MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  subsq Structured version   Visualization version   Unicode version
Theorem subsq 12972
Description: Factor the difference of two squares. (Contributed by NM, 21-Feb-2008.)
Assertion
Ref Expression
subsq |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 2 ) - ( B ^ 2 ) ) = ( ( A + B ) x. ( A - B ) ) )
Proof of Theorem subsq
StepHypRef Expression
1 simpl 473 . . 3 |- ( ( A e. CC /\ B e. CC ) -> A e. CC )
2 simpr 477 . . 3 |- ( ( A e. CC /\ B e. CC ) -> B e. CC )
3 subcl 10280 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
41, 2, 3adddird 10065 . 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) x. ( A - B ) ) = ( ( A x. ( A - B ) ) + ( B x. ( A - B ) ) ) )
5 subdi 10463 . . . . 5 |- ( ( A e. CC /\ A e. CC /\ B e. CC ) -> ( A x. ( A - B ) ) = ( ( A x. A ) - ( A x. B ) ) )
653anidm12 1383 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A x. ( A - B ) ) = ( ( A x. A ) - ( A x. B ) ) )
7 sqval 12922 . . . . . 6 |- ( A e. CC -> ( A ^ 2 ) = ( A x. A ) )
87adantr 481 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A ^ 2 ) = ( A x. A ) )
98oveq1d 6665 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 2 ) - ( A x. B ) ) = ( ( A x. A ) - ( A x. B ) ) )
106, 9eqtr4d 2659 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A x. ( A - B ) ) = ( ( A ^ 2 ) - ( A x. B ) ) )
112, 1, 2subdid 10486 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( B x. ( A - B ) ) = ( ( B x. A ) - ( B x. B ) ) )
12 mulcom 10022 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) = ( B x. A ) )
13 sqval 12922 . . . . . 6 |- ( B e. CC -> ( B ^ 2 ) = ( B x. B ) )
1413adantl 482 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( B ^ 2 ) = ( B x. B ) )
1512, 14oveq12d 6668 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) - ( B ^ 2 ) ) = ( ( B x. A ) - ( B x. B ) ) )
1611, 15eqtr4d 2659 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( B x. ( A - B ) ) = ( ( A x. B ) - ( B ^ 2 ) ) )
1710, 16oveq12d 6668 . 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. ( A - B ) ) + ( B x. ( A - B ) ) ) = ( ( ( A ^ 2 ) - ( A x. B ) ) + ( ( A x. B ) - ( B ^ 2 ) ) ) )
18 sqcl 12925 . . . 4 |- ( A e. CC -> ( A ^ 2 ) e. CC )
1918adantr 481 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A ^ 2 ) e. CC )
20 mulcl 10020 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
21 sqcl 12925 . . . 4 |- ( B e. CC -> ( B ^ 2 ) e. CC )
2221adantl 482 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( B ^ 2 ) e. CC )
2319, 20, 22npncand 10416 . 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A ^ 2 ) - ( A x. B ) ) + ( ( A x. B ) - ( B ^ 2 ) ) ) = ( ( A ^ 2 ) - ( B ^ 2 ) ) )
244, 17, 233eqtrrd 2661 1 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 2 ) - ( B ^ 2 ) ) = ( ( A + B ) x. ( A - B ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990  (class class class)co 6650   CCcc 9934   + caddc 9939   x. cmul 9941   - cmin 10266   2c2 11070   ^cexp 12860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-seq 12802  df-exp 12861
This theorem is referenced by:  subsq2  12973  subsqi  12975  pythagtriplem4  15524  pythagtriplem6  15526  pythagtriplem7  15527  pythagtriplem12  15531  pythagtriplem14  15533  pythagtriplem16  15535  difsqpwdvds  15591  4sqlem8  15649  4sqlem10  15651  4sqlem11  15659  chordthmlem4  24562  heron  24565  dcubic2  24571  cubic  24576  dquart  24580  asinlem2  24596  asinsin  24619  efiatan2  24644  atans2  24658  dvatan  24662  wilthlem1  24794  lgslem1  25022  lgsqrlem2  25072  2sqlem4  25146  2sqblem  25156  rplogsumlem1  25173  2sqmod  29648  pellexlem2  37394  pell1234qrne0  37417  pell1234qrreccl  37418  pell1234qrmulcl  37419  pell14qrdich  37433  rmxyneg  37485  stoweidlem1  40218
  Copyright terms: Public domain W3C validator