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Theorem jm2.27b 35274
Description: Lemma for jm2.27 35276. Expand existential quantifiers for reverse direction. (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 )
Assertion
Ref Expression
jm2.27b  |-  ( ph  ->  C  =  ( A Yrm  B ) )

Proof of Theorem jm2.27b
Dummy variables  p  q  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 jm2.27a11 . . 3  |-  ( ph  ->  ( ( D ^
2 )  -  (
( ( A ^
2 )  -  1 )  x.  ( C ^ 2 ) ) )  =  1 )
2 jm2.27a1 . . . 4  |-  ( ph  ->  A  e.  ( ZZ>= ` 
2 ) )
3 jm2.27a4 . . . 4  |-  ( ph  ->  D  e.  NN0 )
4 jm2.27a3 . . . . 5  |-  ( ph  ->  C  e.  NN )
54nnzd 10925 . . . 4  |-  ( ph  ->  C  e.  ZZ )
6 rmxycomplete 35178 . . . 4  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  D  e.  NN0  /\  C  e.  ZZ )  ->  (
( ( D ^
2 )  -  (
( ( A ^
2 )  -  1 )  x.  ( C ^ 2 ) ) )  =  1  <->  E. p  e.  ZZ  ( D  =  ( A Xrm  p
)  /\  C  =  ( A Yrm  p ) ) ) )
72, 3, 5, 6syl3anc 1228 . . 3  |-  ( ph  ->  ( ( ( D ^ 2 )  -  ( ( ( A ^ 2 )  - 
1 )  x.  ( C ^ 2 ) ) )  =  1  <->  E. p  e.  ZZ  ( D  =  ( A Xrm  p
)  /\  C  =  ( A Yrm  p ) ) ) )
81, 7mpbid 210 . 2  |-  ( ph  ->  E. p  e.  ZZ  ( D  =  ( A Xrm 
p )  /\  C  =  ( A Yrm  p ) ) )
9 jm2.27a12 . . . . 5  |-  ( ph  ->  ( ( F ^
2 )  -  (
( ( A ^
2 )  -  1 )  x.  ( E ^ 2 ) ) )  =  1 )
109adantr 463 . . . 4  |-  ( (
ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  ->  ( ( F ^ 2 )  -  ( ( ( A ^ 2 )  - 
1 )  x.  ( E ^ 2 ) ) )  =  1 )
112adantr 463 . . . . 5  |-  ( (
ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  ->  A  e.  ( ZZ>= `  2 )
)
12 jm2.27a6 . . . . . 6  |-  ( ph  ->  F  e.  NN0 )
1312adantr 463 . . . . 5  |-  ( (
ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  ->  F  e.  NN0 )
14 jm2.27a5 . . . . . . 7  |-  ( ph  ->  E  e.  NN0 )
1514nn0zd 10924 . . . . . 6  |-  ( ph  ->  E  e.  ZZ )
1615adantr 463 . . . . 5  |-  ( (
ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  ->  E  e.  ZZ )
17 rmxycomplete 35178 . . . . 5  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  F  e.  NN0  /\  E  e.  ZZ )  ->  (
( ( F ^
2 )  -  (
( ( A ^
2 )  -  1 )  x.  ( E ^ 2 ) ) )  =  1  <->  E. q  e.  ZZ  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )
1811, 13, 16, 17syl3anc 1228 . . . 4  |-  ( (
ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  ->  ( (
( F ^ 2 )  -  ( ( ( A ^ 2 )  -  1 )  x.  ( E ^
2 ) ) )  =  1  <->  E. q  e.  ZZ  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )
1910, 18mpbid 210 . . 3  |-  ( (
ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  ->  E. q  e.  ZZ  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) )
20 jm2.27a14 . . . . . 6  |-  ( ph  ->  ( ( I ^
2 )  -  (
( ( G ^
2 )  -  1 )  x.  ( H ^ 2 ) ) )  =  1 )
2120ad2antrr 724 . . . . 5  |-  ( ( ( ph  /\  (
p  e.  ZZ  /\  ( D  =  ( A Xrm 
