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

Theorem muleqadd 9408
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 8791 . . . . 5  |-  1  e.  CC
2 mulsub 9218 . . . . . 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 665 . . . . 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 662 . . . 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 8835 . . . . . . 7  |-  ( 1  x.  1 )  =  1
65oveq2i 5831 . . . . . 6  |-  ( ( A  x.  B )  +  ( 1  x.  1 ) )  =  ( ( A  x.  B )  +  1 )
76a1i 10 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  +  ( 1  x.  1 ) )  =  ( ( A  x.  B )  +  1 ) )
8 mulid1 8831 . . . . . 6  |-  ( A  e.  CC  ->  ( A  x.  1 )  =  A )
9 mulid1 8831 . . . . . 6  |-  ( B  e.  CC  ->  ( B  x.  1 )  =  B )
108, 9oveqan12d 5839 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  1 )  +  ( B  x.  1 ) )  =  ( A  +  B ) )
117, 10oveq12d 5838 . . . 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 8817 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
13 addcl 8815 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
14 addsub 9058 . . . . . 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 1265 . . . . 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 642 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  x.  B )  +  1 )  -  ( A  +  B )
)  =  ( ( ( A  x.  B
)  -  ( A  +  B ) )  +  1 ) )
174, 11, 163eqtrd 2320 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  - 
1 )  x.  ( B  -  1 ) )  =  ( ( ( A  x.  B
)  -  ( A  +  B ) )  +  1 ) )
1817eqeq1d 2292 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  -  1 )  x.  ( B  -  1 ) )  =  1  <-> 
( ( ( A  x.  B )  -  ( A  +  B
) )  +  1 )  =  1 ) )
191addid2i 8996 . . . 4  |-  ( 0  +  1 )  =  1
2019eqeq2i 2294 . . 3  |-  ( ( ( ( A  x.  B )  -  ( A  +  B )
)  +  1 )  =  ( 0  +  1 )  <->  ( (
( A  x.  B
)  -  ( A  +  B ) )  +  1 )  =  1 )
2112, 13subcld 9153 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  -  ( A  +  B )
)  e.  CC )
22 0cn 8827 . . . . 5  |-  0  e.  CC
23 addcan2 8993 . . . . 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 1269 . . . 4  |-  ( ( ( A  x.  B
)  -  ( A  +  B ) )  e.  CC  ->  (
( ( ( A  x.  B )  -  ( A  +  B
) )  +  1 )  =  ( 0  +  1 )  <->  ( ( A  x.  B )  -  ( A  +  B ) )  =  0 ) )
2521, 24syl 15 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( ( A  x.  B )  -  ( A  +  B ) )  +  1 )  =  ( 0  +  1 )  <-> 
( ( A  x.  B )  -  ( A  +  B )
)  =  0 ) )
2620, 25syl5rbbr 251 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  x.  B )  -  ( A  +  B
) )  =  0  <-> 
( ( ( A  x.  B )  -  ( A  +  B
) )  +  1 )  =  1 ) )
27 subeq0 9069 . . 3  |-  ( ( ( A  x.  B
)  e.  CC  /\  ( A  +  B
)  e.  CC )  ->  ( ( ( A  x.  B )  -  ( A  +  B ) )  =  0  <->  ( A  x.  B )  =  ( A  +  B ) ) )
2812, 13, 27syl2anc 642 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  x.  B )  -  ( A  +  B
) )  =  0  <-> 
( A  x.  B
)  =  ( A  +  B ) ) )
2918, 26, 283bitr2rd 273 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 176    /\ wa 358    = wceq 1623    e. wcel 1685  (class class class)co 5820   CCcc 8731   0cc0 8733   1c1 8734    + caddc 8736    x. cmul 8738    - cmin 9033
This theorem is referenced by:  conjmul  9473
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-po 4313  df-so 4314  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-iota 6253  df-riota 6300  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-pnf 8865  df-mnf 8866  df-ltxr 8868  df-sub 9035  df-neg 9036
  Copyright terms: Public domain W3C validator