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Theorem jm2.27a 29501
Description: Lemma for jm2.27 29504. 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 10788 . . . . . 6  |-  2  e.  ZZ
3 jm2.27a3 . . . . . . 7  |-  ( ph  ->  C  e.  NN )
43nnzd 10856 . . . . . 6  |-  ( ph  ->  C  e.  ZZ )
5 zmulcl 10803 . . . . . 6  |-  ( ( 2  e.  ZZ  /\  C  e.  ZZ )  ->  ( 2  x.  C
)  e.  ZZ )
62, 4, 5sylancr 663 . . . . 5  |-  ( ph  ->  ( 2  x.  C
)  e.  ZZ )
7 jm2.27a2 . . . . . 6  |-  ( ph  ->  B  e.  NN )
87nnzd 10856 . . . . 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 10855 . . . . . . 7  |-  ( ph  ->  H  e.  ZZ )
13 jm2.27a19 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  C
)  ||  ( H  -  B ) )
14 congsym 29458 . . . . . . . 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 1220 . . . . . . 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 10749 . . . . . . . . . . . . 13  |-  ( ph  ->  0  <_  H )
19 rmy0 29417 . . . . . . . . . . . . . 14  |-  ( G  e.  ( ZZ>= `  2
)  ->  ( G Yrm  0 )  =  0 )
2017, 19syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  ( G Yrm  0 )  =  0 )
21 jm2.27a29 . . . . . . . . . . . . . 14  |-  ( ph  ->  H  =  ( G Yrm  R ) )
2221eqcomd 2462 . . . . . . . . . . . . 13  |-  ( ph  ->  ( G Yrm  R )  =  H )
2318, 20, 223brtr4d 4429 . . . . . . . . . . . 12  |-  ( ph  ->  ( G Yrm  0 )  <_ 
( G Yrm  R ) )
24 0zd 10768 . . . . . . . . . . . . 13  |-  ( ph  ->  0  e.  ZZ )
25 lermy 29445 . . . . . . . . . . . . 13  |-  ( ( G  e.  ( ZZ>= ` 
2 )  /\  0  e.  ZZ  /\  R  e.  ZZ )  ->  (
0  <_  R  <->  ( G Yrm  0 )  <_  ( G Yrm  R
) ) )
2617, 24, 9, 25syl3anc 1219 . . . . . . . . . . . 12  |-  ( ph  ->  ( 0  <_  R  <->  ( G Yrm  0 )  <_  ( G Yrm 
R ) ) )
2723, 26mpbird 232 . . . . . . . . . . 11  |-  ( ph  ->  0  <_  R )
28 elnn0z 10769 . . . . . . . . . . 11  |-  ( R  e.  NN0  <->  ( R  e.  ZZ  /\  0  <_  R ) )
299, 27, 28sylanbrc 664 . . . . . . . . . 10  |-  ( ph  ->  R  e.  NN0 )
30 jm2.16nn0 29500 . . . . . . . . . 10  |-  ( ( G  e.  ( ZZ>= ` 
2 )  /\  R  e.  NN0 )  ->  ( G  -  1 ) 
||  ( ( G Yrm  R )  -  R ) )
3117, 29, 30syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( G  -  1 )  ||  ( ( G Yrm  R )  -  R
) )
3221oveq1d 6214 . . . . . . . . 9  |-  ( ph  ->  ( H  -  R
)  =  ( ( G Yrm  R )  -  R
) )
3331, 32breqtrrd 4425 . . . . . . . 8  |-  ( ph  ->  ( G  -  1 )  ||  ( H  -  R ) )
34 jm2.27a7 . . . . . . . . . . 11  |-  ( ph  ->  G  e.  NN0 )
3534nn0zd 10855 . . . . . . . . . 10  |-  ( ph  ->  G  e.  ZZ )
36 peano2zm 10798 . . . . . . . . . 10  |-  ( G  e.  ZZ  ->  ( G  -  1 )  e.  ZZ )
3735, 36syl 16 . . . . . . . . 9  |-  ( ph  ->  ( G  -  1 )  e.  ZZ )
3812, 9zsubcld 10862 . . . . . . . . 9  |-  ( ph  ->  ( H  -  R
)  e.  ZZ )
39 dvdstr 13683 . . . . . . . . 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 ) ) )