p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  -> 
( ( I ^
2 )  -  (
( ( G ^
2 )  -  1 )  x.  ( H ^ 2 ) ) )  =  1 )
22 jm2.27a13 . . . . . . 7  |-  ( ph  ->  G  e.  ( ZZ>= ` 
2 ) )
2322ad2antrr 724 . . . . . 6  |-  ( ( ( ph  /\  (
p  e.  ZZ  /\  ( D  =  ( A Xrm 
p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  ->  G  e.  ( ZZ>= ` 
2 ) )
24 jm2.27a9 . . . . . . 7  |-  ( ph  ->  I  e.  NN0 )
2524ad2antrr 724 . . . . . 6  |-  ( ( ( ph  /\  (
p  e.  ZZ  /\  ( D  =  ( A Xrm 
p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  ->  I  e.  NN0 )
26 jm2.27a8 . . . . . . . 8  |-  ( ph  ->  H  e.  NN0 )
2726nn0zd 10924 . . . . . . 7  |-  ( ph  ->  H  e.  ZZ )
2827ad2antrr 724 . . . . . 6  |-  ( ( ( ph  /\  (
p  e.  ZZ  /\  ( D  =  ( A Xrm 
p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  ->  H  e.  ZZ )
29 rmxycomplete 35178 . . . . . 6  |-  ( ( G  e.  ( ZZ>= ` 
2 )  /\  I  e.  NN0  /\  H  e.  ZZ )  ->  (
( ( I ^
2 )  -  (
( ( G ^
2 )  -  1 )  x.  ( H ^ 2 ) ) )  =  1  <->  E. r  e.  ZZ  (
I  =  ( G Xrm  r )  /\  H  =  ( G Yrm  r ) ) ) )
3023, 25, 28, 29syl3anc 1228 . . . . 5  |-  ( ( ( ph  /\  (
p  e.  ZZ  /\  ( D  =  ( A Xrm 
p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  -> 
( ( ( I ^ 2 )  -  ( ( ( G ^ 2 )  - 
1 )  x.  ( H ^ 2 ) ) )  =  1  <->  E. r  e.  ZZ  (
I  =  ( G Xrm  r )  /\  H  =  ( G Yrm  r ) ) ) )
3121, 30mpbid 210 . . . 4  |-  ( ( ( ph  /\  (
p  e.  ZZ  /\  ( D  =  ( A Xrm 
p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  ->  E. r  e.  ZZ  ( I  =  ( G Xrm  r )  /\  H  =  ( G Yrm  r ) ) )
322ad3antrrr 728 . . . . 5  |-  ( ( ( ( ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  /\  ( r  e.  ZZ  /\  ( I  =  ( G Xrm  r )  /\  H  =  ( G Yrm  r ) ) ) )  ->  A  e.  ( ZZ>= ` 
2 ) )
33 jm2.27a2 . . . . . 6  |-  ( ph  ->  B  e.  NN )
3433ad3antrrr 728 . . . . 5  |-  ( ( ( ( ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  /\  ( r  e.  ZZ  /\  ( I  =  ( G Xrm  r )  /\  H  =  ( G Yrm  r ) ) ) )  ->  B  e.  NN )
354ad3antrrr 728 . . . . 5  |-  ( ( ( ( ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  /\  ( r  e.  ZZ  /\  ( I  =  ( G Xrm  r )  /\  H  =  ( G Yrm  r ) ) ) )  ->  C  e.  NN )
363ad3antrrr 728 . . . . 5  |-  ( ( ( ( ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  /\  ( r  e.  ZZ  /\  ( I  =  ( G Xrm  r )  /\  H  =  ( G Yrm  r ) ) ) )  ->  D  e.  NN0 )
3714ad3antrrr 728 . . . . 5  |-  ( ( ( ( ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  /\  ( r  e.  ZZ  /\  ( I  =  ( G Xrm  r )  /\  H  =  ( G Yrm  r ) ) ) )  ->  E  e.  NN0 )
3812ad3antrrr 728 . . . . 5  |-  ( ( ( ( ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  /\  ( r  e.  ZZ  /\  ( I  =  ( G Xrm  r )  /\  H  =  ( G Yrm  r ) ) ) )  ->  F  e.  NN0 )
39 jm2.27a7 . . . . . 6  |-  ( ph  ->  G  e.  NN0 )
