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Theorem jm2.27a 26421
Description: Lemma for jm2.27 26424. Reverse direction after existential quantifiers are expanded. (Contributed by Stefan O'Rear, 4-Oct-2014.)
Hypotheses
Ref Expression
jm2.27a1  |-  ( ph  ->  A  e.  ( ZZ>= ` 
2 ) )
jm2.27a2  |-  ( ph  ->  B  e.  NN )
jm2.27a3  |-  ( ph  ->  C  e.  NN )
jm2.27a4  |-  ( ph  ->  D  e.  NN0 )
jm2.27a5  |-  ( ph  ->  E  e.  NN0 )
jm2.27a6  |-  ( ph  ->  F  e.  NN0 )
jm2.27a7  |-  ( ph  ->  G  e.  NN0 )
jm2.27a8  |-  ( ph  ->  H  e.  NN0 )
jm2.27a9  |-  ( ph  ->  I  e.  NN0 )
jm2.27a10  |-  ( ph  ->  J  e.  NN0 )
jm2.27a11  |-  ( ph  ->  ( ( D ^
2 )  -  (
( ( A ^
2 )  -  1 )  x.  ( C ^ 2 ) ) )  =  1 )
jm2.27a12  |-  ( ph  ->  ( ( F ^
2 )  -  (
( ( A ^
2 )  -  1 )  x.  ( E ^ 2 ) ) )  =  1 )
jm2.27a13  |-  ( ph  ->  G  e.  ( ZZ>= ` 
2 ) )
jm2.27a14  |-  ( ph  ->  ( ( I ^
2 )  -  (
( ( G ^
2 )  -  1 )  x.  ( H ^ 2 ) ) )  =  1 )
jm2.27a15  |-  ( ph  ->  E  =  ( ( J  +  1 )  x.  ( 2  x.  ( C ^ 2 ) ) ) )
jm2.27a16  |-  ( ph  ->  F  ||  ( G  -  A ) )
jm2.27a17  |-  ( ph  ->  ( 2  x.  C
)  ||  ( G  -  1 ) )
jm2.27a18  |-  ( ph  ->  F  ||  ( H  -  C ) )
jm2.27a19  |-  ( ph  ->  ( 2  x.  C
)  ||  ( H  -  B ) )
jm2.27a20  |-  ( ph  ->  B  <_  C )
jm2.27a21  |-  ( ph  ->  P  e.  ZZ )
jm2.27a22  |-  ( ph  ->  D  =  ( A Xrm  P ) )
jm2.27a23  |-  ( ph  ->  C  =  ( A Yrm  P ) )
jm2.27a24  |-  ( ph  ->  Q  e.  ZZ )
jm2.27a25  |-  ( ph  ->  F  =  ( A Xrm  Q ) )
jm2.27a26  |-  ( ph  ->  E  =  ( A Yrm  Q ) )
jm2.27a27  |-  ( ph  ->  R  e.  ZZ )
jm2.27a28  |-  ( ph  ->  I  =  ( G Xrm  R ) )
jm2.27a29  |-  ( ph  ->  H  =  ( G Yrm  R ) )
Assertion
Ref Expression
jm2.27a  |-  ( ph  ->  C  =  ( A Yrm  B ) )

Proof of Theorem jm2.27a
StepHypRef Expression
1 jm2.27a23 . 2  |-  ( ph  ->  C  =  ( A Yrm  P ) )
2 2z 10143 . . . . . 6  |-  2  e.  ZZ
3 jm2.27a3 . . . . . . 7  |-  ( ph  ->  C  e.  NN )
43nnzd 10205 . . . . . 6  |-  ( ph  ->  C  e.  ZZ )
5 zmulcl 10155 . . . . . 6  |-  ( ( 2  e.  ZZ  /\  C  e.  ZZ )  ->  ( 2  x.  C
)  e.  ZZ )
62, 4, 5sylancr 644 . . . . 5  |-  ( ph  ->  ( 2  x.  C
)  e.  ZZ )
7 jm2.27a2 . . . . . 6  |-  ( ph  ->  B  e.  NN )
87nnzd 10205 . . . . 5  |-  ( ph  ->  B  e.  ZZ )
9 jm2.27a27 . . . . 5  |-  ( ph  ->  R  e.  ZZ )
10 jm2.27a21 . . . . 5  |-  ( ph  ->  P  e.  ZZ )
11 jm2.27a8 . . . . . . . 8  |-  ( ph  ->  H  e.  NN0 )
1211nn0zd 10204 . . . . . . 7  |-  ( ph  ->  H  e.  ZZ )
13 jm2.27a19 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  C
)  ||  ( H  -  B ) )