406, 37, 38, 39syl3anc 1219 . . . . . . . 8  |-  ( ph  ->  ( ( ( 2  x.  C )  ||  ( G  -  1
)  /\  ( G  -  1 )  ||  ( H  -  R
) )  ->  (
2  x.  C ) 
||  ( H  -  R ) ) )
4116, 33, 40mp2and 679 . . . . . . 7  |-  ( ph  ->  ( 2  x.  C
)  ||  ( H  -  R ) )
42 congtr 29455 . . . . . . 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 )
)
436, 8, 12, 9, 15, 41, 42syl222anc 1235 . . . . . 6  |-  ( ph  ->  ( 2  x.  C
)  ||  ( B  -  R ) )
4443orcd 392 . . . . 5  |-  ( ph  ->  ( ( 2  x.  C )  ||  ( B  -  R )  \/  ( 2  x.  C
)  ||  ( B  -  -u R ) ) )
45 jm2.27a24 . . . . . . 7  |-  ( ph  ->  Q  e.  ZZ )
46 zmulcl 10803 . . . . . . 7  |-  ( ( 2  e.  ZZ  /\  Q  e.  ZZ )  ->  ( 2  x.  Q
)  e.  ZZ )
472, 45, 46sylancr 663 . . . . . 6  |-  ( ph  ->  ( 2  x.  Q
)  e.  ZZ )
48 zsqcl 12052 . . . . . . . . . . . . . 14  |-  ( C  e.  ZZ  ->  ( C ^ 2 )  e.  ZZ )
494, 48syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  ( C ^ 2 )  e.  ZZ )
50 dvdsmul2 13672 . . . . . . . . . . . . 13  |-  ( ( 2  e.  ZZ  /\  ( C ^ 2 )  e.  ZZ )  -> 
( C ^ 2 )  ||  ( 2  x.  ( C ^
2 ) ) )
512, 49, 50sylancr 663 . . . . . . . . . . . 12  |-  ( ph  ->  ( C ^ 2 )  ||  ( 2  x.  ( C ^
2 ) ) )
52 jm2.27a10 . . . . . . . . . . . . . . 15  |-  ( ph  ->  J  e.  NN0 )
5352nn0zd 10855 . . . . . . . . . . . . . 14  |-  ( ph  ->  J  e.  ZZ )
5453peano2zd 10860 . . . . . . . . . . . . 13  |-  ( ph  ->  ( J  +  1 )  e.  ZZ )
55 zmulcl 10803 . . . . . . . . . . . . . 14  |-  ( ( 2  e.  ZZ  /\  ( C ^ 2 )  e.  ZZ )  -> 
( 2  x.  ( C ^ 2 ) )  e.  ZZ )
562, 49, 55sylancr 663 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 2  x.  ( C ^ 2 ) )  e.  ZZ )
57 dvdsmultr2 13685 . . . . . . . . . . . . 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 ) ) ) ) )
5849, 54, 56, 57syl3anc 1219 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( C ^
2 )  ||  (
2  x.  ( C ^ 2 ) )  ->  ( C ^
2 )  ||  (
( J  +  1 )  x.  ( 2  x.  ( C ^
2 ) ) ) ) )
5951, 58mpd 15 . . . . . . . . . . 11  |-  ( ph  ->  ( C ^ 2 )  ||  ( ( J  +  1 )  x.  ( 2  x.  ( C ^ 2 ) ) ) )
601oveq1d 6214 . . . . . . . . . . 11  |-  ( ph  ->  ( C ^ 2 )  =  ( ( A Yrm  P ) ^ 2 ) )
61 jm2.27a15 . . . . . . . . . . . 12  |-  ( ph  ->  E  =  ( ( J  +  1 )  x.  ( 2  x.  ( C ^ 2 ) ) ) )
62 jm2.27a26 . . . . . . . . . . . 12  |-  ( ph  ->  E  =  ( A Yrm  Q ) )
6361, 62eqtr3d 2497 . . . . . . . . . . 11  |-  ( ph  ->  ( ( J  + 
1 )  x.  (
2  x.  ( C ^ 2 ) ) )  =  ( A Yrm  Q ) )
6459, 60, 633brtr3d 4428 . . . . . . . . . 10  |-  ( ph  ->  ( ( A Yrm  P ) ^ 2 )  ||  ( A Yrm  Q ) )
65 jm2.27a1 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  ( ZZ>= ` 
2 ) )
6654zred 10857 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( J  +  1 )  e.  RR )
6756zred 10857 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( 2  x.  ( C ^ 2 ) )  e.  RR )
68 nn0p1nn 10729 . . . . . . . . . . . . . . . . . 18  |-  ( J  e.  NN0  ->  ( J  +  1 )  e.  NN )
