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Theorem acongeq 27173
Description: Two numbers in the fundamental domain are alternating-congruent iff they are equal. TODO: could be used to shorten jm2.26 27198 (Contributed by Stefan O'Rear, 4-Oct-2014.)
Assertion
Ref Expression
acongeq  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( B  =  C  <->  ( ( 2  x.  A )  ||  ( B  -  C
)  \/  ( 2  x.  A )  ||  ( B  -  -u C
) ) ) )

Proof of Theorem acongeq
StepHypRef Expression
1 2z 10070 . . . . . . 7  |-  2  e.  ZZ
2 nnz 10061 . . . . . . . 8  |-  ( A  e.  NN  ->  A  e.  ZZ )
323ad2ant1 976 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  A  e.  ZZ )
4 zmulcl 10082 . . . . . . 7  |-  ( ( 2  e.  ZZ  /\  A  e.  ZZ )  ->  ( 2  x.  A
)  e.  ZZ )
51, 3, 4sylancr 644 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( 2  x.  A )  e.  ZZ )
6 elfzelz 10814 . . . . . . 7  |-  ( B  e.  ( 0 ... A )  ->  B  e.  ZZ )
763ad2ant2 977 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  B  e.  ZZ )
8 congid 27161 . . . . . 6  |-  ( ( ( 2  x.  A
)  e.  ZZ  /\  B  e.  ZZ )  ->  ( 2  x.  A
)  ||  ( B  -  B ) )
95, 7, 8syl2anc 642 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( 2  x.  A )  ||  ( B  -  B )
)
109adantr 451 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  /\  B  =  C )  ->  ( 2  x.  A )  ||  ( B  -  B
) )
11 oveq2 5882 . . . . 5  |-  ( B  =  C  ->  ( B  -  B )  =  ( B  -  C ) )
1211adantl 452 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  /\  B  =  C )  ->  ( B  -  B )  =  ( B  -  C ) )
1310, 12breqtrd 4063 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  /\  B  =  C )  ->  ( 2  x.  A )  ||  ( B  -  C
) )
1413orcd 381 . 2  |-  ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  /\  B  =  C )  ->  ( (
2  x.  A ) 
||  ( B  -  C )  \/  (
2  x.  A ) 
||  ( B  -  -u C ) ) )
15 elfzelz 10814 . . . . . . . . . 10  |-  ( C  e.  ( 0 ... A )  ->  C  e.  ZZ )
16153ad2ant3 978 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  C  e.  ZZ )
177, 16zsubcld 10138 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( B  -  C )  e.  ZZ )
1817zcnd 10134 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( B  -  C )  e.  CC )
1918abscld 11934 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( abs `  ( B  -  C )
)  e.  RR )
20 nnre 9769 . . . . . . . 8  |-  ( A  e.  NN  ->  A  e.  RR )
21203ad2ant1 976 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  A  e.  RR )
22 0re 8854 . . . . . . 7  |-  0  e.  RR
23 resubcl 9127 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  e.  RR )  ->  ( A  -  0 )  e.  RR )
2421, 22, 23sylancl 643 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( A  - 
0 )  e.  RR )
25 2re 9831 . . . . . . 7  |-  2  e.  RR
26 remulcl 8838 . . . . . . 7  |-  ( ( 2  e.  RR  /\  A  e.  RR )  ->  ( 2  x.  A
)  e.  RR )
2725, 21, 26sylancr 644 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( 2  x.  A )  e.  RR )
28 simp2 956 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  B  e.  ( 0 ... A ) )
29 simp3 957 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  C  e.  ( 0 ... A ) )
3024leidd 9355 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( A  - 
0 )  <_  ( A  -  0 ) )
31 fzmaxdif 27171 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ( 0 ... A ) )  /\  ( A  e.  ZZ  /\  C  e.  ( 0 ... A
) )  /\  ( A  -  0 )  <_  ( A  - 
0 ) )  -> 
( abs `  ( B  -  C )
)  <_  ( A  -  0 ) )
323, 28, 3, 29, 30, 31syl221anc 1193 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( abs `  ( B  -  C )
)  <_  ( A  -  0 ) )
33 nnrp 10379 . . . . . . . . 9  |-  ( A  e.  NN  ->  A  e.  RR+ )