14 congsym 26378 . . . . . . . 8  |-  ( ( ( ( 2  x.  C )  e.  ZZ  /\  H  e.  ZZ )  /\  ( B  e.  ZZ  /\  ( 2  x.  C )  ||  ( H  -  B
) ) )  -> 
( 2  x.  C
)  ||  ( B  -  H ) )
156, 12, 8, 13, 14syl22anc 1183 . . . . . . 7  |-  ( ph  ->  ( 2  x.  C
)  ||  ( B  -  H ) )
16 jm2.27a17 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  C
)  ||  ( G  -  1 ) )
17 jm2.27a13 . . . . . . . . . 10  |-  ( ph  ->  G  e.  ( ZZ>= ` 
2 ) )
1811nn0ge0d 10110 . . . . . . . . . . . . 13  |-  ( ph  ->  0  <_  H )
19 rmy0 26337 . . . . . . . . . . . . . 14  |-  ( G  e.  ( ZZ>= `  2
)  ->  ( G Yrm  0 )  =  0 )
2017, 19syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  ( G Yrm  0 )  =  0 )
21 jm2.27a29 . . . . . . . . . . . . . 14  |-  ( ph  ->  H  =  ( G Yrm  R ) )
2221eqcomd 2363 . . . . . . . . . . . . 13  |-  ( ph  ->  ( G Yrm  R )  =  H )
2318, 20, 223brtr4d 4132 . . . . . . . . . . . 12  |-  ( ph  ->  ( G Yrm  0 )  <_ 
( G Yrm  R ) )
24 0z 10124 . . . . . . . . . . . . . 14  |-  0  e.  ZZ
2524a1i 10 . . . . . . . . . . . . 13  |-  ( ph  ->  0  e.  ZZ )
26 lermy 26365 . . . . . . . . . . . . 13  |-  ( ( G  e.  ( ZZ>= ` 
2 )  /\  0  e.  ZZ  /\  R  e.  ZZ )  ->  (
0  <_  R  <->  ( G Yrm  0 )  <_  ( G Yrm  R
) ) )
2717, 25, 9, 26syl3anc 1182 . . . . . . . . . . . 12  |-  ( ph  ->  ( 0  <_  R  <->  ( G Yrm  0 )  <_  ( G Yrm 
R ) ) )
2823, 27mpbird 223 . . . . . . . . . . 11  |-  ( ph  ->  0  <_  R )
29 elnn0z 10125 . . . . . . . . . . 11  |-  ( R  e.  NN0  <->  ( R  e.  ZZ  /\  0  <_  R ) )
309, 28, 29sylanbrc 645 . . . . . . . . . 10  |-  ( ph  ->  R  e.  NN0 )
31 jm2.16nn0 26420 . . . . . . . . . 10  |-  ( ( G  e.  ( ZZ>= ` 
2 )  /\  R  e.  NN0 )  ->  ( G  -  1 ) 
||  ( ( G Yrm  R )  -  R ) )
3217, 30, 31syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( G  -  1 )  ||  ( ( G Yrm  R )  -  R
) )
3321oveq1d 5957 . . . . . . . . 9  |-  ( ph  ->  ( H  -  R
)  =  ( ( G Yrm  R )  -  R
) )
3432, 33breqtrrd 4128 . . . . . . . 8  |-  ( ph  ->  ( G  -  1 )  ||  ( H  -  R ) )
35 jm2.27a7 . . . . . . . . . . 11  |-  ( ph  ->  G  e.  NN0 )
3635nn0zd 10204 . . . . . . . . . 10  |-  ( ph  ->  G  e.  ZZ )
37 peano2zm 10151 . . . . . . . . . 10  |-  ( G  e.  ZZ  ->  ( G  -  1 )  e.  ZZ )
3836, 37syl 15 . . . . . . . . 9  |-  ( ph  ->  ( G  -  1 )  e.  ZZ )
3912, 9zsubcld 10211 . . . . . . . . 9  |-  ( ph  ->  ( H  -  R
)  e.  ZZ )
40 dvdstr 12653 . . . . . . . . 9  |-  ( ( ( 2  x.  C
)  e.  ZZ  /\  ( G  -  1
)  e.  ZZ  /\  ( H  -  R
)  e.  ZZ )  ->  ( ( ( 2  x.  C ) 
||  ( G  - 
1 )  /\  ( G  -  1 ) 
||  ( H  -  R ) )  -> 
( 2  x.  C
)  ||  ( H  -  R ) ) )
416, 38, 39, 40syl3anc 1182 . . . . . . . 8  |-  ( ph  ->  ( ( ( 2  x.  C )  ||  ( G  -  1
