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Theorem dvdsacongtr 27051
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 733 . . . . . 6  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  C ) )  ->  D  ||  A )
2 simpr 449 . . . . . 6  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  C ) )  ->  A  ||  ( B  -  C ) )
3 simprr 735 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  ->  D  e.  ZZ )
43ad2antrr 708 . . . . . . 7  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  C ) )  ->  D  e.  ZZ )
5 simp-4l 744 . . . . . . 7  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  C ) )  ->  A  e.  ZZ )
6 simplr 733 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  ->  B  e.  ZZ )
76ad2antrr 708 . . . . . . . 8  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  C ) )  ->  B  e.  ZZ )
8 simprl 734 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  ->  C  e.  ZZ )
98ad2antrr 708 . . . . . . . 8  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  C ) )  ->  C  e.  ZZ )
107, 9zsubcld 10382 . . . . . . 7  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  C ) )  -> 
( B  -  C
)  e.  ZZ )
11 dvdstr 12885 . . . . . . 7  |-  ( ( D  e.  ZZ  /\  A  e.  ZZ  /\  ( B  -  C )  e.  ZZ )  ->  (
( D  ||  A  /\  A  ||  ( B  -  C ) )  ->  D  ||  ( B  -  C )
) )
124, 5, 10, 11syl3anc 1185 . . . . . 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 662 . . . . 5  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  C ) )  ->  D  ||  ( B  -  C ) )
1413ex 425 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  -> 
( A  ||  ( B  -  C )  ->  D  ||  ( B  -  C ) ) )
15 simplr 733 . . . . . 6  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  -u C ) )  ->  D  ||  A )
16 simpr 449 . . . . . 6  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  -u C ) )  ->  A  ||  ( B  -  -u C ) )
173ad2antrr 708 . . . . . . 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 744 . . . . . . 7  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  -u C ) )  ->  A  e.  ZZ )
196ad2antrr 708 . . . . . . . 8  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  -u C ) )  ->  B  e.  ZZ )
208ad2antrr 708 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  -u C ) )  ->  C  e.  ZZ )
2120znegcld 10379 . . . . . . . 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 10382 . . . . . . 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 12885 . . . . . . 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 1185 . . . . . 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 662 . . . . 5  |-  ( ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ ) )  /\  D  ||  A )  /\  A  ||  ( B  -  -u C ) )  ->  D  ||  ( B  -  -u C ) )
2625ex 425 . . . 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 813 . . 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 588 . 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 1151 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 359    /\ wa 360    /\ w3a 937    e. wcel 1726   class class class wbr 4214  (class class class)co 6083    - cmin 9293   -ucneg 9294   ZZcz 10284    || cdivides 12854
This theorem is referenced by:  jm2.27a  27078
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-n0 10224  df-z 10285  df-dvds 12855
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