34333ad2ant1 976 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  A  e.  RR+ )
3521, 34ltaddrpd 10435 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  A  <  ( A  +  A )
)
3621recnd 8877 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  A  e.  CC )
3736subid1d 9162 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( A  - 
0 )  =  A )
38362timesd 9970 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( 2  x.  A )  =  ( A  +  A ) )
3935, 37, 383brtr4d 4069 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( A  - 
0 )  <  (
2  x.  A ) )
4019, 24, 27, 32, 39lelttrd 8990 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( abs `  ( B  -  C )
)  <  ( 2  x.  A ) )
4140adantr 451 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  /\  ( 2  x.  A )  ||  ( B  -  C )
)  ->  ( abs `  ( B  -  C
) )  <  (
2  x.  A ) )
42 2nn 9893 . . . . . 6  |-  2  e.  NN
43 simpl1 958 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  /\  ( 2  x.  A )  ||  ( B  -  C )
)  ->  A  e.  NN )
44 nnmulcl 9785 . . . . . 6  |-  ( ( 2  e.  NN  /\  A  e.  NN )  ->  ( 2  x.  A
)  e.  NN )
4542, 43, 44sylancr 644 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  /\  ( 2  x.  A )  ||  ( B  -  C )
)  ->  ( 2  x.  A )  e.  NN )
46 simpl2 959 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  /\  ( 2  x.  A )  ||  ( B  -  C )
)  ->  B  e.  ( 0 ... A
) )
4746, 6syl 15 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  /\  ( 2  x.  A )  ||  ( B  -  C )
)  ->  B  e.  ZZ )
48 simpl3 960 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  /\  ( 2  x.  A )  ||  ( B  -  C )
)  ->  C  e.  ( 0 ... A
) )
4948, 15syl 15 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  /\  ( 2  x.  A )  ||  ( B  -  C )
)  ->  C  e.  ZZ )
50 simpr 447 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  /\  ( 2  x.  A )  ||  ( B  -  C )
)  ->  ( 2  x.  A )  ||  ( B  -  C
) )
51 congabseq 27164 . . . . 5  |-  ( ( ( ( 2  x.  A )  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( 2  x.  A
)  ||  ( B  -  C ) )  -> 
( ( abs `  ( B  -  C )
)  <  ( 2  x.  A )  <->  B  =  C ) )
5245, 47, 49, 50, 51syl31anc 1185 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  /\  ( 2  x.  A )  ||  ( B  -  C )
)  ->  ( ( abs `  ( B  -  C ) )  < 
( 2  x.  A
)  <->  B  =  C
) )
5341, 52mpbid 201 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  /\  ( 2  x.  A )  ||  ( B  -  C )
)  ->  B  =  C )
54 simpll2 995 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  B  e.  ( 0 ... A ) )
55 elfzle1 10815 . . . . . . . . . . 11  |-  ( B  e.  ( 0 ... A )  ->  0  <_  B )
5654, 55syl 15 . . . . . . . . . 10  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  0  <_  B
)
577zred 10133 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  B  e.  RR )
5816zred 10133 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  C  e.  RR )
5958renegcld 9226 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  -u C  e.  RR )
6057, 59resubcld 9227 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( B  -  -u C )  e.  RR )
6160recnd 8877 . . . . . . . . . . . . . 14  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( B  -  -u C )  e.  CC )
6261abscld 11934 . . . . . . . . . . . . 13  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( abs `  ( B  -  -u C ) )  e.  RR )
6362ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  ( abs `  ( B  -  -u C ) )  e.  RR )
64 1re 8853 . . . . . . . . . . . . . . . 16  |-  1  e.  RR
65 resubcl 9127 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  RR  /\  1  e.  RR )  ->  ( A  -  1 )  e.  RR )
6621, 64, 65sylancl 643 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( A  - 
1 )  e.  RR )