)  /\  ( G  -  1 )  ||  ( H  -  R
) )  ->  (
2  x.  C ) 
||  ( H  -  R ) ) )
4216, 34, 41mp2and 660 . . . . . . 7  |-  ( ph  ->  ( 2  x.  C
)  ||  ( H  -  R ) )
43 congtr 26375 . . . . . . 7  |-  ( ( ( ( 2  x.  C )  e.  ZZ  /\  B  e.  ZZ )  /\  ( H  e.  ZZ  /\  R  e.  ZZ )  /\  (
( 2  x.  C
)  ||  ( B  -  H )  /\  (
2  x.  C ) 
||  ( H  -  R ) ) )  ->  ( 2  x.  C )  ||  ( B  -  R )
)
446, 8, 12, 9, 15, 42, 43syl222anc 1198 . . . . . 6  |-  ( ph  ->  ( 2  x.  C
)  ||  ( B  -  R ) )
4544orcd 381 . . . . 5  |-  ( ph  ->  ( ( 2  x.  C )  ||  ( B  -  R )  \/  ( 2  x.  C
)  ||  ( B  -  -u R ) ) )
46 jm2.27a24 . . . . . . 7  |-  ( ph  ->  Q  e.  ZZ )
47 zmulcl 10155 . . . . . . 7  |-  ( ( 2  e.  ZZ  /\  Q  e.  ZZ )  ->  ( 2  x.  Q
)  e.  ZZ )
482, 46, 47sylancr 644 . . . . . 6  |-  ( ph  ->  ( 2  x.  Q
)  e.  ZZ )
49 zsqcl 11264 . . . . . . . . . . . . . 14  |-  ( C  e.  ZZ  ->  ( C ^ 2 )  e.  ZZ )
504, 49syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  ( C ^ 2 )  e.  ZZ )
51 dvdsmul2 12642 . . . . . . . . . . . . 13  |-  ( ( 2  e.  ZZ  /\  ( C ^ 2 )  e.  ZZ )  -> 
( C ^ 2 )  ||  ( 2  x.  ( C ^
2 ) ) )
522, 50, 51sylancr 644 . . . . . . . . . . . 12  |-  ( ph  ->  ( C ^ 2 )  ||  ( 2  x.  ( C ^
2 ) ) )
53 jm2.27a10 . . . . . . . . . . . . . . 15  |-  ( ph  ->  J  e.  NN0 )
5453nn0zd 10204 . . . . . . . . . . . . . 14  |-  ( ph  ->  J  e.  ZZ )
5554peano2zd 10209 . . . . . . . . . . . . 13  |-  ( ph  ->  ( J  +  1 )  e.  ZZ )
56 zmulcl 10155 . . . . . . . . . . . . . 14  |-  ( ( 2  e.  ZZ  /\  ( C ^ 2 )  e.  ZZ )  -> 
( 2  x.  ( C ^ 2 ) )  e.  ZZ )
572, 50, 56sylancr 644 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 2  x.  ( C ^ 2 ) )  e.  ZZ )
58 dvdsmultr2 12655 . . . . . . . . . . . . 13  |-  ( ( ( C ^ 2 )  e.  ZZ  /\  ( J  +  1
)  e.  ZZ  /\  ( 2  x.  ( C ^ 2 ) )  e.  ZZ )  -> 
( ( C ^
2 )  ||  (
2  x.  ( C ^ 2 ) )  ->  ( C ^
2 )  ||  (
( J  +  1 )  x.  ( 2  x.  ( C ^
2 ) ) ) ) )
5950, 55, 57, 58syl3anc 1182 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( C ^
2 )  ||  (
2  x.  ( C ^ 2 ) )  ->  ( C ^
2 )  ||  (
( J  +  1 )  x.  ( 2  x.  ( C ^
2 ) ) ) ) )
6052, 59mpd 14 . . . . . . . . . . 11  |-  ( ph  ->  ( C ^ 2 )  ||  ( ( J  +  1 )  x.  ( 2  x.  ( C ^ 2 ) ) ) )
611oveq1d 5957 . . . . . . . . . . 11  |-  ( ph  ->  ( C ^ 2 )  =  ( ( A Yrm  P ) ^ 2 ) )
62 jm2.27a15 . . . . . . . . . . . 12  |-  ( ph  ->  E  =  ( ( J  +  1 )  x.  ( 2  x.  ( C ^ 2 ) ) ) )
63 jm2.27a26 . . . . . . . . . . . 12  |-  ( ph  ->  E  =  ( A Yrm  Q ) )
6462, 63eqtr3d 2392 . . . . . . . . . . 11  |-  ( ph  ->  ( ( J  + 
