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Theorem dvdsacongtr 29332
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 754 . . . . . 6  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  C ) )  ->  D  ||  A )
2 simpr 461 . . . . . 6  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  C ) )  ->  A  ||  ( B  -  C ) )
3 simprr 756 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  ->  D  e.  ZZ )
43ad2antrr 725 . . . . . . 7  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  C ) )  ->  D  e.  ZZ )
5 simp-4l 765 . . . . . . 7  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  C ) )  ->  A  e.  ZZ )
6 simplr 754 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  ->  B  e.  ZZ )
76ad2antrr 725 . . . . . . . 8  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  C ) )  ->  B  e.  ZZ )
8 simprl 755 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  ->  C  e.  ZZ )
98ad2antrr 725 . . . . . . . 8  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  C ) )  ->  C  e.  ZZ )
107, 9zsubcld 10757 . . . . . . 7  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  C ) )  -> 
( B  -  C
)  e.  ZZ )
11 dvdstr 13571 . . . . . . 7  |-  ( ( D  e.  ZZ  /\  A  e.  ZZ  /\  ( B  -  C )  e.  ZZ )  ->  (
( D  ||  A  /\  A  ||  ( B  -  C ) )  ->  D  ||  ( B  -  C )
) )
124, 5, 10, 11syl3anc 1218 . . . . . 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 679 . . . . 5  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  C ) )  ->  D  ||  ( B  -  C ) )
1413ex 434 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  -> 
( A  ||  ( B  -  C )  ->  D  ||  ( B  -  C ) ) )
15 simplr 754 . . . . . 6  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  -u C ) )  ->  D  ||  A )
16 simpr 461 . . . . . 6  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  -u C ) )  ->  A  ||  ( B  -  -u C ) )
173ad2antrr 725 . . . . . . 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 765 . . . . . . 7  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  -u C ) )  ->  A  e.  ZZ )
196ad2antrr 725 . . . . . . . 8  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  -u C ) )  ->  B  e.  ZZ )
208ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  -u C ) )  ->  C  e.  ZZ )
2120znegcld 10754 . . . . . . . 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 10757 . . . . . . 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 13571 . . . . . . 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 1218 . . . . . 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 679 . . . . 5  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  -u C ) )  ->  D  ||  ( B  -  -u C ) )
2625ex 434 . . . 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 834 . . 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 603 . 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 1184 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 setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 965    e. wcel 1756   class class class wbr 4297  (class class class)co 6096    - cmin 9600   -ucneg 9601   ZZcz 10651    || cdivides 13540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-n0 10585  df-z 10652  df-dvds 13541
This theorem is referenced by:  jm2.27a  29359
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