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

Theorem muleqadd 9345
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 8728 . . . . 5  |-  1  e.  CC
2 mulsub 9155 . . . . . 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 668 . . . . 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 665 . . . 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 8772 . . . . . . 7  |-  ( 1  x.  1 )  =  1
65oveq2i 5768 . . . . . 6  |-  ( ( A  x.  B )  +  ( 1  x.  1 ) )  =  ( ( A  x.  B )  +  1 )
76a1i 12 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  +  ( 1  x.  1 ) )  =  ( ( A  x.  B )  +  1 ) )
8 mulid1 8767 . . . . . 6  |-  ( A  e.  CC  ->  ( A  x.  1 )  =  A )
9 mulid1 8767 . . . . . 6  |-  ( B  e.  CC  ->  ( B  x.  1 )  =  B )
108, 9oveqan12d 5776 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  1 )  +  ( B  x.  1 ) )  =  ( A  +  B ) )
117, 10oveq12d 5775 . . . 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 8754 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
13 addcl 8752 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
14 addsub 8995 . . . . . 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 1270 . . . . 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 645 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  x.  B )  +  1 )  -  ( A  +  B )
)  =  ( ( ( A  x.  B
)  -  ( A  +  B ) )  +  1 ) )
174, 11, 163eqtrd 2292 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  - 
1 )  x.  ( B  -  1 ) )  =  ( ( ( A  x.  B
)  -  ( A  +  B ) )  +  1 ) )
1817eqeq1d 2264 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  -  1 )  x.  ( B  -  1 ) )  =  1  <-> 
( ( ( A  x.  B )  -  ( A  +  B
) )  +  1 )  =  1 ) )
191addid2i 8933 . . . 4  |-  ( 0  +  1 )  =  1
2019eqeq2i 2266 . . 3  |-  ( ( ( ( A  x.  B )  -  ( A  +  B )
)  +  1 )  =  ( 0  +  1 )  <->  ( (
( A  x.  B
)  -  ( A  +  B ) )  +  1 )  =  1 )
2112, 13subcld 9090 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  -  ( A  +  B )
)  e.  CC )
22 0cn 8764 . . . . 5  |-  0  e.  CC
23 addcan2 8930 . . . . 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 1274 . . . 4  |-  ( ( ( A  x.  B
)  -  ( A  +  B ) )  e.  CC  ->  (
( ( ( A  x.  B )  -  ( A  +  B
) )  +  1 )  =  ( 0  +  1 )  <->  ( ( A  x.  B )  -  ( A  +  B ) )  =  0 ) )
2521, 24syl 17 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( ( A  x.  B )  -  ( A  +  B ) )  +  1 )  =  ( 0  +  1 )  <-> 
( ( A  x.  B )  -  ( A  +  B )
)  =  0 ) )
2620, 25syl5rbbr 253 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  x.  B )  -  ( A  +  B
) )  =  0  <-> 
( ( ( A  x.  B )  -  ( A  +  B
) )  +  1 )  =  1 ) )
27 subeq0 9006 . . 3  |-  ( ( ( A  x.  B
)  e.  CC  /\  ( A  +  B
)  e.  CC )  ->  ( ( ( A  x.  B )  -  ( A  +  B ) )  =  0  <->  ( A  x.  B )  =  ( A  +  B ) ) )
2812, 13, 27syl2anc 645 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  x.  B )  -  ( A  +  B
) )  =  0  <-> 
( A  x.  B
)  =  ( A  +  B ) ) )
2918, 26, 283bitr2rd 275 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 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621  (class class class)co 5757   CCcc 8668   0cc0 8670   1c1 8671    + caddc 8673    x. cmul 8675    - cmin 8970
This theorem is referenced by:  conjmul  9410
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-po 4251  df-so 4252  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-iota 6190  df-riota 6237  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-pnf 8802  df-mnf 8803  df-ltxr 8805  df-sub 8972  df-neg 8973
  Copyright terms: Public domain W3C validator