1 )  x.  (
2  x.  ( C ^ 2 ) ) )  =  ( A Yrm  Q ) )
6560, 61, 643brtr3d 4131 . . . . . . . . . 10  |-  ( ph  ->  ( ( A Yrm  P ) ^ 2 )  ||  ( A Yrm  Q ) )
66 jm2.27a1 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  ( ZZ>= ` 
2 ) )
6755zred 10206 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( J  +  1 )  e.  RR )
6857zred 10206 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( 2  x.  ( C ^ 2 ) )  e.  RR )
69 nn0p1nn 10092 . . . . . . . . . . . . . . . . . 18  |-  ( J  e.  NN0  ->  ( J  +  1 )  e.  NN )
7053, 69syl 15 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( J  +  1 )  e.  NN )
7170nngt0d 9876 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  0  <  ( J  +  1 ) )
72 2nn 9966 . . . . . . . . . . . . . . . . . 18  |-  2  e.  NN
733nnsqcld 11355 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( C ^ 2 )  e.  NN )
74 nnmulcl 9856 . . . . . . . . . . . . . . . . . 18  |-  ( ( 2  e.  NN  /\  ( C ^ 2 )  e.  NN )  -> 
( 2  x.  ( C ^ 2 ) )  e.  NN )
7572, 73, 74sylancr 644 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( 2  x.  ( C ^ 2 ) )  e.  NN )
7675nngt0d 9876 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  0  <  ( 2  x.  ( C ^
2 ) ) )
7767, 68, 71, 76mulgt0d 9058 . . . . . . . . . . . . . . 15  |-  ( ph  ->  0  <  ( ( J  +  1 )  x.  ( 2  x.  ( C ^ 2 ) ) ) )
7877, 62breqtrrd 4128 . . . . . . . . . . . . . 14  |-  ( ph  ->  0  <  E )
79 rmy0 26337 . . . . . . . . . . . . . . 15  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A Yrm  0 )  =  0 )
8066, 79syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( A Yrm  0 )  =  0 )
8163eqcomd 2363 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( A Yrm  Q )  =  E )
8278, 80, 813brtr4d 4132 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A Yrm  0 )  < 
( A Yrm  Q ) )
83 ltrmy 26362 . . . . . . . . . . . . . 14  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  0  e.  ZZ  /\  Q  e.  ZZ )  ->  (
0  <  Q  <->  ( A Yrm  0 )  <  ( A Yrm  Q ) ) )
8466, 25, 46, 83syl3anc 1182 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 0  <  Q  <->  ( A Yrm  0 )  <  ( A Yrm 
Q ) ) )
8582, 84mpbird 223 . . . . . . . . . . . 12  |-  ( ph  ->  0  <  Q )
86 elnnz 10123 . . . . . . . . . . . 12  |-  ( Q  e.  NN  <->  ( Q  e.  ZZ  /\  0  < 
Q ) )
8746, 85, 86sylanbrc 645 . . . . . . . . . . 11  |-  ( ph  ->  Q  e.  NN )
883nngt0d 9876 . . . . . . . . . . . . . 14  |-  ( ph  ->  0  <  C )
891eqcomd 2363 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( A Yrm  P )  =  C )
9088, 80, 893brtr4d 4132 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A Yrm  0 )  < 
