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Theorem muleqadd 9622
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 9004 . . . . 5  |-  1  e.  CC
2 mulsub 9432 . . . . . 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 9048 . . . . . . 7  |-  ( 1  x.  1 )  =  1
65oveq2i 6051 . . . . . 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 9044 . . . . . 6  |-  ( A  e.  CC  ->  ( A  x.  1 )  =  A )
9 mulid1 9044 . . . . . 6  |-  ( B  e.  CC  ->  ( B  x.  1 )  =  B )
108, 9oveqan12d 6059 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  1 )  +  ( B  x.  1 ) )  =  ( A  +  B ) )
117, 10oveq12d 6058 . . . 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 9030 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
13 addcl 9028 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
14 addsub 9272 . . . . . 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 2440 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  - 
1 )  x.  ( B  -  1 ) )  =  ( ( ( A  x.  B
)  -  ( A  +  B ) )  +  1 ) )
1817eqeq1d 2412 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  -  1 )  x.  ( B  -  1 ) )  =  1  <-> 
( ( ( A  x.  B )  -  ( A  +  B
) )  +  1 )  =  1 ) )
191addid2i 9210 . . . 4  |-  ( 0  +  1 )  =  1
2019eqeq2i 2414 . . 3  |-  ( ( ( ( A  x.  B )  -  ( A  +  B )
)  +  1 )  =  ( 0  +  1 )  <->  ( (
( A  x.  B
)  -  ( A  +  B ) )  +  1 )  =  1 )
2112, 13subcld 9367 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  -  ( A  +  B )
)  e.  CC )
22 0cn 9040 . . . . 5  |-  0  e.  CC
23 addcan2 9207 . . . . 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 9377 . 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 1721  (class class class)co 6040   CCcc 8944   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    - cmin 9247
This theorem is referenced by:  conjmul  9687
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-ltxr 9081  df-sub 9249  df-neg 9250
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