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Theorem jm2.27a 26430
Description: Lemma for jm2.27 26433. 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 9986 . . . . . 6  |-  2  e.  ZZ
3 jm2.27a3 . . . . . . 7  |-  ( ph  ->  C  e.  NN )
43nnzd 10048 . . . . . 6  |-  ( ph  ->  C  e.  ZZ )
5 zmulcl 9998 . . . . . 6  |-  ( ( 2  e.  ZZ  /\  C  e.  ZZ )  ->  ( 2  x.  C
)  e.  ZZ )
62, 4, 5sylancr 647 . . . . 5  |-  ( ph  ->  ( 2  x.  C
)  e.  ZZ )
7 jm2.27a2 . . . . . 6  |-  ( ph  ->  B  e.  NN )
87nnzd 10048 . . . . 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 10047 . . . . . . 7  |-  ( ph  ->  H  e.  ZZ )
13 jm2.27a19 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  C
)  ||  ( H  -  B ) )
14 congsym 26387 . . . . . . . 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 1188 . . . . . . 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 9953 . . . . . . . . . . . . 13  |-  ( ph  ->  0  <_  H )
19 rmy0 26346 . . . . . . . . . . . . . 14  |-  ( G  e.  ( ZZ>= `  2
)  ->  ( G Yrm  0 )  =  0 )
2017, 19syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  ( G Yrm  0 )  =  0 )
21 jm2.27a29 . . . . . . . . . . . . . 14  |-  ( ph  ->  H  =  ( G Yrm  R ) )
2221eqcomd 2261 . . . . . . . . . . . . 13  |-  ( ph  ->  ( G Yrm  R )  =  H )
2318, 20, 223brtr4d 3993 . . . . . . . . . . . 12  |-  ( ph  ->  ( G Yrm  0 )  <_ 
( G Yrm  R ) )
24 0z 9967 . . . . . . . . . . . . . 14  |-  0  e.  ZZ
2524a1i 12 . . . . . . . . . . . . 13  |-  ( ph  ->  0  e.  ZZ )
26 lermy 26374 . . . . . . . . . . . . 13  |-  ( ( G  e.  ( ZZ>= ` 
2 )  /\  0  e.  ZZ  /\  R  e.  ZZ )  ->  (
0  <_  R  <->  ( G Yrm  0 )  <_  ( G Yrm  R
) ) )
2717, 25, 9, 26syl3anc 1187 . . . . . . . . . . . 12  |-  ( ph  ->  ( 0  <_  R  <->  ( G Yrm  0 )  <_  ( G Yrm 
R ) ) )
2823, 27mpbird 225 . . . . . . . . . . 11  |-  ( ph  ->  0  <_  R )
29 elnn0z 9968 . . . . . . . . . . 11  |-  ( R  e.  NN0  <->  ( R  e.  ZZ  /\  0  <_  R ) )
309, 28, 29sylanbrc 648 . . . . . . . . . 10  |-  ( ph  ->  R  e.  NN0 )
31 jm2.16nn0 26429 . . . . . . . . . 10  |-  ( ( G  e.  ( ZZ>= ` 
2 )  /\  R  e.  NN0 )  ->  ( G  -  1 ) 
||  ( ( G Yrm  R )  -  R ) )
3217, 30, 31syl2anc 645 . . . . . . . . 9  |-  ( ph  ->  ( G  -  1 )  ||  ( ( G Yrm  R )  -  R
) )
3321oveq1d 5772 . . . . . . . . 9  |-  ( ph  ->  ( H  -  R
)  =  ( ( G Yrm  R )  -  R
) )
3432, 33breqtrrd 3989 . . . . . . . 8  |-  ( ph  ->  ( G  -  1 )  ||  ( H  -  R ) )
35 jm2.27a7 . . . . . . . . . . 11  |-  ( ph  ->  G  e.  NN0 )
3635nn0zd 10047 . . . . . . . . . 10  |-  ( ph  ->  G  e.  ZZ )
37 peano2zm 9994 . . . . . . . . . 10  |-  ( G  e.  ZZ  ->  ( G  -  1 )  e.  ZZ )