6766renegcld 9226 . . . . . . . . . . . . . 14  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  -u ( A  - 
1 )  e.  RR )
6821, 67resubcld 9227 . . . . . . . . . . . . 13  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( A  -  -u ( A  -  1 ) )  e.  RR )
6968ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  ( A  -  -u ( A  -  1 ) )  e.  RR )
7027ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  ( 2  x.  A )  e.  RR )
717ad2antrr 706 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  B  e.  ZZ )
7271zcnd 10134 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  B  e.  CC )
7316znegcld 10135 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  -u C  e.  ZZ )
7473ad2antrr 706 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  -u C  e.  ZZ )
7574zcnd 10134 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  -u C  e.  CC )
7672, 75abssubd 11951 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  ( abs `  ( B  -  -u C ) )  =  ( abs `  ( -u C  -  B ) ) )
77 elfzel1 10813 . . . . . . . . . . . . . . 15  |-  ( C  e.  ( 0 ... ( A  -  1 ) )  ->  0  e.  ZZ )
7877adantl 452 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  0  e.  ZZ )
79 simpr 447 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  C  e.  ( 0 ... ( A  -  1 ) ) )
80 0z 10051 . . . . . . . . . . . . . . . . . . 19  |-  0  e.  ZZ
8180a1i 10 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  0  e.  ZZ )
82 1z 10069 . . . . . . . . . . . . . . . . . . 19  |-  1  e.  ZZ
83 zsubcl 10077 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  ZZ  /\  1  e.  ZZ )  ->  ( A  -  1 )  e.  ZZ )
843, 82, 83sylancl 643 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( A  - 
1 )  e.  ZZ )
85 fzneg 27172 . . . . . . . . . . . . . . . . . 18  |-  ( ( C  e.  ZZ  /\  0  e.  ZZ  /\  ( A  -  1 )  e.  ZZ )  -> 
( C  e.  ( 0 ... ( A  -  1 ) )  <->  -u C  e.  ( -u ( A  -  1 ) ... -u 0
) ) )
8616, 81, 84, 85syl3anc 1182 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( C  e.  ( 0 ... ( A  -  1 ) )  <->  -u C  e.  (
-u ( A  - 
1 ) ... -u 0
) ) )
8786ad2antrr 706 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  ( C  e.  ( 0 ... ( A  -  1 ) )  <->  -u C  e.  (
-u ( A  - 
1 ) ... -u 0
) ) )
8879, 87mpbid 201 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  -u C  e.  (
-u ( A  - 
1 ) ... -u 0
) )
89 neg0 9109 . . . . . . . . . . . . . . . . 17  |-  -u 0  =  0
9089a1i 10 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  -u 0  =  0 )
9190oveq2d 5890 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  ( -u ( A  -  1 ) ... -u 0 )  =  ( -u ( A  -  1 ) ... 0 ) )
9288, 91eleqtrd 2372 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  -u C  e.  (
-u ( A  - 
1 ) ... 0
) )
933ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  A  e.  ZZ )
94 simp1 955 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  A  e.  NN )
9542, 94, 44sylancr 644 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( 2  x.  A )  e.  NN )
96 nnm1nn0 10021 . . . . . . . . . . . . . . . . . 18  |-  ( ( 2  x.  A )  e.  NN  ->  (
( 2  x.  A
)  -  1 )  e.  NN0 )
9795, 96syl 15 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( ( 2  x.  A )  - 
1 )  e.  NN0 )
9897nn0ge0d 10037 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  0  <_  (
( 2  x.  A
)  -  1 ) )
99 0cn 8847 . . . . . . . . . . . . . . . . . 18  |-  0  e.  CC
10099subid1i 9134 . . . . . . . . . . . . . . . . 17  |-  ( 0  -  0 )  =  0
101100a1i 10 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( 0  -  0 )  =  0 )
102 ax-1cn 8811 . . . . . . . . . . . . . . . . . . 19  |-  1  e.  CC
103102a1i 10 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  1  e.  CC )