( A Yrm  P ) )
91 ltrmy 26362 . . . . . . . . . . . . . 14  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  0  e.  ZZ  /\  P  e.  ZZ )  ->  (
0  <  P  <->  ( A Yrm  0 )  <  ( A Yrm  P ) ) )
9266, 25, 10, 91syl3anc 1182 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 0  <  P  <->  ( A Yrm  0 )  <  ( A Yrm 
P ) ) )
9390, 92mpbird 223 . . . . . . . . . . . 12  |-  ( ph  ->  0  <  P )
94 elnnz 10123 . . . . . . . . . . . 12  |-  ( P  e.  NN  <->  ( P  e.  ZZ  /\  0  < 
P ) )
9510, 93, 94sylanbrc 645 . . . . . . . . . . 11  |-  ( ph  ->  P  e.  NN )
96 jm2.20nn 26413 . . . . . . . . . . 11  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  Q  e.  NN  /\  P  e.  NN )  ->  (
( ( A Yrm  P ) ^ 2 )  ||  ( A Yrm  Q )  <->  ( P  x.  ( A Yrm  P ) ) 
||  Q ) )
9766, 87, 95, 96syl3anc 1182 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( A Yrm  P ) ^ 2 ) 
||  ( A Yrm  Q )  <-> 
( P  x.  ( A Yrm 
P ) )  ||  Q ) )
9865, 97mpbid 201 . . . . . . . . 9  |-  ( ph  ->  ( P  x.  ( A Yrm 
P ) )  ||  Q )
991, 4eqeltrrd 2433 . . . . . . . . . 10  |-  ( ph  ->  ( A Yrm  P )  e.  ZZ )
100 muldvds2 12645 . . . . . . . . . 10  |-  ( ( P  e.  ZZ  /\  ( A Yrm  P )  e.  ZZ  /\  Q  e.  ZZ )  ->  (
( P  x.  ( A Yrm 
P ) )  ||  Q  ->  ( A Yrm  P ) 
||  Q ) )
10110, 99, 46, 100syl3anc 1182 . . . . . . . . 9  |-  ( ph  ->  ( ( P  x.  ( A Yrm  P ) ) 
||  Q  ->  ( A Yrm 
P )  ||  Q
) )
10298, 101mpd 14 . . . . . . . 8  |-  ( ph  ->  ( A Yrm  P )  ||  Q )
1031, 102eqbrtrd 4122 . . . . . . 7  |-  ( ph  ->  C  ||  Q )
1042a1i 10 . . . . . . . 8  |-  ( ph  ->  2  e.  ZZ )
105 dvdscmul 12646 . . . . . . . 8  |-  ( ( C  e.  ZZ  /\  Q  e.  ZZ  /\  2  e.  ZZ )  ->  ( C  ||  Q  ->  (
2  x.  C ) 
||  ( 2  x.  Q ) ) )
1064, 46, 104, 105syl3anc 1182 . . . . . . 7  |-  ( ph  ->  ( C  ||  Q  ->  ( 2  x.  C
)  ||  ( 2  x.  Q ) ) )
107103, 106mpd 14 . . . . . 6  |-  ( ph  ->  ( 2  x.  C
)  ||  ( 2  x.  Q ) )
108 jm2.27a25 . . . . . . . . . 10  |-  ( ph  ->  F  =  ( A Xrm  Q ) )
109 jm2.27a6 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  NN0 )
110109nn0zd 10204 . . . . . . . . . 10  |-  ( ph  ->  F  e.  ZZ )
111108, 110eqeltrrd 2433 . . . . . . . . 9  |-  ( ph  ->  ( A Xrm  Q )  e.  ZZ )
112 frmy 26322 . . . . . . . . . . 11  |- Yrm  : (
( ZZ>= `  2 )  X.  ZZ ) --> ZZ
113112fovcl 6033 . . . . . . . . . 10  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  R  e.  ZZ )  ->  ( A Yrm 
R )  e.  ZZ )
11466, 9, 113syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( A Yrm  R )  e.  ZZ )
11521, 12eqeltrrd 2433 . . . . . . . . 9  |-  ( ph  ->  ( G Yrm  R )  e.  ZZ )
