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Theorem congtr 26463
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 979 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( B  -  C )  /\  A  ||  ( C  -  D
) ) )  ->  A  e.  ZZ )
2 simp1r 980 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( B  -  C )  /\  A  ||  ( C  -  D
) ) )  ->  B  e.  ZZ )
3 simp2l 981 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( B  -  C )  /\  A  ||  ( C  -  D
) ) )  ->  C  e.  ZZ )
42, 3zsubcld 10118 . . 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 10057 . . . 4  |-  ( ( C  e.  ZZ  /\  D  e.  ZZ )  ->  ( C  -  D
)  e.  ZZ )
653ad2ant2 977 . . 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 957 . . 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 12556 . . . 4  |-  ( ( A  e.  ZZ  /\  ( B  -  C
)  e.  ZZ  /\  ( C  -  D
)  e.  ZZ )  ->  ( ( A 
||  ( B  -  C )  /\  A  ||  ( C  -  D
) )  ->  A  ||  ( ( B  -  C )  +  ( C  -  D ) ) ) )
98imp 418 . . 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 1185 . 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 10025 . . . . 5  |-  ( B  e.  ZZ  ->  B  e.  CC )
1211adantl 452 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  B  e.  CC )
13123ad2ant1 976 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( B  -  C )  /\  A  ||  ( C  -  D
) ) )  ->  B  e.  CC )
14 zcn 10025 . . . . 5  |-  ( C  e.  ZZ  ->  C  e.  CC )
1514adantr 451 . . . 4  |-  ( ( C  e.  ZZ  /\  D  e.  ZZ )  ->  C  e.  CC )
16153ad2ant2 977 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( B  -  C )  /\  A  ||  ( C  -  D
) ) )  ->  C  e.  CC )
17 zcn 10025 . . . . 5  |-  ( D  e.  ZZ  ->  D  e.  CC )
1817adantl 452 . . . 4  |-  ( ( C  e.  ZZ  /\  D  e.  ZZ )  ->  D  e.  CC )
19183ad2ant2 977 . . 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 9177 . 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 4048 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 4    /\ wa 358    /\ w3a 934    e. wcel 1685   class class class wbr 4024  (class class class)co 5820   CCcc 8731    + caddc 8736    - cmin 9033   ZZcz 10020    || cdivides 12527
This theorem is referenced by:  congmul  26465  acongtr  26476  jm2.18  26492  jm2.27a  26509
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  ax-pre-mulgt0 8810
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-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  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-recs 6384  df-rdg 6419  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-nn 9743  df-n0 9962  df-z 10021  df-dvds 12528
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