10436, 36, 103addsubassd 9193 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( ( A  +  A )  - 
1 )  =  ( A  +  ( A  -  1 ) ) )
10538oveq1d 5889 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( ( 2  x.  A )  - 
1 )  =  ( ( A  +  A
)  -  1 ) )
106 subcl 9067 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A  -  1 )  e.  CC )
10736, 102, 106sylancl 643 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( A  - 
1 )  e.  CC )
10836, 107subnegd 9180 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( A  -  -u ( A  -  1 ) )  =  ( A  +  ( A  -  1 ) ) )
109104, 105, 1083eqtr4rd 2339 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( A  -  -u ( A  -  1 ) )  =  ( ( 2  x.  A
)  -  1 ) )
11098, 101, 1093brtr4d 4069 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( 0  -  0 )  <_  ( A  -  -u ( A  -  1 ) ) )
111110ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  ( 0  -  0 )  <_  ( A  -  -u ( A  -  1 ) ) )
112 fzmaxdif 27171 . . . . . . . . . . . . . 14  |-  ( ( ( 0  e.  ZZ  /\  -u C  e.  ( -u ( A  -  1 ) ... 0 ) )  /\  ( A  e.  ZZ  /\  B  e.  ( 0 ... A
) )  /\  (
0  -  0 )  <_  ( A  -  -u ( A  -  1 ) ) )  -> 
( abs `  ( -u C  -  B ) )  <_  ( A  -  -u ( A  - 
1 ) ) )
11378, 92, 93, 54, 111, 112syl221anc 1193 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  ( abs `  ( -u C  -  B ) )  <_  ( A  -  -u ( A  - 
1 ) ) )
11476, 113eqbrtrd 4059 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  ( abs `  ( B  -  -u C ) )  <_  ( A  -  -u ( A  - 
1 ) ) )
11527ltm1d 9705 . . . . . . . . . . . . . 14  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( ( 2  x.  A )  - 
1 )  <  (
2  x.  A ) )
116109, 115eqbrtrd 4059 . . . . . . . . . . . . 13  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( A  -  -u ( A  -  1 ) )  <  (
2  x.  A ) )
117116ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  ( A  -  -u ( A  -  1 ) )  <  (
2  x.  A ) )
11863, 69, 70, 114, 117lelttrd 8990 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  ( abs `  ( B  -  -u C ) )  <  ( 2  x.  A ) )
11995ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  ( 2  x.  A )  e.  NN )
120 simplr 731 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  ( 2  x.  A )  ||  ( B  -  -u C ) )
121 congabseq 27164 . . . . . . . . . . . 12  |-  ( ( ( ( 2  x.  A )  e.  NN  /\  B  e.  ZZ  /\  -u C  e.  ZZ )  /\  ( 2  x.  A )  ||  ( B  -  -u C ) )  ->  ( ( abs `  ( B  -  -u C ) )  < 
( 2  x.  A
)  <->  B  =  -u C
) )
122119, 71, 74, 120, 121syl31anc 1185 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  ( ( abs `  ( B  -  -u C
) )  <  (
2  x.  A )  <-> 
B  =  -u C
) )
123118, 122mpbid 201 . . . . . . . . . 10  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  B  =  -u C )
12456, 123breqtrd 4063 . . . . . . . . 9  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  0  <_  -u C
)
125 elfzelz 10814 . . . . . . . . . . . 12  |-  ( C  e.  ( 0 ... ( A  -  1 ) )  ->  C  e.  ZZ )
126125zred 10133 . . . . . . . . . . 11  |-  ( C  e.  ( 0 ... ( A  -  1 ) )  ->  C  e.  RR )
127126adantl 452 . . . . . . . . . 10  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  C  e.  RR )
128127le0neg1d 9360 . . . . . . . . 9  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  ( C  <_ 
0  <->  0  <_  -u C
) )
129124, 128mpbird 223 . . . . . . . 8  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  C  <_  0
)
130 elfzle1 10815 . . . . . . . . 9  |-  ( C  e.  ( 0 ... ( A  -  1 ) )  ->  0  <_  C )
131130adantl 452 . . . . . . . 8  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  0  <_  C
)
132 letri3 8923 . . . . . . . . 9  |-  ( ( C  e.  RR  /\  0  e.  RR )  ->  ( C  =  0  <-> 
( C  <_  0  /\  0  <_  C ) ) )