116 eluzelz 10327 . . . . . . . . . . . . 13  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  ZZ )
11766, 116syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  ZZ )
118 jm2.27a16 . . . . . . . . . . . 12  |-  ( ph  ->  F  ||  ( G  -  A ) )
119 congsym 26378 . . . . . . . . . . . 12  |-  ( ( ( F  e.  ZZ  /\  G  e.  ZZ )  /\  ( A  e.  ZZ  /\  F  ||  ( G  -  A
) ) )  ->  F  ||  ( A  -  G ) )
120110, 36, 117, 118, 119syl22anc 1183 . . . . . . . . . . 11  |-  ( ph  ->  F  ||  ( A  -  G ) )
121108, 120eqbrtrrd 4124 . . . . . . . . . 10  |-  ( ph  ->  ( A Xrm  Q )  ||  ( A  -  G
) )
122 jm2.15nn0 26419 . . . . . . . . . . 11  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  G  e.  ( ZZ>= `  2 )  /\  R  e.  NN0 )  ->  ( A  -  G )  ||  (
( A Yrm  R )  -  ( G Yrm  R ) ) )
12366, 17, 30, 122syl3anc 1182 . . . . . . . . . 10  |-  ( ph  ->  ( A  -  G
)  ||  ( ( A Yrm 
R )  -  ( G Yrm 
R ) ) )
124117, 36zsubcld 10211 . . . . . . . . . . 11  |-  ( ph  ->  ( A  -  G
)  e.  ZZ )
125114, 115zsubcld 10211 . . . . . . . . . . 11  |-  ( ph  ->  ( ( A Yrm  R )  -  ( G Yrm  R ) )  e.  ZZ )
126 dvdstr 12653 . . . . . . . . . . 11  |-  ( ( ( A Xrm  Q )  e.  ZZ  /\  ( A  -  G )  e.  ZZ  /\  ( ( A Yrm  R )  -  ( G Yrm 
R ) )  e.  ZZ )  ->  (
( ( A Xrm  Q ) 
||  ( A  -  G )  /\  ( A  -  G )  ||  ( ( A Yrm  R )  -  ( G Yrm  R ) ) )  ->  ( A Xrm 
Q )  ||  (
( A Yrm  R )  -  ( G Yrm  R ) ) ) )
127111, 124, 125, 126syl3anc 1182 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( A Xrm  Q )  ||  ( A  -  G )  /\  ( A  -  G
)  ||  ( ( A Yrm 
R )  -  ( G Yrm 
R ) ) )  ->  ( A Xrm  Q ) 
||  ( ( A Yrm  R )  -  ( G Yrm  R ) ) ) )
128121, 123, 127mp2and 660 . . . . . . . . 9  |-  ( ph  ->  ( A Xrm  Q )  ||  ( ( A Yrm  R )  -  ( G Yrm  R ) ) )
129 jm2.27a18 . . . . . . . . . 10  |-  ( ph  ->  F  ||  ( H  -  C ) )
13021, 1oveq12d 5960 . . . . . . . . . 10  |-  ( ph  ->  ( H  -  C
)  =  ( ( G Yrm  R )  -  ( A Yrm 
P ) ) )
131129, 108, 1303brtr3d 4131 . . . . . . . . 9  |-  ( ph  ->  ( A Xrm  Q )  ||  ( ( G Yrm  R )  -  ( A Yrm  P ) ) )
132 congtr 26375 . . . . . . . . 9  |-  ( ( ( ( A Xrm  Q )  e.  ZZ  /\  ( A Yrm 
R )  e.  ZZ )  /\  ( ( G Yrm  R )  e.  ZZ  /\  ( A Yrm  P )  e.  ZZ )  /\  (
( A Xrm  Q )  ||  ( ( A Yrm  R )  -  ( G Yrm  R ) )  /\  ( A Xrm  Q )  ||  ( ( G Yrm  R )  -  ( A Yrm 
P ) ) ) )  ->  ( A Xrm  Q
)  ||  ( ( A Yrm 
R )  -  ( A Yrm 
P ) ) )
133111, 114, 115, 99, 128, 131, 132syl222anc 1198 . . . . . . . 8  |-  ( ph  ->  ( A Xrm  Q )  ||  ( ( A Yrm  R )  -  ( A Yrm  P ) ) )
134133orcd 381 . . . . . . 7  |-  ( ph  ->  ( ( A Xrm  Q ) 
