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Theorem congtr 26384
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 10054 . . 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 9993 . . . 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 12487 . . . 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 9961 . . . . 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 9961 . . . . 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 9961 . . . . 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 9114 . 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 3987 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 3963  (class class class)co 5757   CCcc 8668    + caddc 8673    - cmin 8970   ZZcz 9956    || cdivides 12458
This theorem is referenced by:  congmul  26386  acongtr  26397  jm2.18  26413  jm2.27a  26430
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  ax-pre-mulgt0 8747
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-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  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-recs 6321  df-rdg 6356  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-n 9680  df-n0 9898  df-z 9957  df-divides 12459
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