3836, 37syl 17 . . . . . . . . 9  |-  ( ph  ->  ( G  -  1 )  e.  ZZ )
3912, 9zsubcld 10054 . . . . . . . . 9  |-  ( ph  ->  ( H  -  R
)  e.  ZZ )
40 dvdstr 12489 . . . . . . . . 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 1187 . . . . . . . 8  |-  ( ph  ->  ( ( ( 2  x.  C )  ||  ( G  -  1
)  /\  ( G  -  1 )  ||  ( H  -  R
) )  ->  (
2  x.  C ) 
||  ( H  -  R ) ) )
4216, 34, 41mp2and 663 . . . . . . 7  |-  ( ph  ->  ( 2  x.  C
)  ||  ( H  -  R ) )
43 congtr 26384 . . . . . . 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 1203 . . . . . 6  |-  ( ph  ->  ( 2  x.  C
)  ||  ( B  -  R ) )
4544orcd 383 . . . . 5  |-  ( ph  ->  ( ( 2  x.  C )  ||  ( B  -  R )  \/  ( 2  x.  C
)  ||  ( B  -  -u R ) ) )
46 jm2.27a24 . . . . . . 7  |-  ( ph  ->  Q  e.  ZZ )
47 zmulcl 9998 . . . . . . 7  |-  ( ( 2  e.  ZZ  /\  Q  e.  ZZ )  ->  ( 2  x.  Q
)  e.  ZZ )
482, 46, 47sylancr 647 . . . . . 6  |-  ( ph  ->  ( 2  x.  Q
)  e.  ZZ )
49 zsqcl 11105 . . . . . . . . . . . . . 14  |-  ( C  e.  ZZ  ->  ( C ^ 2 )  e.  ZZ )
504, 49syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  ( C ^ 2 )  e.  ZZ )
51 dvdsmul2 12478 . . . . . . . . . . . . 13  |-  ( ( 2  e.  ZZ  /\  ( C ^ 2 )  e.  ZZ )  -> 
( C ^ 2 )  ||  ( 2  x.  ( C ^
2 ) ) )
522, 50, 51sylancr 647 . . . . . . . . . . . 12  |-  ( ph  ->  ( C ^ 2 )  ||  ( 2  x.  ( C ^
2 ) ) )
53 jm2.27a10 . . . . . . . . . . . . . . 15  |-  ( ph  ->  J  e.  NN0 )
5453nn0zd 10047 . . . . . . . . . . . . . 14  |-  ( ph  ->  J  e.  ZZ )
5554peano2zd 10052 . . . . . . . . . . . . 13  |-  ( ph  ->  ( J  +  1 )  e.  ZZ )
56 zmulcl 9998 . . . . . . . . . . . . . 14  |-  ( ( 2  e.  ZZ  /\  ( C ^ 2 )  e.  ZZ )  -> 
( 2  x.  ( C ^ 2 ) )  e.  ZZ )
572, 50, 56sylancr 647 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 2  x.  ( C ^ 2 ) )  e.  ZZ )
58 dvdsmultr2 12491 . . . . . . . . . . . . 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 1187 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( C ^
2 )  ||  (
2  x.  ( C ^ 2 ) )  ->  ( C ^
2 )  ||  (
( J  +  1 )  x.  ( 2  x.  ( C ^
2 ) ) ) ) )
6052, 59mpd 16 . . . . . . . . . . 11  |-  ( ph  ->  ( C ^ 2 )  ||  ( ( J  +  1 )  x.  ( 2  x.  ( C ^ 2 ) ) ) )
611oveq1d 5772 . . . . . . . . . . 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 2290 . . . . . . . . . . 11  |-  ( ph  ->  ( ( J  + 
1 )  x.  (
2  x.  ( C ^ 2 ) ) )  =  ( A Yrm  Q ) )
6560, 61, 643brtr3d 3992 . . . . . . . . . 10  |-  ( ph  ->  ( ( A Yrm  P ) ^ 2 )  ||  ( A Yrm  Q ) )
66 jm2.27a1 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  ( ZZ>= ` 
2 ) )
6755zred 10049 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( J  +  1 )  e.  RR )
