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Theorem congtr 35520
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 1029 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( B  -  C )  /\  A  ||  ( C  -  D
) ) )  ->  A  e.  ZZ )
2 simp1r 1030 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( B  -  C )  /\  A  ||  ( C  -  D
) ) )  ->  B  e.  ZZ )
3 simp2l 1031 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( B  -  C )  /\  A  ||  ( C  -  D
) ) )  ->  C  e.  ZZ )
42, 3zsubcld 11045 . . 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 10979 . . . 4  |-  ( ( C  e.  ZZ  /\  D  e.  ZZ )  ->  ( C  -  D
)  e.  ZZ )
653ad2ant2 1027 . . 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 1007 . . 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 14312 . . . 4  |-  ( ( A  e.  ZZ  /\  ( B  -  C
)  e.  ZZ  /\  ( C  -  D
)  e.  ZZ )  ->  ( ( A 
||  ( B  -  C )  /\  A  ||  ( C  -  D
) )  ->  A  ||  ( ( B  -  C )  +  ( C  -  D ) ) ) )
98imp 430 . . 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 1267 . 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 10942 . . . . 5  |-  ( B  e.  ZZ  ->  B  e.  CC )
1211adantl 467 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  B  e.  CC )
13123ad2ant1 1026 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( B  -  C )  /\  A  ||  ( C  -  D
) ) )  ->  B  e.  CC )
14 zcn 10942 . . . . 5  |-  ( C  e.  ZZ  ->  C  e.  CC )
1514adantr 466 . . . 4  |-  ( ( C  e.  ZZ  /\  D  e.  ZZ )  ->  C  e.  CC )
16153ad2ant2 1027 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( B  -  C )  /\  A  ||  ( C  -  D
) ) )  ->  C  e.  CC )
17 zcn 10942 . . . . 5  |-  ( D  e.  ZZ  ->  D  e.  CC )
1817adantl 467 . . . 4  |-  ( ( C  e.  ZZ  /\  D  e.  ZZ )  ->  D  e.  CC )
19183ad2ant2 1027 . . 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 10009 . 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 4450 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 370    /\ w3a 982    e. wcel 1870   class class class wbr 4426  (class class class)co 6305   CCcc 9536    + caddc 9541    - cmin 9859   ZZcz 10937    || cdvds 14283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-n0 10870  df-z 10938  df-dvds 14284
This theorem is referenced by:  congmul  35522  acongtr  35533  jm2.18  35548  jm2.27a  35565
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