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

Theorem muleqadd 9591
Description: Property of numbers whose product equals their sum. Equation 5 of [Kreyszig] p. 12. (Contributed by NM, 13-Nov-2006.)
Assertion
Ref Expression
muleqadd  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  =  ( A  +  B )  <-> 
( ( A  - 
1 )  x.  ( B  -  1 ) )  =  1 ) )

Proof of Theorem muleqadd
StepHypRef Expression
1 ax-1cn 8974 . . . . 5  |-  1  e.  CC
2 mulsub 9401 . . . . . 6  |-  ( ( ( A  e.  CC  /\  1  e.  CC )  /\  ( B  e.  CC  /\  1  e.  CC ) )  -> 
( ( A  - 
1 )  x.  ( B  -  1 ) )  =  ( ( ( A  x.  B
)  +  ( 1  x.  1 ) )  -  ( ( A  x.  1 )  +  ( B  x.  1 ) ) ) )
31, 2mpanr2 666 . . . . 5  |-  ( ( ( A  e.  CC  /\  1  e.  CC )  /\  B  e.  CC )  ->  ( ( A  -  1 )  x.  ( B  -  1 ) )  =  ( ( ( A  x.  B )  +  ( 1  x.  1 ) )  -  ( ( A  x.  1 )  +  ( B  x.  1 ) ) ) )
41, 3mpanl2 663 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  - 
1 )  x.  ( B  -  1 ) )  =  ( ( ( A  x.  B
)  +  ( 1  x.  1 ) )  -  ( ( A  x.  1 )  +  ( B  x.  1 ) ) ) )
51mulid1i 9018 . . . . . . 7  |-  ( 1  x.  1 )  =  1
65oveq2i 6024 . . . . . 6  |-  ( ( A  x.  B )  +  ( 1  x.  1 ) )  =  ( ( A  x.  B )  +  1 )
76a1i 11 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  +  ( 1  x.  1 ) )  =  ( ( A  x.  B )  +  1 ) )
8 mulid1 9014 . . . . . 6  |-  ( A  e.  CC  ->  ( A  x.  1 )  =  A )
9 mulid1 9014 . . . . . 6  |-  ( B  e.  CC  ->  ( B  x.  1 )  =  B )
108, 9oveqan12d 6032 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  1 )  +  ( B  x.  1 ) )  =  ( A  +  B ) )
117, 10oveq12d 6031 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  x.  B )  +  ( 1  x.  1 ) )  -  (
( A  x.  1 )  +  ( B  x.  1 ) ) )  =  ( ( ( A  x.  B
)  +  1 )  -  ( A  +  B ) ) )
12 mulcl 9000 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
13 addcl 8998 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
14 addsub 9241 . . . . . 6  |-  ( ( ( A  x.  B
)  e.  CC  /\  1  e.  CC  /\  ( A  +  B )  e.  CC )  ->  (
( ( A  x.  B )  +  1 )  -  ( A  +  B ) )  =  ( ( ( A  x.  B )  -  ( A  +  B ) )  +  1 ) )
151, 14mp3an2 1267 . . . . 5  |-  ( ( ( A  x.  B
)  e.  CC  /\  ( A  +  B
)  e.  CC )  ->  ( ( ( A  x.  B )  +  1 )  -  ( A  +  B
) )  =  ( ( ( A  x.  B )  -  ( A  +  B )
)  +  1 ) )
1612, 13, 15syl2anc 643 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  x.  B )  +  1 )  -  ( A  +  B )
)  =  ( ( ( A  x.  B
)  -  ( A  +  B ) )  +  1 ) )
174, 11, 163eqtrd 2416 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  - 
1 )  x.  ( B  -  1 ) )  =  ( ( ( A  x.  B
)  -  ( A  +  B ) )  +  1 ) )
1817eqeq1d 2388 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  -  1 )  x.  ( B  -  1 ) )  =  1  <-> 
( ( ( A  x.  B )  -  ( A  +  B
) )  +  1 )  =  1 ) )
191addid2i 9179 . . . 4  |-  ( 0  +  1 )  =  1
2019eqeq2i 2390 . . 3  |-  ( ( ( ( A  x.  B )  -  ( A  +  B )
)  +  1 )  =  ( 0  +  1 )  <->  ( (
( A  x.  B
)  -  ( A  +  B ) )  +  1 )  =  1 )
2112, 13subcld 9336 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  -  ( A  +  B )
)  e.  CC )
22 0cn 9010 . . . . 5  |-  0  e.  CC
23 addcan2 9176 . . . . 5  |-  ( ( ( ( A  x.  B )  -  ( A  +  B )
)  e.  CC  /\  0  e.  CC  /\  1  e.  CC )  ->  (
( ( ( A  x.  B )  -  ( A  +  B
) )  +  1 )  =  ( 0  +  1 )  <->  ( ( A  x.  B )  -  ( A  +  B ) )  =  0 ) )
2422, 1, 23mp3an23 1271 . . . 4  |-  ( ( ( A  x.  B
)  -  ( A  +  B ) )  e.  CC  ->  (
( ( ( A  x.  B )  -  ( A  +  B
) )  +  1 )  =  ( 0  +  1 )  <->  ( ( A  x.  B )  -  ( A  +  B ) )  =  0 ) )
2521, 24syl 16 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( ( A  x.  B )  -  ( A  +  B ) )  +  1 )  =  ( 0  +  1 )  <-> 
( ( A  x.  B )  -  ( A  +  B )
)  =  0 ) )
2620, 25syl5rbbr 252 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  x.  B )  -  ( A  +  B
) )  =  0  <-> 
( ( ( A  x.  B )  -  ( A  +  B
) )  +  1 )  =  1 ) )
2712, 13subeq0ad 9346 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  x.  B )  -  ( A  +  B
) )  =  0  <-> 
( A  x.  B
)  =  ( A  +  B ) ) )
2818, 26, 273bitr2rd 274 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  =  ( A  +  B )  <-> 
( ( A  - 
1 )  x.  ( B  -  1 ) )  =  1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717  (class class class)co 6013   CCcc 8914   0cc0 8916   1c1 8917    + caddc 8919    x. cmul 8921    - cmin 9216
This theorem is referenced by:  conjmul  9656
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-po 4437  df-so 4438  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-riota 6478  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-ltxr 9051  df-sub 9218  df-neg 9219
  Copyright terms: Public domain W3C validator