6857zred 10049 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( 2  x.  ( C ^ 2 ) )  e.  RR )
69 nn0p1nn 9935 . . . . . . . . . . . . . . . . . 18  |-  ( J  e.  NN0  ->  ( J  +  1 )  e.  NN )
7053, 69syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( J  +  1 )  e.  NN )
7170nngt0d 9722 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  0  <  ( J  +  1 ) )
72 2nn 9809 . . . . . . . . . . . . . . . . . 18  |-  2  e.  NN
733nnsqcld 11196 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( C ^ 2 )  e.  NN )
74 nnmulcl 9702 . . . . . . . . . . . . . . . . . 18  |-  ( ( 2  e.  NN  /\  ( C ^ 2 )  e.  NN )  -> 
( 2  x.  ( C ^ 2 ) )  e.  NN )
7572, 73, 74sylancr 647 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( 2  x.  ( C ^ 2 ) )  e.  NN )
7675nngt0d 9722 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  0  <  ( 2  x.  ( C ^
2 ) ) )
7767, 68, 71, 76mulgt0d 8904 . . . . . . . . . . . . . . 15  |-  ( ph  ->  0  <  ( ( J  +  1 )  x.  ( 2  x.  ( C ^ 2 ) ) ) )
7877, 62breqtrrd 3989 . . . . . . . . . . . . . 14  |-  ( ph  ->  0  <  E )
79 rmy0 26346 . . . . . . . . . . . . . . 15  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A Yrm  0 )  =  0 )
8066, 79syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( A Yrm  0 )  =  0 )
8163eqcomd 2261 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( A Yrm  Q )  =  E )
8278, 80, 813brtr4d 3993 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A Yrm  0 )  < 
( A Yrm  Q ) )
83 ltrmy 26371 . . . . . . . . . . . . . 14  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  0  e.  ZZ  /\  Q  e.  ZZ )  ->  (
0  <  Q  <->  ( A Yrm  0 )  <  ( A Yrm  Q ) ) )
8466, 25, 46, 83syl3anc 1187 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 0  <  Q  <->  ( A Yrm  0 )  <  ( A Yrm 
Q ) ) )
8582, 84mpbird 225 . . . . . . . . . . . 12  |-  ( ph  ->  0  <  Q )
86 elnnz 9966 . . . . . . . . . . . 12  |-  ( Q  e.  NN  <->  ( Q  e.  ZZ  /\  0  < 
Q ) )
8746, 85, 86sylanbrc 648 . . . . . . . . . . 11  |-  ( ph  ->  Q  e.  NN )
883nngt0d 9722 . . . . . . . . . . . . . 14  |-  ( ph  ->  0  <  C )
891eqcomd 2261 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( A Yrm  P )  =  C )
9088, 80, 893brtr4d 3993 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A Yrm  0 )  < 
( A Yrm  P ) )
91 ltrmy 26371 . . . . . . . . . . . . . 14  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  0  e.  ZZ  /\  P  e.  ZZ )  ->  (
0  <  P  <->  ( A Yrm  0 )  <  ( A Yrm  P ) ) )
9266, 25, 10, 91syl3anc 1187 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 0  <  P  <->  ( A Yrm  0 )  <  ( A Yrm 
P ) ) )
