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Theorem acongneg2 26996
Description: Negate right side of alternating congruence. Makes essential use of the "alternating" part. (Contributed by Stefan O'Rear, 3-Oct-2014.)
Assertion
Ref Expression
acongneg2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( A  ||  ( B  -  -u C )  \/  A  ||  ( B  -  -u -u C
) ) )  -> 
( A  ||  ( B  -  C )  \/  A  ||  ( B  -  -u C ) ) )

Proof of Theorem acongneg2
StepHypRef Expression
1 zcn 10277 . . . . . . . . 9  |-  ( C  e.  ZZ  ->  C  e.  CC )
213ad2ant3 980 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  C  e.  CC )
32negnegd 9392 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  -u -u C  =  C )
43oveq2d 6089 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( B  -  -u -u C
)  =  ( B  -  C ) )
54breq2d 4216 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( A  ||  ( B  -  -u -u C )  <->  A  ||  ( B  -  C )
) )
65biimpd 199 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( A  ||  ( B  -  -u -u C )  ->  A  ||  ( B  -  C
) ) )
76orim2d 814 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  (
( A  ||  ( B  -  -u C )  \/  A  ||  ( B  -  -u -u C
) )  ->  ( A  ||  ( B  -  -u C )  \/  A  ||  ( B  -  C
) ) ) )
87imp 419 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( A  ||  ( B  -  -u C )  \/  A  ||  ( B  -  -u -u C
) ) )  -> 
( A  ||  ( B  -  -u C )  \/  A  ||  ( B  -  C )
) )
98orcomd 378 1  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( A  ||  ( B  -  -u C )  \/  A  ||  ( B  -  -u -u C
) ) )  -> 
( A  ||  ( B  -  C )  \/  A  ||  ( B  -  -u C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    /\ w3a 936    e. wcel 1725   class class class wbr 4204  (class class class)co 6073   CCcc 8978    - cmin 9281   -ucneg 9282   ZZcz 10272    || cdivides 12842
This theorem is referenced by:  jm2.26a  27025
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9112  df-mnf 9113  df-ltxr 9115  df-sub 9283  df-neg 9284  df-z 10273
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