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Theorem dvdsacongtr 26939
Description: Alternating congruence passes from a base to a dividing base. (Contributed by Stefan O'Rear, 4-Oct-2014.)
Assertion
Ref Expression
dvdsacongtr  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( D  ||  A  /\  ( A  ||  ( B  -  C )  \/  A  ||  ( B  -  -u C
) ) ) )  ->  ( D  ||  ( B  -  C
)  \/  D  ||  ( B  -  -u C
) ) )

Proof of Theorem dvdsacongtr
StepHypRef Expression
1 simplr 732 . . . . . 6  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  C ) )  ->  D  ||  A )
2 simpr 448 . . . . . 6  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  C ) )  ->  A  ||  ( B  -  C ) )
3 simprr 734 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  ->  D  e.  ZZ )
43ad2antrr 707 . . . . . . 7  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  C ) )  ->  D  e.  ZZ )
5 simp-4l 743 . . . . . . 7  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  C ) )  ->  A  e.  ZZ )
6 simplr 732 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  ->  B  e.  ZZ )
76ad2antrr 707 . . . . . . . 8  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  C ) )  ->  B  e.  ZZ )
8 simprl 733 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  ->  C  e.  ZZ )
98ad2antrr 707 . . . . . . . 8  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  C ) )  ->  C  e.  ZZ )
107, 9zsubcld 10336 . . . . . . 7  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  C ) )  -> 
( B  -  C
)  e.  ZZ )
11 dvdstr 12838 . . . . . . 7  |-  ( ( D  e.  ZZ  /\  A  e.  ZZ  /\  ( B  -  C )  e.  ZZ )  ->  (
( D  ||  A  /\  A  ||  ( B  -  C ) )  ->  D  ||  ( B  -  C )
) )
124, 5, 10, 11syl3anc 1184 . . . . . 6  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  C ) )  -> 
( ( D  ||  A  /\  A  ||  ( B  -  C )
)  ->  D  ||  ( B  -  C )
) )
131, 2, 12mp2and 661 . . . . 5  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  C ) )  ->  D  ||  ( B  -  C ) )
1413ex 424 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  -> 
( A  ||  ( B  -  C )  ->  D  ||  ( B  -  C ) ) )
15 simplr 732 . . . . . 6  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  -u C ) )  ->  D  ||  A )
16 simpr 448 . . . . . 6  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  -u C ) )  ->  A  ||  ( B  -  -u C ) )
173ad2antrr 707 . . . . . . 7  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  -u C ) )  ->  D  e.  ZZ )
18 simp-4l 743 . . . . . . 7  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  -u C ) )  ->  A  e.  ZZ )
196ad2antrr 707 . . . . . . . 8  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  -u C ) )  ->  B  e.  ZZ )
208ad2antrr 707 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  -u C ) )  ->  C  e.  ZZ )
2120znegcld 10333 . . . . . . . 8  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  -u C ) )  ->  -u C  e.  ZZ )
2219, 21zsubcld 10336 . . . . . . 7  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  -u C ) )  -> 
( B  -  -u C
)  e.  ZZ )
23 dvdstr 12838 . . . . . . 7  |-  ( ( D  e.  ZZ  /\  A  e.  ZZ  /\  ( B  -  -u C )  e.  ZZ )  -> 
( ( D  ||  A  /\  A  ||  ( B  -  -u C ) )  ->  D  ||  ( B  -  -u C ) ) )
2417, 18, 22, 23syl3anc 1184 . . . . . 6  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  -u C ) )  -> 
( ( D  ||  A  /\  A  ||  ( B  -  -u C ) )  ->  D  ||  ( B  -  -u C ) ) )
2515, 16, 24mp2and 661 . . . . 5  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  -u C ) )  ->  D  ||  ( B  -  -u C ) )
2625ex 424 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  -> 
( A  ||  ( B  -  -u C )  ->  D  ||  ( B  -  -u C ) ) )
2714, 26orim12d 812 . . 3  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  -> 
( ( A  ||  ( B  -  C
)  \/  A  ||  ( B  -  -u C
) )  ->  ( D  ||  ( B  -  C )  \/  D  ||  ( B  -  -u C
) ) ) )
2827expimpd 587 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  -> 
( ( D  ||  A  /\  ( A  ||  ( B  -  C
)  \/  A  ||  ( B  -  -u C
) ) )  -> 
( D  ||  ( B  -  C )  \/  D  ||  ( B  -  -u C ) ) ) )
29283impia 1150 1  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( D  ||  A  /\  ( A  ||  ( B  -  C )  \/  A  ||  ( B  -  -u C
) ) ) )  ->  ( D  ||  ( B  -  C
)  \/  D  ||  ( B  -  -u C
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    /\ w3a 936    e. wcel 1721   class class class wbr 4172  (class class class)co 6040    - cmin 9247   -ucneg 9248   ZZcz 10238    || cdivides 12807
This theorem is referenced by:  jm2.27a  26966
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-dvds 12808
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