6952, 68syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( J  +  1 )  e.  NN )
7069nngt0d 10475 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  0  <  ( J  +  1 ) )
71 2nn 10589 . . . . . . . . . . . . . . . . . 18  |-  2  e.  NN
723nnsqcld 12144 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( C ^ 2 )  e.  NN )
73 nnmulcl 10455 . . . . . . . . . . . . . . . . . 18  |-  ( ( 2  e.  NN  /\  ( C ^ 2 )  e.  NN )  -> 
( 2  x.  ( C ^ 2 ) )  e.  NN )
7471, 72, 73sylancr 663 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( 2  x.  ( C ^ 2 ) )  e.  NN )
7574nngt0d 10475 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  0  <  ( 2  x.  ( C ^
2 ) ) )
7666, 67, 70, 75mulgt0d 9636 . . . . . . . . . . . . . . 15  |-  ( ph  ->  0  <  ( ( J  +  1 )  x.  ( 2  x.  ( C ^ 2 ) ) ) )
7776, 61breqtrrd 4425 . . . . . . . . . . . . . 14  |-  ( ph  ->  0  <  E )
78 rmy0 29417 . . . . . . . . . . . . . . 15  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A Yrm  0 )  =  0 )
7965, 78syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( A Yrm  0 )  =  0 )
8062eqcomd 2462 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( A Yrm  Q )  =  E )
8177, 79, 803brtr4d 4429 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A Yrm  0 )  < 
( A Yrm  Q ) )
82 ltrmy 29442 . . . . . . . . . . . . . 14  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  0  e.  ZZ  /\  Q  e.  ZZ )  ->  (
0  <  Q  <->  ( A Yrm  0 )  <  ( A Yrm  Q ) ) )
8365, 24, 45, 82syl3anc 1219 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 0  <  Q  <->  ( A Yrm  0 )  <  ( A Yrm 
Q ) ) )
8481, 83mpbird 232 . . . . . . . . . . . 12  |-  ( ph  ->  0  <  Q )
85 elnnz 10766 . . . . . . . . . . . 12  |-  ( Q  e.  NN  <->  ( Q  e.  ZZ  /\  0  < 
Q ) )
8645, 84, 85sylanbrc 664 . . . . . . . . . . 11  |-  ( ph  ->  Q  e.  NN )
873nngt0d 10475 . . . . . . . . . . . . . 14  |-  ( ph  ->  0  <  C )
881eqcomd 2462 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( A Yrm  P )  =  C )
8987, 79, 883brtr4d 4429 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A Yrm  0 )  < 
( A Yrm  P ) )
90 ltrmy 29442 . . . . . . . . . . . . . 14  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  0  e.  ZZ  /\  P  e.  ZZ )  ->  (
0  <  P  <->  ( A Yrm  0 )  <  ( A Yrm  P ) ) )
9165, 24, 10, 90syl3anc 1219 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 0  <  P  <->  ( A Yrm  0 )  <  ( A Yrm 
P ) ) )
9289, 91mpbird 232 . . . . . . . . . . . 12  |-  ( ph  ->  0  <  P )
93 elnnz 10766 . . . . . . . . . . . 12  |-  ( P  e.  NN  <->  ( P  e.  ZZ  /\  0  < 
P ) )
9410, 92, 93sylanbrc 664 . . . . . . . . . . 11  |-  ( ph  ->  P  e.  NN )
95 jm2.20nn 29493 . . . . . . . . . . 11  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  Q  e.  NN  /\  P  e.  NN )  ->  (
