MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  binom2i Structured version   Unicode version

Theorem binom2i 12251
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 9598 . . . 4  |-  ( A  +  B )  e.  CC
43, 1, 2adddii 9604 . . 3  |-  ( ( A  +  B )  x.  ( A  +  B ) )  =  ( ( ( A  +  B )  x.  A )  +  ( ( A  +  B
)  x.  B ) )
51, 2, 1adddiri 9605 . . . . . 6  |-  ( ( A  +  B )  x.  A )  =  ( ( A  x.  A )  +  ( B  x.  A ) )
62, 1mulcomi 9600 . . . . . . 7  |-  ( B  x.  A )  =  ( A  x.  B
)
76oveq2i 6288 . . . . . 6  |-  ( ( A  x.  A )  +  ( B  x.  A ) )  =  ( ( A  x.  A )  +  ( A  x.  B ) )
85, 7eqtri 2470 . . . . 5  |-  ( ( A  +  B )  x.  A )  =  ( ( A  x.  A )  +  ( A  x.  B ) )
91, 2, 2adddiri 9605 . . . . 5  |-  ( ( A  +  B )  x.  B )  =  ( ( A  x.  B )  +  ( B  x.  B ) )
108, 9oveq12i 6289 . . . 4  |-  ( ( ( A  +  B
)  x.  A )  +  ( ( A  +  B )  x.  B ) )  =  ( ( ( A  x.  A )  +  ( A  x.  B
) )  +  ( ( A  x.  B
)  +  ( B  x.  B ) ) )
111, 1mulcli 9599 . . . . . 6  |-  ( A  x.  A )  e.  CC
121, 2mulcli 9599 . . . . . 6  |-  ( A  x.  B )  e.  CC
1311, 12addcli 9598 . . . . 5  |-  ( ( A  x.  A )  +  ( A  x.  B ) )  e.  CC
142, 2mulcli 9599 . . . . 5  |-  ( B  x.  B )  e.  CC
1513, 12, 14addassi 9602 . . . 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 9602 . . . . 5  |-  ( ( ( A  x.  A
)  +  ( A  x.  B ) )  +  ( A  x.  B ) )  =  ( ( A  x.  A )  +  ( ( A  x.  B
)  +  ( A  x.  B ) ) )
1716oveq1i 6287 . . . 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 2476 . . 3  |-  ( ( ( A  +  B
)  x.  A )  +  ( ( A  +  B )  x.  B ) )  =  ( ( ( A  x.  A )  +  ( ( A  x.  B )  +  ( A  x.  B ) ) )  +  ( B  x.  B ) )
194, 18eqtri 2470 . 2  |-  ( ( A  +  B )  x.  ( A  +  B ) )  =  ( ( ( A  x.  A )  +  ( ( A  x.  B )  +  ( A  x.  B ) ) )  +  ( B  x.  B ) )
203sqvali 12221 . 2  |-  ( ( A  +  B ) ^ 2 )  =  ( ( A  +  B )  x.  ( A  +  B )
)
211sqvali 12221 . . . 4  |-  ( A ^ 2 )  =  ( A  x.  A
)
22122timesi 10657 . . . 4  |-  ( 2  x.  ( A  x.  B ) )  =  ( ( A  x.  B )  +  ( A  x.  B ) )
2321, 22oveq12i 6289 . . 3  |-  ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  B
) ) )  =  ( ( A  x.  A )  +  ( ( A  x.  B
)  +  ( A  x.  B ) ) )
242sqvali 12221 . . 3  |-  ( B ^ 2 )  =  ( B  x.  B
)
2523, 24oveq12i 6289 . 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 2480 1  |-  ( ( A  +  B ) ^ 2 )  =  ( ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  B )
) )  +  ( B ^ 2 ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1381    e. wcel 1802  (class class class)co 6277   CCcc 9488    + caddc 9493    x. cmul 9495   2c2 10586   ^cexp 12140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-2nd 6782  df-recs 7040  df-rdg 7074  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-n0 10797  df-z 10866  df-uz 11086  df-seq 12082  df-exp 12141
This theorem is referenced by:  binom2  12257  nn0opthlem1  12322  ax5seglem7  24103  norm-ii-i  25919  quad3  28890
  Copyright terms: Public domain W3C validator