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

Theorem congtr 26420
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 984 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( B  -  C )  /\  A  ||  ( C  -  D
) ) )  ->  A  e.  ZZ )
2 simp1r 985 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( B  -  C )  /\  A  ||  ( C  -  D
) ) )  ->  B  e.  ZZ )
3 simp2l 986 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( B  -  C )  /\  A  ||  ( C  -  D
) ) )  ->  C  e.  ZZ )
42, 3zsubcld 10090 . . 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 10029 . . . 4  |-  ( ( C  e.  ZZ  /\  D  e.  ZZ )  ->  ( C  -  D
)  e.  ZZ )
653ad2ant2 982 . . 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 962 . . 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 12523 . . . 4  |-  ( ( A  e.  ZZ  /\  ( B  -  C
)  e.  ZZ  /\  ( C  -  D
)  e.  ZZ )  ->  ( ( A 
||  ( B  -  C )  /\  A  ||  ( C  -  D
) )  ->  A  ||  ( ( B  -  C )  +  ( C  -  D ) ) ) )
98imp 420 . . 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 1190 . 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 9997 . . . . 5  |-  ( B  e.  ZZ  ->  B  e.  CC )
1211adantl 454 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  B  e.  CC )
13123ad2ant1 981 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( B  -  C )  /\  A  ||  ( C  -  D
) ) )  ->  B  e.  CC )
14 zcn 9997 . . . . 5  |-  ( C  e.  ZZ  ->  C  e.  CC )
1514adantr 453 . . . 4  |-  ( ( C  e.  ZZ  /\  D  e.  ZZ )  ->  C  e.  CC )
16153ad2ant2 982 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( B  -  C )  /\  A  ||  ( C  -  D
) ) )  ->  C  e.  CC )
17 zcn 9997 . . . . 5  |-  ( D  e.  ZZ  ->  D  e.  CC )
1817adantl 454 . . . 4  |-  ( ( C  e.  ZZ  /\  D  e.  ZZ )  ->  D  e.  CC )
19183ad2ant2 982 . . 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 9149 . 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 4021 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 set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 939    e. wcel 1621   class class class wbr 3997  (class class class)co 5792   CCcc 8703    + caddc 8708    - cmin 9005   ZZcz 9992    || cdivides 12494
This theorem is referenced by:  congmul  26422  acongtr  26433  jm2.18  26449  jm2.27a  26466
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 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-n 9715  df-n0 9934  df-z 9993  df-divides 12495
  Copyright terms: Public domain W3C validator