( ( A Yrm  P ) ^ 2 )  ||  ( A Yrm  Q )  <->  ( P  x.  ( A Yrm  P ) ) 
||  Q ) )
9665, 86, 94, 95syl3anc 1219 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( A Yrm  P ) ^ 2 ) 
||  ( A Yrm  Q )  <-> 
( P  x.  ( A Yrm 
P ) )  ||  Q ) )
9764, 96mpbid 210 . . . . . . . . 9  |-  ( ph  ->  ( P  x.  ( A Yrm 
P ) )  ||  Q )
981, 4eqeltrrd 2543 . . . . . . . . . 10  |-  ( ph  ->  ( A Yrm  P )  e.  ZZ )
99 muldvds2 13675 . . . . . . . . . 10  |-  ( ( P  e.  ZZ  /\  ( A Yrm  P )  e.  ZZ  /\  Q  e.  ZZ )  ->  (
( P  x.  ( A Yrm 
P ) )  ||  Q  ->  ( A Yrm  P ) 
||  Q ) )
10010, 98, 45, 99syl3anc 1219 . . . . . . . . 9  |-  ( ph  ->  ( ( P  x.  ( A Yrm  P ) ) 
||  Q  ->  ( A Yrm 
P )  ||  Q
) )
10197, 100mpd 15 . . . . . . . 8  |-  ( ph  ->  ( A Yrm  P )  ||  Q )
1021, 101eqbrtrd 4419 . . . . . . 7  |-  ( ph  ->  C  ||  Q )
1032a1i 11 . . . . . . . 8  |-  ( ph  ->  2  e.  ZZ )
104 dvdscmul 13676 . . . . . . . 8  |-  ( ( C  e.  ZZ  /\  Q  e.  ZZ  /\  2  e.  ZZ )  ->  ( C  ||  Q  ->  (
2  x.  C ) 
||  ( 2  x.  Q ) ) )
1054, 45, 103, 104syl3anc 1219 . . . . . . 7  |-  ( ph  ->  ( C  ||  Q  ->  ( 2  x.  C
)  ||  ( 2  x.  Q ) ) )
106102, 105mpd 15 . . . . . 6  |-  ( ph  ->  ( 2  x.  C
)  ||  ( 2  x.  Q ) )
107 jm2.27a25 . . . . . . . . . 10  |-  ( ph  ->  F  =  ( A Xrm  Q ) )
108 jm2.27a6 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  NN0 )
109108nn0zd 10855 . . . . . . . . . 10  |-  ( ph  ->  F  e.  ZZ )
110107, 109eqeltrrd 2543 . . . . . . . . 9  |-  ( ph  ->  ( A Xrm  Q )  e.  ZZ )
111 frmy 29402 . . . . . . . . . . 11  |- Yrm  : (
( ZZ>= `  2 )  X.  ZZ ) --> ZZ
112111fovcl 6304 . . . . . . . . . 10  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  R  e.  ZZ )  ->  ( A Yrm 
R )  e.  ZZ )
11365, 9, 112syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( A Yrm  R )  e.  ZZ )
11421, 12eqeltrrd 2543 . . . . . . . . 9  |-  ( ph  ->  ( G Yrm  R )  e.  ZZ )
115 eluzelz 10980 . . . . . . . . . . . . 13  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  ZZ )
11665, 115syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  ZZ )
117 jm2.27a16 . . . . . . . . . . . 12  |-  ( ph  ->  F  ||  ( G  -  A ) )
118 congsym 29458 . . . . . . . . . . . 12  |-  ( ( ( F  e.  ZZ  /\  G  e.  ZZ )  /\  ( A  e.  ZZ  /\  F  ||  ( G  -  A
) ) )  ->  F  ||  ( A  -  G ) )
119109, 35, 116, 117, 118syl22anc 1220 . . . . . . . . . . 11  |-  ( ph  ->  F  ||  ( A  -  G ) )
120107, 119eqbrtrrd 4421 . . . . . . . . . 10  |-  ( ph  ->  ( A Xrm  Q )  ||  ( A  -  G
) )
121 jm2.15nn0 29499 . . . . . . . . . . 11  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  G  e.  ( ZZ>= `  2 )  /\  R  e.  NN0 )  ->  ( A  -  G )  ||  (
( A Yrm  R )  -  ( G Yrm  R ) ) )
12265, 17, 29, 121syl3anc 1219 . . . . . . . . . 10  |-  ( ph  ->  ( A  -  G
)  ||  ( ( A Yrm 
R )  -  ( G Yrm 
R ) ) )
123116, 35zsubcld 10862 . . . . . . . . . . 11  |-  ( ph  ->  ( A  -  G