133127, 22, 132sylancl 643 . . . . . . . 8  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  ( C  =  0  <->  ( C  <_ 
0  /\  0  <_  C ) ) )
134129, 131, 133mpbir2and 888 . . . . . . 7  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  C  =  0 )
135134negeqd 9062 . . . . . 6  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  -u C  =  -u
0 )
136135, 90eqtrd 2328 . . . . 5  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  -u C  =  0 )
137136, 123, 1343eqtr4d 2338 . . . 4  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  e.  ( 0 ... ( A  - 
1 ) ) )  ->  B  =  C )
138 oveq2 5882 . . . . . . . . 9  |-  ( C  =  A  ->  ( B  -  C )  =  ( B  -  A ) )
139138adantl 452 . . . . . . . 8  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  =  A )  ->  ( B  -  C
)  =  ( B  -  A ) )
140139fveq2d 5545 . . . . . . 7  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  =  A )  ->  ( abs `  ( B  -  C )
)  =  ( abs `  ( B  -  A
) ) )
14140ad2antrr 706 . . . . . . 7  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  =  A )  ->  ( abs `  ( B  -  C )
)  <  ( 2  x.  A ) )
142140, 141eqbrtrrd 4061 . . . . . 6  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  =  A )  ->  ( abs `  ( B  -  A )
)  <  ( 2  x.  A ) )
14395ad2antrr 706 . . . . . . 7  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  =  A )  ->  ( 2  x.  A
)  e.  NN )
1447ad2antrr 706 . . . . . . 7  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  =  A )  ->  B  e.  ZZ )
1453ad2antrr 706 . . . . . . 7  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  =  A )  ->  A  e.  ZZ )
146 simplr 731 . . . . . . . . 9  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  =  A )  ->  ( 2  x.  A
)  ||  ( B  -  -u C ) )
1477zcnd 10134 . . . . . . . . . . . . . 14  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  B  e.  CC )
14836, 36, 147ppncand 9213 . . . . . . . . . . . . 13  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( ( A  +  A )  +  ( B  -  A
) )  =  ( A  +  B ) )
14936, 147addcomd 9030 . . . . . . . . . . . . 13  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( A  +  B )  =  ( B  +  A ) )
150148, 149eqtrd 2328 . . . . . . . . . . . 12  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( ( A  +  A )  +  ( B  -  A
) )  =  ( B  +  A ) )
151150ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  =  A )  ->  ( ( A  +  A )  +  ( B  -  A ) )  =  ( B  +  A ) )
152 oveq2 5882 . . . . . . . . . . . 12  |-  ( C  =  A  ->  ( B  +  C )  =  ( B  +  A ) )
153152adantl 452 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  =  A )  ->  ( B  +  C
)  =  ( B  +  A ) )
154151, 153eqtr4d 2331 . . . . . . . . . 10  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  =  A )  ->  ( ( A  +  A )  +  ( B  -  A ) )  =  ( B  +  C ) )
15538oveq1d 5889 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( ( 2  x.  A )  +  ( B  -  A
) )  =  ( ( A  +  A
)  +  ( B  -  A ) ) )
156155ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  =  A )  ->  ( ( 2  x.  A )  +  ( B  -  A ) )  =  ( ( A  +  A )  +  ( B  -  A ) ) )
15716zcnd 10134 . . . . . . . . . . . 12  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  C  e.  CC )
158147, 157subnegd 9180 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( B  -  -u C )  =  ( B  +  C ) )
159158ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  =  A )  ->  ( B  -  -u C
)  =  ( B  +  C ) )
160154, 156, 1593eqtr4d 2338 . . . . . . . . 9  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  =  A )  ->  ( ( 2  x.  A )  +  ( B  -  A ) )  =  ( B  -  -u C ) )
161146, 160breqtrrd 4065 . . . . . . . 8  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  =  A )  ->  ( 2  x.  A