9390, 92mpbird 225 . . . . . . . . . . . 12  |-  ( ph  ->  0  <  P )
94 elnnz 9966 . . . . . . . . . . . 12  |-  ( P  e.  NN  <->  ( P  e.  ZZ  /\  0  < 
P ) )
9510, 93, 94sylanbrc 648 . . . . . . . . . . 11  |-  ( ph  ->  P  e.  NN )
96 jm2.20nn 26422 . . . . . . . . . . 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 1187 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( A Yrm  P ) ^ 2 ) 
||  ( A Yrm  Q )  <-> 
( P  x.  ( A Yrm 
P ) )  ||  Q ) )
9865, 97mpbid 203 . . . . . . . . 9  |-  ( ph  ->  ( P  x.  ( A Yrm 
P ) )  ||  Q )
991, 4eqeltrrd 2331 . . . . . . . . . 10  |-  ( ph  ->  ( A Yrm  P )  e.  ZZ )
100 muldvds2 12481 . . . . . . . . . 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 1187 . . . . . . . . 9  |-  ( ph  ->  ( ( P  x.  ( A Yrm  P ) ) 
||  Q  ->  ( A Yrm 
P )  ||  Q
) )
10298, 101mpd 16 . . . . . . . 8  |-  ( ph  ->  ( A Yrm  P )  ||  Q )
1031, 102eqbrtrd 3983 . . . . . . 7  |-  ( ph  ->  C  ||  Q )
1042a1i 12 . . . . . . . 8  |-  ( ph  ->  2  e.  ZZ )
105 dvdscmul 12482 . . . . . . . 8  |-  ( ( C  e.  ZZ  /\  Q  e.  ZZ  /\  2  e.  ZZ )  ->  ( C  ||  Q  ->  (
2  x.  C ) 
||  ( 2  x.  Q ) ) )
1064, 46, 104, 105syl3anc 1187 . . . . . . 7  |-  ( ph  ->  ( C  ||  Q  ->  ( 2  x.  C
)  ||  ( 2  x.  Q ) ) )
107103, 106mpd 16 . . . . . 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 10047 . . . . . . . . . 10  |-  ( ph  ->  F  e.  ZZ )
111108, 110eqeltrrd 2331 . . . . . . . . 9  |-  ( ph  ->  ( A Xrm  Q )  e.  ZZ )
112 frmy 26331 . . . . . . . . . . 11  |- Yrm  : (
( ZZ>= `  2 )  X.  ZZ ) --> ZZ
113112fovcl 5848 . . . . . . . . . 10  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  R  e.  ZZ )  ->  ( A Yrm 
R )  e.  ZZ )
11466, 9, 113syl2anc 645 . . . . . . . . 9  |-  ( ph  ->  ( A Yrm  R )  e.  ZZ )
11521, 12eqeltrrd 2331 . . . . . . . . 9  |-  ( ph  ->  ( G Yrm  R )  e.  ZZ )
116 eluzelz 10170 . . . . . . . . . . . . 13  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  ZZ )
11766, 116syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  ZZ )
118 jm2.27a16 . . . . . . . . . . . 12  |-  ( ph  ->  F  ||  ( G  -  A ) )
119 congsym 26387 . . . . . . . . . . . 12  |-  ( ( ( F  e.  ZZ  /\  G  e.  ZZ )  /\  ( A  e.  ZZ  /\  F  ||  ( G  -  A
) ) )  ->  F  ||  ( A  -  G ) )
120110, 36, 117, 118, 119syl22anc 1188 . . . . . . . . . . 11  |-  ( ph  ->  F  ||  ( A  -  G ) )
121108, 120eqbrtrrd 3985 . . . . . . . . . 10  |-  ( ph  ->  ( A Xrm  Q )  ||  ( A  -  G
) )
122 jm2.15nn0 26428 . . . . . . . . . . 11  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  G  e.  ( ZZ>= `  2 )  /\  R  e.  NN0 )  ->  ( A  -  G )  ||  (
( A Yrm  R )  -  ( G Yrm  R ) ) )
