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Theorem jm2.25lem1 27194
Description: Lemma for jm2.26 27198. (Contributed by Stefan O'Rear, 2-Oct-2014.)
Assertion
Ref Expression
jm2.25lem1  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( C  -  D )  \/  A  ||  ( C  -  -u D
) ) )  -> 
( ( A  ||  ( D  -  B
)  \/  A  ||  ( D  -  -u B
) )  <->  ( A  ||  ( C  -  B
)  \/  A  ||  ( C  -  -u B
) ) ) )

Proof of Theorem jm2.25lem1
StepHypRef Expression
1 simpl1l 1006 . . 3  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( C  -  D )  \/  A  ||  ( C  -  -u D
) ) )  /\  ( A  ||  ( D  -  B )  \/  A  ||  ( D  -  -u B ) ) )  ->  A  e.  ZZ )
2 simpl2l 1008 . . 3  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( C  -  D )  \/  A  ||  ( C  -  -u D
) ) )  /\  ( A  ||  ( D  -  B )  \/  A  ||  ( D  -  -u B ) ) )  ->  C  e.  ZZ )
3 simpl2r 1009 . . 3  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( C  -  D )  \/  A  ||  ( C  -  -u D
) ) )  /\  ( A  ||  ( D  -  B )  \/  A  ||  ( D  -  -u B ) ) )  ->  D  e.  ZZ )
4 simpl1r 1007 . . 3  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( C  -  D )  \/  A  ||  ( C  -  -u D
) ) )  /\  ( A  ||  ( D  -  B )  \/  A  ||  ( D  -  -u B ) ) )  ->  B  e.  ZZ )
5 simpl3 960 . . 3  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( C  -  D )  \/  A  ||  ( C  -  -u D
) ) )  /\  ( A  ||  ( D  -  B )  \/  A  ||  ( D  -  -u B ) ) )  ->  ( A  ||  ( C  -  D
)  \/  A  ||  ( C  -  -u D
) ) )
6 simpr 447 . . 3  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( C  -  D )  \/  A  ||  ( C  -  -u D
) ) )  /\  ( A  ||  ( D  -  B )  \/  A  ||  ( D  -  -u B ) ) )  ->  ( A  ||  ( D  -  B
)  \/  A  ||  ( D  -  -u B
) ) )
7 acongtr 27168 . . 3  |-  ( ( ( A  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  B  e.  ZZ )  /\  (
( A  ||  ( C  -  D )  \/  A  ||  ( C  -  -u D ) )  /\  ( A  ||  ( D  -  B
)  \/  A  ||  ( D  -  -u B
) ) ) )  ->  ( A  ||  ( C  -  B
)  \/  A  ||  ( C  -  -u B
) ) )
81, 2, 3, 4, 5, 6, 7syl222anc 1198 . 2  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( C  -  D )  \/  A  ||  ( C  -  -u D
) ) )  /\  ( A  ||  ( D  -  B )  \/  A  ||  ( D  -  -u B ) ) )  ->  ( A  ||  ( C  -  B
)  \/  A  ||  ( C  -  -u B
) ) )
9 simpl1l 1006 . . 3  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( C  -  D )  \/  A  ||  ( C  -  -u D
) ) )  /\  ( A  ||  ( C  -  B )  \/  A  ||  ( C  -  -u B ) ) )  ->  A  e.  ZZ )
10 simpl2r 1009 . . 3  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( C  -  D )  \/  A  ||  ( C  -  -u D
) ) )  /\  ( A  ||  ( C  -  B )  \/  A  ||  ( C  -  -u B ) ) )  ->  D  e.  ZZ )
11 simpl2l 1008 . . 3  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( C  -  D )  \/  A  ||  ( C  -  -u D
) ) )  /\  ( A  ||  ( C  -  B )  \/  A  ||  ( C  -  -u B ) ) )  ->  C  e.  ZZ )
12 simpl1r 1007 . . 3  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( C  -  D )  \/  A  ||  ( C  -  -u D
) ) )  /\  ( A  ||  ( C  -  B )  \/  A  ||  ( C  -  -u B ) ) )  ->  B  e.  ZZ )
13 simpl3 960 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( C  -  D )  \/  A  ||  ( C  -  -u D
) ) )  /\  ( A  ||  ( C  -  B )  \/  A  ||  ( C  -  -u B ) ) )  ->  ( A  ||  ( C  -  D
)  \/  A  ||  ( C  -  -u D
) ) )
14 acongsym 27166 . . . 4  |-  ( ( ( A  e.  ZZ  /\  C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( C  -  D )  \/  A  ||  ( C  -  -u D ) ) )  ->  ( A  ||  ( D  -  C
)  \/  A  ||  ( D  -  -u C
) ) )
159, 11, 10, 13, 14syl31anc 1185 . . 3  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( C  -  D )  \/  A  ||  ( C  -  -u D
) ) )  /\  ( A  ||  ( C  -  B )  \/  A  ||  ( C  -  -u B ) ) )  ->  ( A  ||  ( D  -  C
)  \/  A  ||  ( D  -  -u C
) ) )
16 simpr 447 . . 3  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( C  -  D )  \/  A  ||  ( C  -  -u D
) ) )  /\  ( A  ||  ( C  -  B )  \/  A  ||  ( C  -  -u B ) ) )  ->  ( A  ||  ( C  -  B
)  \/  A  ||  ( C  -  -u B
) ) )
17 acongtr 27168 . . 3  |-  ( ( ( A  e.  ZZ  /\  D  e.  ZZ )  /\  ( C  e.  ZZ  /\  B  e.  ZZ )  /\  (
( A  ||  ( D  -  C )  \/  A  ||  ( D  -  -u C ) )  /\  ( A  ||  ( C  -  B
)  \/  A  ||  ( C  -  -u B
) ) ) )  ->  ( A  ||  ( D  -  B
)  \/  A  ||  ( D  -  -u B
) ) )
189, 10, 11, 12, 15, 16, 17syl222anc 1198 . 2  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( C  -  D )  \/  A  ||  ( C  -  -u D
) ) )  /\  ( A  ||  ( C  -  B )  \/  A  ||  ( C  -  -u B ) ) )  ->  ( A  ||  ( D  -  B
)  \/  A  ||  ( D  -  -u B
) ) )
198, 18impbida 805 1  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( C  -  D )  \/  A  ||  ( C  -  -u D
) ) )  -> 
( ( A  ||  ( D  -  B
)  \/  A  ||  ( D  -  -u B
) )  <->  ( A  ||  ( C  -  B
)  \/  A  ||  ( C  -  -u B
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    e. wcel 1696   class class class wbr 4039  (class class class)co 5874    - cmin 9053   -ucneg 9054   ZZcz 10040    || cdivides 12547
This theorem is referenced by:  jm2.25  27195
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-dvds 12548
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