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Theorem binom2i 12232
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 9589 . . . 4  |-  ( A  +  B )  e.  CC
43, 1, 2adddii 9595 . . 3  |-  ( ( A  +  B )  x.  ( A  +  B ) )  =  ( ( ( A  +  B )  x.  A )  +  ( ( A  +  B
)  x.  B ) )
51, 2, 1adddiri 9596 . . . . . 6  |-  ( ( A  +  B )  x.  A )  =  ( ( A  x.  A )  +  ( B  x.  A ) )
62, 1mulcomi 9591 . . . . . . 7  |-  ( B  x.  A )  =  ( A  x.  B
)
76oveq2i 6286 . . . . . 6  |-  ( ( A  x.  A )  +  ( B  x.  A ) )  =  ( ( A  x.  A )  +  ( A  x.  B ) )
85, 7eqtri 2489 . . . . 5  |-  ( ( A  +  B )  x.  A )  =  ( ( A  x.  A )  +  ( A  x.  B ) )
91, 2, 2adddiri 9596 . . . . 5  |-  ( ( A  +  B )  x.  B )  =  ( ( A  x.  B )  +  ( B  x.  B ) )
108, 9oveq12i 6287 . . . 4  |-  ( ( ( A  +  B
)  x.  A )  +  ( ( A  +  B )  x.  B ) )  =  ( ( ( A  x.  A )  +  ( A  x.  B
) )  +  ( ( A  x.  B
)  +  ( B  x.  B ) ) )
111, 1mulcli 9590 . . . . . 6  |-  ( A  x.  A )  e.  CC
121, 2mulcli 9590 . . . . . 6  |-  ( A  x.  B )  e.  CC
1311, 12addcli 9589 . . . . 5  |-  ( ( A  x.  A )  +  ( A  x.  B ) )  e.  CC
142, 2mulcli 9590 . . . . 5  |-  ( B  x.  B )  e.  CC
1513, 12, 14addassi 9593 . . . 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 9593 . . . . 5  |-  ( ( ( A  x.  A
)  +  ( A  x.  B ) )  +  ( A  x.  B ) )  =  ( ( A  x.  A )  +  ( ( A  x.  B
)  +  ( A  x.  B ) ) )
1716oveq1i 6285 . . . 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 2495 . . 3  |-  ( ( ( A  +  B
)  x.  A )  +  ( ( A  +  B )  x.  B ) )  =  ( ( ( A  x.  A )  +  ( ( A  x.  B )  +  ( A  x.  B ) ) )  +  ( B  x.  B ) )
194, 18eqtri 2489 . 2  |-  ( ( A  +  B )  x.  ( A  +  B ) )  =  ( ( ( A  x.  A )  +  ( ( A  x.  B )  +  ( A  x.  B ) ) )  +  ( B  x.  B ) )
203sqvali 12202 . 2  |-  ( ( A  +  B ) ^ 2 )  =  ( ( A  +  B )  x.  ( A  +  B )
)
211sqvali 12202 . . . 4  |-  ( A ^ 2 )  =  ( A  x.  A
)
22122timesi 10645 . . . 4  |-  ( 2  x.  ( A  x.  B ) )  =  ( ( A  x.  B )  +  ( A  x.  B ) )
2321, 22oveq12i 6287 . . 3  |-  ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  B
) ) )  =  ( ( A  x.  A )  +  ( ( A  x.  B
)  +  ( A  x.  B ) ) )
242sqvali 12202 . . 3  |-  ( B ^ 2 )  =  ( B  x.  B
)
2523, 24oveq12i 6287 . 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 2499 1  |-  ( ( A  +  B ) ^ 2 )  =  ( ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  B )
) )  +  ( B ^ 2 ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1374    e. wcel 1762  (class class class)co 6275   CCcc 9479    + caddc 9484    x. cmul 9486   2c2 10574   ^cexp 12122
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-n0 10785  df-z 10854  df-uz 11072  df-seq 12064  df-exp 12123
This theorem is referenced by:  binom2  12238  nn0opthlem1  12303  ax5seglem7  23907  norm-ii-i  25716  quad3  28485
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