||  ( ( A Yrm  R )  -  ( A Yrm  P ) )  \/  ( A Xrm 
Q )  ||  (
( A Yrm  R )  -  -u ( A Yrm  P ) ) ) )
135 jm2.26 26418 . . . . . . . 8  |-  ( ( ( A  e.  (
ZZ>= `  2 )  /\  Q  e.  NN )  /\  ( R  e.  ZZ  /\  P  e.  ZZ ) )  ->  ( (
( A Xrm  Q )  ||  ( ( A Yrm  R )  -  ( A Yrm  P ) )  \/  ( A Xrm  Q )  ||  ( ( A Yrm  R )  -  -u ( A Yrm 
P ) ) )  <-> 
( ( 2  x.  Q )  ||  ( R  -  P )  \/  ( 2  x.  Q
)  ||  ( R  -  -u P ) ) ) )
13666, 87, 9, 10, 135syl22anc 1183 . . . . . . 7  |-  ( ph  ->  ( ( ( A Xrm  Q )  ||  ( ( A Yrm  R )  -  ( A Yrm 
P ) )  \/  ( A Xrm  Q )  ||  ( ( A Yrm  R )  -  -u ( A Yrm  P ) ) )  <->  ( (
2  x.  Q ) 
||  ( R  -  P )  \/  (
2  x.  Q ) 
||  ( R  -  -u P ) ) ) )
137134, 136mpbid 201 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  Q )  ||  ( R  -  P )  \/  ( 2  x.  Q
)  ||  ( R  -  -u P ) ) )
138 dvdsacongtr 26394 . . . . . 6  |-  ( ( ( ( 2  x.  Q )  e.  ZZ  /\  R  e.  ZZ )  /\  ( P  e.  ZZ  /\  ( 2  x.  C )  e.  ZZ )  /\  (
( 2  x.  C
)  ||  ( 2  x.  Q )  /\  ( ( 2  x.  Q )  ||  ( R  -  P )  \/  ( 2  x.  Q
)  ||  ( R  -  -u P ) ) ) )  ->  (
( 2  x.  C
)  ||  ( R  -  P )  \/  (
2  x.  C ) 
||  ( R  -  -u P ) ) )
13948, 9, 10, 6, 107, 137, 138syl222anc 1198 . . . . 5  |-  ( ph  ->  ( ( 2  x.  C )  ||  ( R  -  P )  \/  ( 2  x.  C
)  ||  ( R  -  -u P ) ) )
140 acongtr 26388 . . . . 5  |-  ( ( ( ( 2  x.  C )  e.  ZZ  /\  B  e.  ZZ )  /\  ( R  e.  ZZ  /\  P  e.  ZZ )  /\  (
( ( 2  x.  C )  ||  ( B  -  R )  \/  ( 2  x.  C
)  ||  ( B  -  -u R ) )  /\  ( ( 2  x.  C )  ||  ( R  -  P
)  \/  ( 2  x.  C )  ||  ( R  -  -u P
) ) ) )  ->  ( ( 2  x.  C )  ||  ( B  -  P
)  \/  ( 2  x.  C )  ||  ( B  -  -u P
) ) )
1416, 8, 9, 10, 45, 139, 140syl222anc 1198 . . . 4  |-  ( ph  ->  ( ( 2  x.  C )  ||  ( B  -  P )  \/  ( 2  x.  C
)  ||  ( B  -  -u P ) ) )
1427nnnn0d 10107 . . . . . 6  |-  ( ph  ->  B  e.  NN0 )
1433nnnn0d 10107 . . . . . 6  |-  ( ph  ->  C  e.  NN0 )
144 jm2.27a20 . . . . . 6  |-  ( ph  ->  B  <_  C )
145 elfz2nn0 10910 . . . . . 6  |-  ( B  e.  ( 0 ... C )  <->  ( B  e.  NN0  /\  C  e. 
NN0  /\  B  <_  C ) )
146142, 143, 144, 145syl3anbrc 1136 . . . . 5  |-  ( ph  ->  B  e.  ( 0 ... C ) )
14795nnnn0d 10107 . . . . . 6  |-  ( ph  ->  P  e.  NN0 )
148 rmygeid 26374 . . . . . . . 8  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  P  e.  NN0 )  ->  P  <_  ( A Yrm  P ) )
14966, 147, 148syl2anc 642 . . . . . . 7  |-  ( ph  ->  P  <_  ( A Yrm  P
) )
150149, 1breqtrrd 4128 . . . . . 6  |-  ( ph  ->  P  <_  C )
151 elfz2nn0 10910 . . . . . 6  |-  ( P  e.  ( 0 ... C )  <->  ( P  e.  NN0  /\  C  e. 
