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Theorem acongeq 27070
Description: Two numbers in the fundamental domain are alternating-congruent iff they are equal. TODO: could be used to shorten jm2.26 27095 (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 10054 . . . . . . 7  |-  2  e.  ZZ
2 nnz 10045 . . . . . . . 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 10066 . . . . . . 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 10798 . . . . . . 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 27058 . . . . . 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 5866 . . . . 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 4047 . . 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 10798 . . . . . . . . . 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 10122 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( B  -  C )  e.  ZZ )
1817zcnd 10118 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( B  -  C )  e.  CC )
1918abscld 11918 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( abs `  ( B  -  C )
)  e.  RR )
20 nnre 9753 . . . . . . . 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 8838 . . . . . . 7  |-  0  e.  RR
23 resubcl 9111 . . . . . . 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 9815 . . . . . . 7  |-  2  e.  RR
26 remulcl 8822 . . . . . . 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 9339 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( A  - 
0 )  <_  ( A  -  0 ) )
31 fzmaxdif 27068 . . . . . . 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 10363 . . . . . . . . 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 10419 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  A  <  ( A  +  A )
)
3621recnd 8861 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  A  e.  CC )
3736subid1d 9146 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( A  - 
0 )  =  A )
38362timesd 9954 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( 2  x.  A )  =  ( A  +  A ) )
3935, 37, 383brtr4d 4053 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( A  - 
0 )  <  (
2  x.  A ) )
4019, 24, 27, 32, 39lelttrd 8974 . . . . 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 9877 . . . . . 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 9769 . . . . . 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 27061 . . . . 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 10799 . . . . . . . . . . 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 10117 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  B  e.  RR )
5816zred 10117 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  C  e.  RR )
5958renegcld 9210 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  -u C  e.  RR )
6057, 59resubcld 9211 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( B  -  -u C )  e.  RR )
6160recnd 8861 . . . . . . . . . . . . . 14  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( B  -  -u C )  e.  CC )
6261abscld 11918 . . . . . . . . . . . . 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 8837 . . . . . . . . . . . . . . . 16  |-  1  e.  RR
65 resubcl 9111 . . . . . . . . . . . . . . . 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 9210 . . . . . . . . . . . . . 14  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  -u ( A  - 
1 )  e.  RR )
6821, 67resubcld 9211 . . . . . . . . . . . . 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 10118 . . . . . . . . . . . . . 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 10119 . . . . . . . . . . . . . . . 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 10118 . . . . . . . . . . . . . 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 11935 . . . . . . . . . . . . 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 10797 . . . . . . . . . . . . . . 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 10035 . . . . . . . . . . . . . . . . . . 19  |-  0  e.  ZZ
8180a1i 10 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  0  e.  ZZ )
82 1z 10053 . . . . . . . . . . . . . . . . . . 19  |-  1  e.  ZZ
83 zsubcl 10061 . . . . . . . . . . . . . . . . . . 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 27069 . . . . . . . . . . . . . . . . . 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 9093 . . . . . . . . . . . . . . . . 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 5874 . . . . . . . . . . . . . . 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 2359 . . . . . . . . . . . . . 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 10005 . . . . . . . . . . . . . . . . . 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 10021 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  0  <_  (
( 2  x.  A
)  -  1 ) )
99 0cn 8831 . . . . . . . . . . . . . . . . . 18  |-  0  e.  CC
10099subid1i 9118 . . . . . . . . . . . . . . . . 17  |-  ( 0  -  0 )  =  0
101100a1i 10 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( 0  -  0 )  =  0 )
102 ax-1cn 8795 . . . . . . . . . . . . . . . . . . 19  |-  1  e.  CC
103102a1i 10 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  1  e.  CC )
10436, 36, 103addsubassd 9177 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( ( A  +  A )  - 
1 )  =  ( A  +  ( A  -  1 ) ) )
10538oveq1d 5873 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( ( 2  x.  A )  - 
1 )  =  ( ( A  +  A
)  -  1 ) )
106 subcl 9051 . . . . . . . . . . . . . . . . . . 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 9164 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( A  -  -u ( A  -  1 ) )  =  ( A  +  ( A  -  1 ) ) )
109104, 105, 1083eqtr4rd 2326 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( A  -  -u ( A  -  1 ) )  =  ( ( 2  x.  A
)  -  1 ) )
11098, 101, 1093brtr4d 4053 . . . . . . . . . . . . . . 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 27068 . . . . . . . . . . . . . 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 4043 . . . . . . . . . . . 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 9689 . . . . . . . . . . . . . 14  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( ( 2  x.  A )  - 
1 )  <  (
2  x.  A ) )
116109, 115eqbrtrd 4043 . . . . . . . . . . . . 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 8974 . . . . . . . . . . 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 27061 . . . . . . . . . . . 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 4047 . . . . . . . . 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 10798 . . . . . . . . . . . 12  |-  ( C  e.  ( 0 ... ( A  -  1 ) )  ->  C  e.  ZZ )
126125zred 10117 . . . . . . . . . . 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 9344 . . . . . . . . 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 10799 . . . . . . . . 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 8907 . . . . . . . . 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 9046 . . . . . 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 2315 . . . . 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 2325 . . . 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 5866 . . . . . . . . 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 5529 . . . . . . 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 4045 . . . . . 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 10118 . . . . . . . . . . . . . 14  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  B  e.  CC )
14836, 36, 147ppncand 9197 . . . . . . . . . . . . 13  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( ( A  +  A )  +  ( B  -  A
) )  =  ( A  +  B ) )
14936, 147addcomd 9014 . . . . . . . . . . . . 13  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  ( A  +  B )  =  ( B  +  A ) )
150148, 149eqtrd 2315 . . . . . . . . . . . 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 5866 . . . . . . . . . . . 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 2318 . . . . . . . . . 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 5873 . . . . . . . . . . 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 10118 . . . . . . . . . . . 12  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  C  e.  CC )
158147, 157subnegd 9164 . . . . . . . . . . 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 2325 . . . . . . . . 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 4049 . . . . . . . 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 10122 . . . . . . . . . 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 12567 . . . . . . . . 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 27061 . . . . . . 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 2318 . . . 4  |-  ( ( ( ( A  e.  NN  /\  B  e.  ( 0 ... A
)  /\  C  e.  ( 0 ... A
) )  /\  (
2  x.  A ) 
||  ( B  -  -u C ) )  /\  C  =  A )  ->  B  =  C )
173 nnnn0 9972 . . . . . . . 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 10262 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
176174, 175syl6eleq 2373 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  ( 0 ... A )  /\  C  e.  ( 0 ... A ) )  ->  A  e.  (
ZZ>= `  0 ) )
177 fzm1 10862 . . . . . . 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 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868    - cmin 9037   -ucneg 9038   NNcn 9746   2c2 9795   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   RR+crp 10354   ...cfz 10782   abscabs 11719    || cdivides 12531
This theorem is referenced by:  jm2.27a  27098
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532
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