Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  congtr Structured version   Unicode version

Theorem congtr 30799
Description: A wff of the form  A  ||  ( B  -  C
) is interpreted as a congruential equation. This is similar to  ( B  mod  A
)  =  ( C  mod  A ), but is defined such that behavior is regular for zero and negative values of  A. To use this concept effectively, we need to show that congruential equations behave similarly to normal equations; first a transitivity law. Idea for the future: If there was a congruential equation symbol, it could incorporate type constraints, so that most of these would not need them. (Contributed by Stefan O'Rear, 1-Oct-2014.)
Assertion
Ref Expression
congtr  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( B  -  C )  /\  A  ||  ( C  -  D
) ) )  ->  A  ||  ( B  -  D ) )

Proof of Theorem congtr
StepHypRef Expression
1 simp1l 1020 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( B  -  C )  /\  A  ||  ( C  -  D
) ) )  ->  A  e.  ZZ )
2 simp1r 1021 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( B  -  C )  /\  A  ||  ( C  -  D
) ) )  ->  B  e.  ZZ )
3 simp2l 1022 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( B  -  C )  /\  A  ||  ( C  -  D
) ) )  ->  C  e.  ZZ )
42, 3zsubcld 10981 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( B  -  C )  /\  A  ||  ( C  -  D
) ) )  -> 
( B  -  C
)  e.  ZZ )
5 zsubcl 10915 . . . 4  |-  ( ( C  e.  ZZ  /\  D  e.  ZZ )  ->  ( C  -  D
)  e.  ZZ )
653ad2ant2 1018 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( B  -  C )  /\  A  ||  ( C  -  D
) ) )  -> 
( C  -  D
)  e.  ZZ )
7 simp3 998 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( B  -  C )  /\  A  ||  ( C  -  D
) ) )  -> 
( A  ||  ( B  -  C )  /\  A  ||  ( C  -  D ) ) )
8 dvds2add 13888 . . . 4  |-  ( ( A  e.  ZZ  /\  ( B  -  C
)  e.  ZZ  /\  ( C  -  D
)  e.  ZZ )  ->  ( ( A 
||  ( B  -  C )  /\  A  ||  ( C  -  D
) )  ->  A  ||  ( ( B  -  C )  +  ( C  -  D ) ) ) )
98imp 429 . . 3  |-  ( ( ( A  e.  ZZ  /\  ( B  -  C
)  e.  ZZ  /\  ( C  -  D
)  e.  ZZ )  /\  ( A  ||  ( B  -  C
)  /\  A  ||  ( C  -  D )
) )  ->  A  ||  ( ( B  -  C )  +  ( C  -  D ) ) )
101, 4, 6, 7, 9syl31anc 1231 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( B  -  C )  /\  A  ||  ( C  -  D
) ) )  ->  A  ||  ( ( B  -  C )  +  ( C  -  D
) ) )
11 zcn 10879 . . . . 5  |-  ( B  e.  ZZ  ->  B  e.  CC )
1211adantl 466 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  B  e.  CC )
13123ad2ant1 1017 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( B  -  C )  /\  A  ||  ( C  -  D
) ) )  ->  B  e.  CC )
14 zcn 10879 . . . . 5  |-  ( C  e.  ZZ  ->  C  e.  CC )
1514adantr 465 . . . 4  |-  ( ( C  e.  ZZ  /\  D  e.  ZZ )  ->  C  e.  CC )
16153ad2ant2 1018 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( B  -  C )  /\  A  ||  ( C  -  D
) ) )  ->  C  e.  CC )
17 zcn 10879 . . . . 5  |-  ( D  e.  ZZ  ->  D  e.  CC )
1817adantl 466 . . . 4  |-  ( ( C  e.  ZZ  /\  D  e.  ZZ )  ->  D  e.  CC )
19183ad2ant2 1018 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( B  -  C )  /\  A  ||  ( C  -  D
) ) )  ->  D  e.  CC )
2013, 16, 19npncand 9964 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( B  -  C )  /\  A  ||  ( C  -  D
) ) )  -> 
( ( B  -  C )  +  ( C  -  D ) )  =  ( B  -  D ) )
2110, 20breqtrd 4476 1  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( B  -  C )  /\  A  ||  ( C  -  D
) ) )  ->  A  ||  ( B  -  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    e. wcel 1767   class class class wbr 4452  (class class class)co 6294   CCcc 9500    + caddc 9505    - cmin 9815   ZZcz 10874    || cdivides 13859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-recs 7052  df-rdg 7086  df-er 7321  df-en 7527  df-dom 7528  df-sdom 7529  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-nn 10547  df-n0 10806  df-z 10875  df-dvds 13860
This theorem is referenced by:  congmul  30801  acongtr  30812  jm2.18  30826  jm2.27a  30843
  Copyright terms: Public domain W3C validator