4039ad3antrrr 728 . . . . 5  |-  ( ( ( ( ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  /\  ( r  e.  ZZ  /\  ( I  =  ( G Xrm  r )  /\  H  =  ( G Yrm  r ) ) ) )  ->  G  e.  NN0 )
4126ad3antrrr 728 . . . . 5  |-  ( ( ( ( ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  /\  ( r  e.  ZZ  /\  ( I  =  ( G Xrm  r )  /\  H  =  ( G Yrm  r ) ) ) )  ->  H  e.  NN0 )
4224ad3antrrr 728 . . . . 5  |-  ( ( ( ( ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  /\  ( r  e.  ZZ  /\  ( I  =  ( G Xrm  r )  /\  H  =  ( G Yrm  r ) ) ) )  ->  I  e.  NN0 )
43 jm2.27a10 . . . . . 6  |-  ( ph  ->  J  e.  NN0 )
4443ad3antrrr 728 . . . . 5  |-  ( ( ( ( ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  /\  ( r  e.  ZZ  /\  ( I  =  ( G Xrm  r )  /\  H  =  ( G Yrm  r ) ) ) )  ->  J  e.  NN0 )
451ad3antrrr 728 . . . . 5  |-  ( ( ( ( ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  /\  ( r  e.  ZZ  /\  ( I  =  ( G Xrm  r )  /\  H  =  ( G Yrm  r ) ) ) )  -> 
( ( D ^
2 )  -  (
( ( A ^
2 )  -  1 )  x.  ( C ^ 2 ) ) )  =  1 )
469ad3antrrr 728 . . . . 5  |-  ( ( ( ( ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  /\  ( r  e.  ZZ  /\  ( I  =  ( G Xrm  r )  /\  H  =  ( G Yrm  r ) ) ) )  -> 
( ( F ^
2 )  -  (
( ( A ^
2 )  -  1 )  x.  ( E ^ 2 ) ) )  =  1 )
4722ad3antrrr 728 . . . . 5  |-  ( ( ( ( ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  /\  ( r  e.  ZZ  /\  ( I  =  ( G Xrm  r )  /\  H  =  ( G Yrm  r ) ) ) )  ->  G  e.  ( ZZ>= ` 
2 ) )
4820ad3antrrr 728 . . . . 5  |-  ( ( ( ( ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  /\  ( r  e.  ZZ  /\  ( I  =  ( G Xrm  r )  /\  H  =  ( G Yrm  r ) ) ) )  -> 
( ( I ^
2 )  -  (
( ( G ^
2 )  -  1 )  x.  ( H ^ 2 ) ) )  =  1 )
49 jm2.27a15 . . . . . 6  |-  ( ph  ->  E  =  ( ( J  +  1 )  x.  ( 2  x.  ( C ^ 2 ) ) ) )
5049ad3antrrr 728 . . . . 5  |-  ( ( ( ( ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  /\  ( r  e.  ZZ  /\  ( I  =  ( G Xrm  r )  /\  H  =  ( G Yrm  r ) ) ) )  ->  E  =  ( ( J  +  1 )  x.  ( 2  x.  ( C ^ 2 ) ) ) )
51 jm2.27a16 . . . . . 6  |-  ( ph  ->  F  ||  ( G  -  A ) )
5251ad3antrrr 728 . . . . 5  |-  ( ( ( ( ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  /\  ( r  e.  ZZ  /\  ( I  =  ( G Xrm  r )  /\  H  =  ( G Yrm  r ) ) ) )  ->  F  ||  ( G  -  A ) )
53 jm2.27a17 . . . . . 6  |-  ( ph  ->  ( 2  x.  C
)  ||  ( G  -  1 ) )
5453ad3antrrr 728 . . . . 5  |-  ( ( ( ( ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  /\  ( r  e.  ZZ  /\  ( I  =  ( G Xrm  r )  /\  H  =  ( G Yrm  r ) ) ) )  -> 
( 2  x.  C
)  ||  ( G  -  1 ) )
55 jm2.27a18 . . . . . 6  |-  ( ph  ->  F  ||  ( H  -  C ) )
5655ad3antrrr 728 . . . . 5  |-  ( ( ( ( ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  /\  ( r  e.  ZZ  /\  ( I  =  ( G Xrm  r )  /\  H  =  ( G Yrm  r ) ) ) )  ->  F  ||  ( H  -  C ) )
57 jm2.27a19 . . . . . 6  |-  ( ph  ->  ( 2  x.  C
)  ||  ( H  -  B ) )
