcomi 7791 . . . . . . 7  |-  ( B  x.  A )  =  ( A  x.  B
)
76oveq2i 5788 . . . . . 6  |-  ( ( A  x.  A )  +  ( B  x.  A ) )  =  ( ( A  x.  A )  +  ( A  x.  B ) )
85, 7eqtri 2160 . . . . 5  |-  ( ( A  +  B )  x.  A )  =  ( ( A  x.  A )  +  ( A  x.  B ) )
91, 2, 2adddiri 7796 . . . . 5  |-  ( ( A  +  B )  x.  B )  =  ( ( A  x.  B )  +  ( B  x.  B ) )
108, 9oveq12i 5789 . . . 4  |-  ( ( ( A  +  B
)  x.  A )  +  ( ( A  +  B )  x.  B ) )  =  ( ( ( A  x.  A )  +  ( A  x.  B
) )  +  ( ( A  x.  B
)  +  ( B  x.  B ) ) )
111, 1mulcli 7790 . . . . . 6  |-  ( A  x.  A )  e.  CC
121, 2mulcli 7790 . . . . . 6  |-  ( A  x.  B )  e.  CC
1311, 12addcli 7789 . . . . 5  |-  ( ( A  x.  A )  +  ( A  x.  B ) )  e.  CC
142, 2mulcli 7790 . . . . 5  |-  ( B  x.  B )  e.  CC
1513, 12, 14addassi 7793 . . . 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 7793 . . . . 5  |-  ( ( ( A  x.  A
)  +  ( A  x.  B ) )  +  ( A  x.  B ) )  =  ( ( A  x.  A )  +  ( ( A  x.  B
)  +  ( A  x.  B ) ) )
1716oveq1i 5787 . . . 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 2166 . . 3  |-  ( ( ( A  +  B
)  x.  A )  +  ( ( A  +  B )  x.  B ) )  =  ( ( ( A  x.  A )  +  ( ( A  x.  B )  +  ( A  x.  B ) ) )  +  ( B  x.  B ) )
194, 18eqtri 2160 . 2  |-  ( ( A  +  B )  x.  ( A  +  B ) )  =  ( ( ( A  x.  A )  +  ( ( A  x.  B )  +  ( A  x.  B ) ) )  +  ( B  x.  B ) )
203sqvali 10396 . 2  |-  ( ( A  +  B ) ^ 2 )  =  ( ( A  +  B )  x.  ( A  +  B )
)
211sqvali 10396 . . . 4  |-  ( A ^ 2 )  =  ( A  x.  A
)
22122timesi 8869 . . . 4  |-  ( 2  x.  ( A  x.  B ) )  =  ( ( A  x.  B )  +  ( A  x.  B ) )
2321, 22oveq12i 5789 . . 3  |-  ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  B
) ) )  =  ( ( A  x.  A )  +  ( ( A  x.  B
)  +  ( A  x.  B ) ) )
242sqvali 10396 . . 3  |-  ( B ^ 2 )  =  ( B  x.  B
)
2523, 24oveq12i 5789 . 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 2170 1  |-  ( ( A  +  B ) ^ 2 )  =  ( ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  B )
) )  +  ( B ^ 2 ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1331    e. wcel 1480  (class class class)co 5777   CCcc 7637    + caddc 7642    x. cmul 7644   2c2 8790   ^cexp 10316
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4046  ax-sep 4049  ax-nul 4057  ax-pow 4101  ax-pr 4134  ax-un 4358  ax-setind 4455  ax-iinf 4505  ax-cnex 7730  ax-resscn 7731  ax-1cn 7732  ax-1re 7733  ax-icn 7734  ax-addcl 7735  ax-addrcl 7736  ax-mulcl 7737  ax-mulrcl 7738  ax-addcom 7739  ax-mulcom 7740  ax-addass 7741  ax-mulass 7742  ax-distr 7743  ax-i2m1 7744  ax-0lt1 7745  ax-1rid 7746  ax-0id 7747  ax-rnegex 7748  ax-precex 7749  ax-cnre 7750  ax-pre-ltirr 7751  ax-pre-ltwlin 7752  ax-pre-lttrn 7753  ax-pre-apti 7754  ax-pre-ltadd 7755  ax-pre-mulgt0 7756  ax-pre-mulext 7757
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-int 3775  df-iun 3818  df-br 3933  df-opab 3993  df-mpt 3994  df-tr 4030  df-id 4218  df-po 4221  df-iso 4222  df-iord 4291  df-on 4293  df-ilim 4294  df-suc 4296  df-iom 4508  df-xp 4548  df-rel 4549  df-cnv 4550  df-co 4551  df-dm 4552  df-rn 4553  df-res 4554  df-ima 4555  df-iota 5091  df-fun 5128  df-fn 5129  df-f 5130  df-f1 5131  df-fo 5132  df-f1o 5133  df-fv 5134  df-riota 5733  df-ov 5780  df-oprab 5781  df-mpo 5782  df-1st 6041  df-2nd 6042  df-recs 6205  df-frec 6291  df-pnf 7821  df-mnf 7822  df-xr 7823  df-ltxr 7824  df-le 7825  df-sub 7954  df-neg 7955  df-reap 8356  df-ap 8363  df-div 8452  df-inn 8740  df-2 8798  df-n0 8997  df-z 9074  df-uz 9346  df-seqfrec 10243  df-exp 10317
This theorem is referenced by: (None)
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