NN0  /\  P  <_  C ) )
152147, 143, 150, 151syl3anbrc 1136 . . . . 5  |-  ( ph  ->  P  e.  ( 0 ... C ) )
153 acongeq 26393 . . . . 5  |-  ( ( C  e.  NN  /\  B  e.  ( 0 ... C )  /\  P  e.  ( 0 ... C ) )  ->  ( B  =  P  <->  ( ( 2  x.  C )  ||  ( B  -  P
)  \/  ( 2  x.  C )  ||  ( B  -  -u P
) ) ) )
1543, 146, 152, 153syl3anc 1182 . . . 4  |-  ( ph  ->  ( B  =  P  <-> 
( ( 2  x.  C )  ||  ( B  -  P )  \/  ( 2  x.  C
)  ||  ( B  -  -u P ) ) ) )
155141, 154mpbird 223 . . 3  |-  ( ph  ->  B  =  P )
156155oveq2d 5958 . 2  |-  ( ph  ->  ( A Yrm  B )  =  ( A Yrm  P ) )
1571, 156eqtr4d 2393 1  |-  ( ph  ->  C  =  ( A Yrm  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1642    e. wcel 1710   class class class wbr 4102   ` cfv 5334  (class class class)co 5942   0cc0 8824   1c1 8825    + caddc 8827    x. cmul 8829    < clt 8954    <_ cle 8955    - cmin 9124   -ucneg 9125   NNcn 9833   2c2 9882   NN0cn0 10054   ZZcz 10113   ZZ>=cuz 10319   ...cfz 10871   ^cexp 11194    || cdivides 12622   Xrm crmx 26308   Yrm crmy 26309
This theorem is referenced by:  jm2.27b  26422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-inf2 7429  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902  ax-addf 8903  ax-mulf 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-iin 3987  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-se 4432  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-isom 5343  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-of 6162  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-2o 6564  df-oadd 6567  df-omul 6568  df-er 6744  df-map 6859  df-pm 6860  df-ixp 6903  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-fi 7252  df-sup 7281  df-oi 7312  df-card 7659  df-acn 7662  df-cda 7881  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-3 9892  df-4 9893  df-5 9894  df-6 9895  df-7 9896  df-8 9897  df-9 9898  df-10 9899  df-n0 10055  df-z 10114  df-dec 10214  df-uz 10320  df-q 10406  df-rp 10444  df-xneg 10541  df-xadd 10542  df-xmul 10543  df-ioo 10749  df-ioc 10750  df-ico 10751  df-icc 10752  df-fz 10872  df-fzo 10960  df-fl 11014  df-mod 11063  df-seq 11136  df-exp 11195  df-fac 11379  df-bc 11406  df-hash 11428  df-shft 11652  df-cj 11674  df-re 11675  df-im 11676  df-sqr 11810  df-abs 11811  df-limsup 12035  df-clim 12052  df-rlim 12053  df-sum 12250  df-ef 12440  df-sin 12442  df-cos 12443  df-pi 12445  df-dvds 12623  df-gcd 12777  df-prm 12850  df-numer 12897  df-denom 12898  df-struct 13241  df-ndx 13242  df-slot 13243  df-base 13244  df-sets 13245  df-ress 13246  df-plusg 13312  df-mulr 13313  df-starv 13314  df-sca 13315  df-vsca 13316  df-tset 13318  df-ple 13319  df-ds 13321  df-unif 13322  df-hom 13323  df-cco 13324  df-rest 13420  df-topn 13421  df-topgen 13437  df-pt 13438  df-prds 13441  df-xrs 13496  df-0g 13497  df-gsum 13498  df-qtop 13503  df-imas 13504  df-xps 13506  df-mre 13581  df-mrc 13582  df-acs 13584  df-mnd 14460  df-submnd 14509  df-mulg 14585  df-cntz 14886  df-cmn 15184  df-xmet 16469  df-met 16470  df-bl 16471  df-mopn 16472  df-fbas 16473  df-fg 16474  df-cnfld 16477  df-top 16736  df-bases 16738  df-topon 16739  df-topsp 16740  df-cld 16856  df-ntr 16857  df-cls 16858  df-nei 16935  df-lp 16968  df-perf 16969  df-cn 17057  df-cnp 17058  df-haus 17143  df-tx 17357  df-hmeo 17546  df-fil 17637  df-fm 17729  df-flim 17730  df-flf 17731  df-xms 17981  df-ms 17982  df-tms 17983  df-cncf 18479  df-limc 19314  df-dv 19315  df-log 20015  df-squarenn 26249  df-pell1qr 26250  df-pell14qr 26251  df-pell1234qr 26252  df-pellfund 26253  df-rmx 26310  df-rmy 26311
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