12366, 17, 30, 122syl3anc 1187 . . . . . . . . . 10  |-  ( ph  ->  ( A  -  G
)  ||  ( ( A Yrm 
R )  -  ( G Yrm 
R ) ) )
124117, 36zsubcld 10054 . . . . . . . . . . 11  |-  ( ph  ->  ( A  -  G
)  e.  ZZ )
125114, 115zsubcld 10054 . . . . . . . . . . 11  |-  ( ph  ->  ( ( A Yrm  R )  -  ( G Yrm  R ) )  e.  ZZ )
126 dvdstr 12489 . . . . . . . . . . 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 1187 . . . . . . . . . 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 663 . . . . . . . . 9  |-  ( ph  ->  ( A Xrm  Q )  ||  ( ( A Yrm  R )  -  ( G Yrm  R ) ) )
129 jm2.27a18 . . . . . . . . . 10  |-  ( ph  ->  F  ||  ( H  -  C ) )
13021, 1oveq12d 5775 . . . . . . . . . 10  |-  ( ph  ->  ( H  -  C
)  =  ( ( G Yrm  R )  -  ( A Yrm 
P ) ) )
131129, 108, 1303brtr3d 3992 . . . . . . . . 9  |-  ( ph  ->  ( A Xrm  Q )  ||  ( ( G Yrm  R )  -  ( A Yrm  P ) ) )
132 congtr 26384 . . . . . . . . 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 1203 . . . . . . . 8  |-  ( ph  ->  ( A Xrm  Q )  ||  ( ( A Yrm  R )  -  ( A Yrm  P ) ) )
134133orcd 383 . . . . . . 7  |-  ( ph  ->  ( ( A Xrm  Q ) 
||  ( ( A Yrm  R )  -  ( A Yrm  P ) )  \/  ( A Xrm 
Q )  ||  (
( A Yrm  R )  -  -u ( A Yrm  P ) ) ) )
135 jm2.26 26427 . . . . . . . 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 1188 . . . . . . 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 203 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  Q )  ||  ( R  -  P )  \/  ( 2  x.  Q
)  ||  ( R  -  -u P ) ) )
138 dvdsacongtr 26403 . . . . . 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 1203 . . . . 5  |-  ( ph  ->  ( ( 2  x.  C )  ||  ( R  -  P )  \/  ( 2  x.  C
)  ||  ( R  -  -u P ) ) )
140 acongtr 26397 . . . . 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 1203 . . . 4  |-  ( ph  ->  ( ( 2  x.  C )  ||  ( B  -  P )  \/  ( 2  x.  C
)  ||  ( B  -  -u P ) ) )
1427nnnn0d 9950 . . . . . 6  |-  ( ph  ->  B  e.  NN0 )
1433nnnn0d 9950 . . . . . 6  |-  ( ph  ->  C  e.  NN0 )
144 jm2.27a20 . . . . . 6  |-  ( ph  ->  B  <_  C )
145 elfz2nn0 10752 . . . . . 6  |-  ( B  e.  ( 0 ... C )  <->  ( B  e.  NN0  /\  C  e. 
NN0  /\  B  <_  C ) )
146142, 143, 144, 145syl3anbrc 1141 . . . . 5  |-  ( ph  ->  B  e.  ( 0 ... C ) )
14795nnnn0d 9950 . . . . . 6  |-  ( ph  ->  P  e.  NN0 )
148 rmygeid 26383 . . . . . . . 8  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  P  e.  NN0 )  ->  P  <_  ( A Yrm  P ) )
14966, 147, 148syl2anc 645 . . . . . . 7  |-  ( ph  ->  P  <_  ( A Yrm  P
) )
150149, 1breqtrrd 3989 . . . . . 6  |-  ( ph  ->  P  <_  C )
151 elfz2nn0 10752 . . . . . 6  |-  ( P  e.  ( 0 ... C )  <->  ( P  e.  NN0  /\  C  e. 
