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Theorem binom2sub 10978
Description: Expand the square of a subtraction. (Contributed by Scott Fenton, 10-Jun-2013.)
Assertion
Ref Expression
binom2sub |- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) ^ 2 ) = ( ( ( A ^ 2 ) - ( 2 x. ( A x. B ) ) ) + ( B ^ 2 ) ) )
Proof of Theorem binom2sub
StepHypRef Expression
1 negcl 8438 . . . 4 |- ( B e. CC -> -u B e. CC )
2 binom2 10976 . . . 4 |- ( ( A e. CC /\ -u B e. CC ) -> ( ( A + -u B ) ^ 2 ) = ( ( ( A ^ 2 ) + ( 2 x. ( A x. -u B ) ) ) + ( -u B ^ 2 ) ) )
31, 2sylan2 286 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + -u B ) ^ 2 ) = ( ( ( A ^ 2 ) + ( 2 x. ( A x. -u B ) ) ) + ( -u B ^ 2 ) ) )
4 negsub 8486 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + -u B ) = ( A - B ) )
54oveq1d 6043 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + -u B ) ^ 2 ) = ( ( A - B ) ^ 2 ) )
63, 5eqtr3d 2266 . 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A ^ 2 ) + ( 2 x. ( A x. -u B ) ) ) + ( -u B ^ 2 ) ) = ( ( A - B ) ^ 2 ) )
7 mulneg2 8634 . . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> ( A x. -u B ) = -u ( A x. B ) )
87oveq2d 6044 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( 2 x. ( A x. -u B ) ) = ( 2 x. -u ( A x. B ) ) )
9 2cn 9273 . . . . . . 7 |- 2 e. CC
10 mulcl 8219 . . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
11 mulneg2 8634 . . . . . . 7 |- ( ( 2 e. CC /\ ( A x. B ) e. CC ) -> ( 2 x. -u ( A x. B ) ) = -u ( 2 x. ( A x. B ) ) )
129, 10, 11sylancr 414 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( 2 x. -u ( A x. B ) ) = -u ( 2 x. ( A x. B ) ) )
138, 12eqtr2d 2265 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> -u ( 2 x. ( A x. B ) ) = ( 2 x. ( A x. -u B ) ) )
1413oveq2d 6044 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 2 ) + -u ( 2 x. ( A x. B ) ) ) = ( ( A ^ 2 ) + ( 2 x. ( A x. -u B ) ) ) )
15 sqcl 10925 . . . . . 6 |- ( A e. CC -> ( A ^ 2 ) e. CC )
1615adantr 276 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A ^ 2 ) e. CC )
17 mulcl 8219 . . . . . 6 |- ( ( 2 e. CC /\ ( A x. B ) e. CC ) -> ( 2 x. ( A x. B ) ) e. CC )
189, 10, 17sylancr 414 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( 2 x. ( A x. B ) ) e. CC )
1916, 18negsubd 8555 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 2 ) + -u ( 2 x. ( A x. B ) ) ) = ( ( A ^ 2 ) - ( 2 x. ( A x. B ) ) ) )
2014, 19eqtr3d 2266 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 2 ) + ( 2 x. ( A x. -u B ) ) ) = ( ( A ^ 2 ) - ( 2 x. ( A x. B ) ) ) )
21 sqneg 10923 . . . 4 |- ( B e. CC -> ( -u B ^ 2 ) = ( B ^ 2 ) )
2221adantl 277 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( -u B ^ 2 ) = ( B ^ 2 ) )
2320, 22oveq12d 6046 . 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A ^ 2 ) + ( 2 x. ( A x. -u B ) ) ) + ( -u B ^ 2 ) ) = ( ( ( A ^ 2 ) - ( 2 x. ( A x. B ) ) ) + ( B ^ 2 ) ) )
246, 23eqtr3d 2266 1 |- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) ^ 2 ) = ( ( ( A ^ 2 ) - ( 2 x. ( A x. B ) ) ) + ( B ^ 2 ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202  (class class class)co 6028   CCcc 8090   + caddc 8095   x. cmul 8097   - cmin 8409   -ucneg 8410   2c2 9253   ^cexp 10863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210
This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-n0 9462  df-z 9541  df-uz 9817  df-seqfrec 10773  df-exp 10864
This theorem is referenced by:  binom2sub1  10979  binom2subi  10980  resqrexlemover  11650  resqrexlemcalc1  11654  amgm2  11758  bdtrilem  11879  pythagtriplem1  12918  pythagtriplem14  12930  tangtx  15649  qdiff  16781
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