)  e.  ZZ )
124113, 114zsubcld 10862 . . . . . . . . . . 11  |-  ( ph  ->  ( ( A Yrm  R )  -  ( G Yrm  R ) )  e.  ZZ )
125 dvdstr 13683 . . . . . . . . . . 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 ) ) ) )
126110, 123, 124, 125syl3anc 1219 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( A Xrm  Q )  ||  ( A  -  G )  /\  ( A  -  G
)  ||  ( ( A Yrm 
R )  -  ( G Yrm 
R ) ) )  ->  ( A Xrm  Q ) 
||  ( ( A Yrm  R )  -  ( G Yrm  R ) ) ) )
127120, 122, 126mp2and 679 . . . . . . . . 9  |-  ( ph  ->  ( A Xrm  Q )  ||  ( ( A Yrm  R )  -  ( G Yrm  R ) ) )
128 jm2.27a18 . . . . . . . . . 10  |-  ( ph  ->  F  ||  ( H  -  C ) )
12921, 1oveq12d 6217 . . . . . . . . . 10  |-  ( ph  ->  ( H  -  C
)  =  ( ( G Yrm  R )  -  ( A Yrm 
P ) ) )
130128, 107, 1293brtr3d 4428 . . . . . . . . 9  |-  ( ph  ->  ( A Xrm  Q )  ||  ( ( G Yrm  R )  -  ( A Yrm  P ) ) )
131 congtr 29455 . . . . . . . . 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 ) ) )
132110, 113, 114, 98, 127, 130, 131syl222anc 1235 . . . . . . . 8  |-  ( ph  ->  ( A Xrm  Q )  ||  ( ( A Yrm  R )  -  ( A Yrm  P ) ) )
133132orcd 392 . . . . . . 7  |-  ( ph  ->  ( ( A Xrm  Q ) 
||  ( ( A Yrm  R )  -  ( A Yrm  P ) )  \/  ( A Xrm 
Q )  ||  (
( A Yrm  R )  -  -u ( A Yrm  P ) ) ) )
134 jm2.26 29498 . . . . . . . 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 ) ) ) )
13565, 86, 9, 10, 134syl22anc 1220 . . . . . . 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 ) ) ) )
136133, 135mpbid 210 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  Q )  ||  ( R  -  P )  \/  ( 2  x.  Q
)  ||  ( R  -  -u P ) ) )
137 dvdsacongtr 29474 . . . . . 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 ) ) )
13847, 9, 10, 6, 106, 136, 137syl222anc 1235 . . . . 5  |-  ( ph  ->  ( ( 2  x.  C )  ||  ( R  -  P )  \/  ( 2  x.  C
)  ||  ( R  -  -u P ) ) )
139 acongtr 29468 . . . . 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
) ) )
1406, 8, 9, 10, 44, 138, 139syl222anc 1235 . . . 4  |-  ( ph  ->  ( ( 2  x.  C )  ||  ( B  -  P )  \/  ( 2  x.  C
)  ||  ( B  -  -u P ) ) )
1417nnnn0d 10746 . . . . . 6  |-  ( ph  ->  B  e.  NN0 )
1423nnnn0d 10746 . . . . . 6  |-  ( ph  ->  C  e.  NN0 )
143 jm2.27a20 . . . . . 6  |-  ( ph  ->  B  <_  C )
144 elfz2nn0 11596 . . . . . 6  |-  ( B  e.  ( 0 ... C )  <->  ( B  e.  NN0  /\  C  e. 
NN0  /\  B  <_  C ) )
145141, 142, 143, 144syl3anbrc 1172 . . . . 5  |-  ( ph  ->  B  e.  ( 0 ... C ) )
14694nnnn0d 10746 . . . . . 6  |-  ( ph  ->  P  e.  NN0 )
147 rmygeid 29454 . . . . . . . 8  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  P  e.  NN0 )  ->  P  <_  ( A Yrm  P ) )
14865, 146, 147syl2anc 661 . . . . . . 7  |-  ( ph  ->  P  <_  ( A Yrm  P
) )
149148, 1breqtrrd 4425 . . . . . 6  |-  ( ph  ->  P  <_  C )
150 elfz2nn0 11596 . . . . . 6  |-  ( P  e.  ( 0 ... C )  <->  ( P  e.  NN0  /\  C  e. 