NN0  /\  P  <_  C ) )
152147, 143, 150, 151syl3anbrc 1141 . . . . 5  |-  ( ph  ->  P  e.  ( 0 ... C ) )
153 acongeq 26402 . . . . 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 1187 . . . 4  |-  ( ph  ->  ( B  =  P  <-> 
( ( 2  x.  C )  ||  ( B  -  P )  \/  ( 2  x.  C
)  ||  ( B  -  -u P ) ) ) )
155141, 154mpbird 225 . . 3  |-  ( ph  ->  B  =  P )
156155oveq2d 5773 . 2  |-  ( ph  ->  ( A Yrm  B )  =  ( A Yrm  P ) )
1571, 156eqtr4d 2291 1  |-  ( ph  ->  C  =  ( A Yrm  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1619    e. wcel 1621   class class class wbr 3963   ` cfv 4638  (class class class)co 5757   0cc0 8670   1c1 8671    + caddc 8673    x. cmul 8675    < clt 8800    <_ cle 8801    - cmin 8970   -ucneg 8971   NNcn 9679   2c2 9728   NN0cn0 9897   ZZcz 9956   ZZ>=cuz 10162   ...cfz 10713   ^cexp 11035    || cdivides 12458   Xrm crmx 26317   Yrm crmy 26318
This theorem is referenced by:  jm2.27b  26431
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748  ax-addf 8749  ax-mulf 8750
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-of 5977  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-2o 6413  df-oadd 6416  df-omul 6417  df-er 6593  df-map 6707  df-pm 6708  df-ixp 6751  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-fi 7098  df-sup 7127  df-oi 7158  df-card 7505  df-acn 7508  df-cda 7727  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-3 9738  df-4 9739  df-5 9740  df-6 9741  df-7 9742  df-8 9743  df-9 9744  df-10 9745  df-n0 9898  df-z 9957  df-dec 10057  df-uz 10163  df-q 10249  df-rp 10287  df-xneg 10384  df-xadd 10385  df-xmul 10386  df-ioo 10591  df-ioc 10592  df-ico 10593  df-icc 10594  df-fz 10714  df-fzo 10802  df-fl 10856  df-mod 10905  df-seq 10978  df-exp 11036  df-fac 11220  df-bc 11247  df-hash 11269  df-shft 11492  df-cj 11514  df-re 11515  df-im 11516  df-sqr 11650  df-abs 11651  df-limsup 11875  df-clim 11892  df-rlim 11893  df-sum 12089  df-ef 12276  df-sin 12278  df-cos 12279  df-pi 12281  df-divides 12459  df-gcd 12613  df-prime 12686  df-numer 12733  df-denom 12734  df-struct 13077  df-ndx 13078  df-slot 13079  df-base 13080  df-sets 13081  df-ress 13082  df-plusg 13148  df-mulr 13149  df-starv 13150  df-sca 13151  df-vsca 13152  df-tset 13154  df-ple 13155  df-ds 13157  df-hom 13159  df-cco 13160  df-rest 13254  df-topn 13255  df-topgen 13271  df-pt 13272  df-prds 13275  df-xrs 13330  df-0g 13331  df-gsum 13332  df-qtop 13337  df-imas 13338  df-xps 13340  df-mre 13415  df-mrc 13416  df-acs 13418  df-mnd 14294  df-submnd 14343  df-mulg 14419  df-cntz 14720  df-cmn 15018  df-xmet 16300  df-met 16301  df-bl 16302  df-mopn 16303  df-cnfld 16305  df-top 16563  df-bases 16565  df-topon 16566  df-topsp 16567  df-cld 16683  df-ntr 16684  df-cls 16685  df-nei 16762  df-lp 16795  df-perf 16796  df-cn 16884  df-cnp 16885  df-haus 16970  df-tx 17184  df-hmeo 17373  df-fbas 17447  df-fg 17448  df-fil 17468  df-fm 17560  df-flim 17561  df-flf 17562  df-xms 17812  df-ms 17813  df-tms 17814  df-cncf 18309  df-limc 19143  df-dv 19144  df-log 19841  df-squarenn 26258  df-pell1qr 26259  df-pell14qr 26260  df-pell1234qr 26261  df-pellfund 26262  df-rmx 26319  df-rmy 26320
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