)  ||  ( (
2  x.  A )  +  ( B  -  A ) ) )
1625ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  =  A )  ->  ( 2  x.  A
)  e.  ZZ )
1637, 3zsubcld 10138 . . . . . . . . . 10  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( B  -  A )  e.  ZZ )
164163ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  =  A )  ->  ( B  -  A
)  e.  ZZ )
165 dvdsadd 12583 . . . . . . . . 9  |-  ( ( ( 2  x.  A
)  e.  ZZ  /\  ( B  -  A
)  e.  ZZ )  ->  ( ( 2  x.  A )  ||  ( B  -  A
)  <->  ( 2  x.  A )  ||  (
( 2  x.  A
)  +  ( B  -  A ) ) ) )
166162, 164, 165syl2anc 642 . . . . . . . 8  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  =  A )  ->  ( ( 2  x.  A )  ||  ( B  -  A )  <->  ( 2  x.  A ) 
||  ( ( 2  x.  A )  +  ( B  -  A
) ) ) )
167161, 166mpbird 223 . . . . . . 7  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  =  A )  ->  ( 2  x.  A
)  ||  ( B  -  A ) )
168 congabseq 27164 . . . . . . 7  |-  ( ( ( ( 2  x.  A )  e.  NN  /\  B  e.  ZZ  /\  A  e.  ZZ )  /\  ( 2  x.  A
)  ||  ( B  -  A ) )  -> 
( ( abs `  ( B  -  A )
)  <  ( 2  x.  A )  <->  B  =  A ) )
169143, 144, 145, 167, 168syl31anc 1185 . . . . . 6  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  =  A )  ->  ( ( abs `  ( B  -  A )
)  <  ( 2  x.  A )  <->  B  =  A ) )
170142, 169mpbid 201 . . . . 5  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  =  A )  ->  B  =  A )
171 simpr 447 . . . . 5  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  =  A )  ->  C  =  A )
172170, 171eqtr4d 2331 . . . 4  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  =  A )  ->  B  =  C )
173 nnnn0 9988 . . . . . . . 8  |-  ( A  e.  NN  ->  A  e.  NN0 )
1741733ad2ant1 976 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  A  e.  NN0 )
175 nn0uz 10278 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
176174, 175syl6eleq 2386 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  A  e.  (
ZZ>= `  0 ) )
177 fzm1 10878 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  0
)  ->  ( C  e.  ( 0 ... A
)  <->  ( C  e.  ( 0 ... ( A  -  1 ) )  \/  C  =  A ) ) )
178177biimpa 470 . . . . . 6  |-  ( ( A  e.  ( ZZ>= ` 
0 )  /\  C  e.  ( 0 ... A
) )  ->  ( C  e.  ( 0 ... ( A  - 
1 ) )  \/  C  =  A ) )
179176, 29, 178syl2anc 642 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( C  e.  ( 0 ... ( A  -  1 ) )  \/  C  =  A ) )
180179adantr 451 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  /\  ( 2  x.  A )  ||  ( B  -  -u C ) )  ->  ( C  e.  ( 0 ... ( A  -  1 ) )  \/  C  =  A ) )
181137, 172, 180mpjaodan 761 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  /\  ( 2  x.  A )  ||  ( B  -  -u C ) )  ->  B  =  C )
18253, 181jaodan 760 . 2  |-  ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  /\  ( ( 2  x.  A )  ||  ( B  -  C
)  \/  ( 2  x.  A )  ||  ( B  -  -u C
) ) )  ->  B  =  C )
18314, 182impbida 805 1  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( B  =  C  <->  ( ( 2  x.  A )  ||  ( B  -  C
)  \/  ( 2  x.  A )  ||  ( B  -  -u C
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    <_ cle 8884    - cmin 9053   -ucneg 9054   NNcn 9762   2c2 9811   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   RR+crp 10370   ...cfz 10798   abscabs 11735    || cdivides 12547
This theorem is referenced by:  jm2.27a  27201
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-cnex 8809  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  ax-pre-sup 8831
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-rmo 2564  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-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548
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