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Theorem acongsym 26734
Description: Symmetry of alternating congruence. (Contributed by Stefan O'Rear, 2-Oct-2014.)
Assertion
Ref Expression
acongsym  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( A  ||  ( B  -  C )  \/  A  ||  ( B  -  -u C ) ) )  ->  ( A  ||  ( C  -  B
)  \/  A  ||  ( C  -  -u B
) ) )

Proof of Theorem acongsym
StepHypRef Expression
1 congsym 26726 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  A  ||  ( B  -  C
) ) )  ->  A  ||  ( C  -  B ) )
21exp32 589 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( C  e.  ZZ  ->  ( A  ||  ( B  -  C )  ->  A  ||  ( C  -  B ) ) ) )
323impia 1150 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( A  ||  ( B  -  C )  ->  A  ||  ( C  -  B
) ) )
4 zcn 10221 . . . . . . . . 9  |-  ( B  e.  ZZ  ->  B  e.  CC )
543ad2ant2 979 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  B  e.  CC )
65negnegd 9336 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  -u -u B  =  B )
76oveq1d 6037 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( -u -u B  -  -u C
)  =  ( B  -  -u C ) )
84negcld 9332 . . . . . . . 8  |-  ( B  e.  ZZ  ->  -u B  e.  CC )
983ad2ant2 979 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  -u B  e.  CC )
10 zcn 10221 . . . . . . . 8  |-  ( C  e.  ZZ  ->  C  e.  CC )
11103ad2ant3 980 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  C  e.  CC )
129, 11neg2subd 9362 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( -u -u B  -  -u C
)  =  ( C  -  -u B ) )
137, 12eqtr3d 2423 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( B  -  -u C )  =  ( C  -  -u B ) )
1413breq2d 4167 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( A  ||  ( B  -  -u C )  <->  A  ||  ( C  -  -u B ) ) )
1514biimpd 199 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( A  ||  ( B  -  -u C )  ->  A  ||  ( C  -  -u B
) ) )
163, 15orim12d 812 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  (
( A  ||  ( B  -  C )  \/  A  ||  ( B  -  -u C ) )  ->  ( A  ||  ( C  -  B
)  \/  A  ||  ( C  -  -u B
) ) ) )
1716imp 419 1  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( A  ||  ( B  -  C )  \/  A  ||  ( B  -  -u C ) ) )  ->  ( A  ||  ( C  -  B
)  \/  A  ||  ( C  -  -u B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    /\ w3a 936    e. wcel 1717   class class class wbr 4155  (class class class)co 6022   CCcc 8923    - cmin 9225   -ucneg 9226   ZZcz 10216    || cdivides 12781
This theorem is referenced by:  jm2.25lem1  26762  jm2.26  26766
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-n0 10156  df-z 10217  df-dvds 12782
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