NN0  /\  P  <_  C ) )
151146, 142, 149, 150syl3anbrc 1172 . . . . 5  |-  ( ph  ->  P  e.  ( 0 ... C ) )
152 acongeq 29473 . . . . 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
) ) ) )
1533, 145, 151, 152syl3anc 1219 . . . 4  |-  ( ph  ->  ( B  =  P  <-> 
( ( 2  x.  C )  ||  ( B  -  P )  \/  ( 2  x.  C
)  ||  ( B  -  -u P ) ) ) )
154140, 153mpbird 232 . . 3  |-  ( ph  ->  B  =  P )
155154oveq2d 6215 . 2  |-  ( ph  ->  ( A Yrm  B )  =  ( A Yrm  P ) )
1561, 155eqtr4d 2498 1  |-  ( ph  ->  C  =  ( A Yrm  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758   class class class wbr 4399   ` cfv 5525  (class class class)co 6199   0cc0 9392   1c1 9393    + caddc 9395    x. cmul 9397    < clt 9528    <_ cle 9529    - cmin 9705   -ucneg 9706   NNcn 10432   2c2 10481   NN0cn0 10689   ZZcz 10756   ZZ>=cuz 10971   ...cfz 11553   ^cexp 11981    || cdivides 13652   Xrm crmx 29388   Yrm crmy 29389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-inf2 7957  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470  ax-addf 9471  ax-mulf 9472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-of 6429  df-om 6586  df-1st 6686  df-2nd 6687  df-supp 6800  df-recs 6941  df-rdg 6975  df-1o 7029  df-2o 7030  df-oadd 7033  df-omul 7034  df-er 7210  df-map 7325  df-pm 7326  df-ixp 7373  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-fsupp 7731  df-fi 7771  df-sup 7801  df-oi 7834  df-card 8219  df-acn 8222  df-cda 8447  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-4 10492  df-5 10493  df-6 10494  df-7 10495  df-8 10496  df-9 10497  df-10 10498  df-n0 10690  df-z 10757  df-dec 10866  df-uz 10972  df-q 11064  df-rp 11102  df-xneg 11199  df-xadd 11200  df-xmul 11201  df-ioo 11414  df-ioc 11415  df-ico 11416  df-icc 11417  df-fz 11554  df-fzo 11665  df-fl 11758  df-mod 11825  df-seq 11923  df-exp 11982  df-fac 12168  df-bc 12195  df-hash 12220  df-shft 12673  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-limsup 13066  df-clim 13083  df-rlim 13084  df-sum 13281  df-ef 13470  df-sin 13472  df-cos 13473  df-pi 13475  df-dvds 13653  df-gcd 13808  df-prm 13881  df-numer 13930  df-denom 13931  df-struct 14293  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-plusg 14369  df-mulr 14370  df-starv 14371  df-sca 14372  df-vsca 14373  df-ip 14374  df-tset 14375  df-ple 14376  df-ds 14378  df-unif 14379  df-hom 14380  df-cco 14381  df-rest 14479  df-topn 14480  df-0g 14498  df-gsum 14499  df-topgen 14500  df-pt 14501  df-prds 14504  df-xrs 14558  df-qtop 14563  df-imas 14564  df-xps 14566  df-mre 14642  df-mrc 14643  df-acs 14645  df-mnd 15533  df-submnd 15583  df-mulg 15666  df-cntz 15953  df-cmn 16399  df-psmet 17933  df-xmet 17934  df-met 17935  df-bl 17936  df-mopn 17937  df-fbas 17938  df-fg 17939  df-cnfld 17943  df-top 18634  df-bases 18636  df-topon 18637  df-topsp 18638  df-cld 18754  df-ntr 18755  df-cls 18756  df-nei 18833  df-lp 18871  df-perf 18872  df-cn 18962  df-cnp 18963  df-haus 19050  df-tx 19266  df-hmeo 19459  df-fil 19550  df-fm 19642  df-flim 19643  df-flf 19644  df-xms 20026  df-ms 20027  df-tms 20028  df-cncf 20585  df-limc 21473  df-dv 21474  df-log 22140  df-squarenn 29329  df-pell1qr 29330  df-pell14qr 29331  df-pell1234qr 29332  df-pellfund 29333  df-rmx 29390  df-rmy 29391
This theorem is referenced by:  jm2.27b  29502
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