5857ad3antrrr 728 . . . . 5  |-  ( ( ( ( ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  /\  ( r  e.  ZZ  /\  ( I  =  ( G Xrm  r )  /\  H  =  ( G Yrm  r ) ) ) )  -> 
( 2  x.  C
)  ||  ( H  -  B ) )
59 jm2.27a20 . . . . . 6  |-  ( ph  ->  B  <_  C )
6059ad3antrrr 728 . . . . 5  |-  ( ( ( ( ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  /\  ( r  e.  ZZ  /\  ( I  =  ( G Xrm  r )  /\  H  =  ( G Yrm  r ) ) ) )  ->  B  <_  C )
61 simprl 755 . . . . . 6  |-  ( (
ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  ->  p  e.  ZZ )
6261ad2antrr 724 . . . . 5  |-  ( ( ( ( ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  /\  ( r  e.  ZZ  /\  ( I  =  ( G Xrm  r )  /\  H  =  ( G Yrm  r ) ) ) )  ->  p  e.  ZZ )
63 simprrl 764 . . . . . 6  |-  ( (
ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  ->  D  =  ( A Xrm  p ) )
6463ad2antrr 724 . . . . 5  |-  ( ( ( ( ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  /\  ( r  e.  ZZ  /\  ( I  =  ( G Xrm  r )  /\  H  =  ( G Yrm  r ) ) ) )  ->  D  =  ( A Xrm  p
) )
65 simprrr 765 . . . . . 6  |-  ( (
ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  ->  C  =  ( A Yrm  p ) )
6665ad2antrr 724 . . . . 5  |-  ( ( ( ( ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  /\  ( r  e.  ZZ  /\  ( I  =  ( G Xrm  r )  /\  H  =  ( G Yrm  r ) ) ) )  ->  C  =  ( A Yrm  p
) )
67 simplrl 760 . . . . 5  |-  ( ( ( ( ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  /\  ( r  e.  ZZ  /\  ( I  =  ( G Xrm  r )  /\  H  =  ( G Yrm  r ) ) ) )  -> 
q  e.  ZZ )
68 simprl 755 . . . . . 6  |-  ( ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) )  ->  F  =  ( A Xrm  q ) )
6968ad2antlr 725 . . . . 5  |-  ( ( ( ( ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  /\  ( r  e.  ZZ  /\  ( I  =  ( G Xrm  r )  /\  H  =  ( G Yrm  r ) ) ) )  ->  F  =  ( A Xrm  q ) )
70 simprr 756 . . . . . 6  |-  ( ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) )  ->  E  =  ( A Yrm  q ) )
7170ad2antlr 725 . . . . 5  |-  ( ( ( ( ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  /\  ( r  e.  ZZ  /\  ( I  =  ( G Xrm  r )  /\  H  =  ( G Yrm  r ) ) ) )  ->  E  =  ( A Yrm  q ) )
72 simprl 755 . . . . 5  |-  ( ( ( ( ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  /\  ( r  e.  ZZ  /\  ( I  =  ( G Xrm  r )  /\  H  =  ( G Yrm  r ) ) ) )  -> 
r  e.  ZZ )
73 simprrl 764 . . . . 5  |-  ( ( ( ( ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  /\  ( r  e.  ZZ  /\  ( I  =  ( G Xrm  r )  /\  H  =  ( G Yrm  r ) ) ) )  ->  I  =  ( G Xrm  r ) )
74 simprrr 765 . . . . 5  |-  ( ( ( ( ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  /\  ( r  e.  ZZ  /\  ( I  =  ( G Xrm  r )  /\  H  =  ( G Yrm  r ) ) ) )  ->  H  =  ( G Yrm  r ) )
7532, 34, 35, 36, 37, 38, 40, 41, 42, 44, 45, 46, 47, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 67, 69, 71, 72, 73, 74jm2.27a 35273 . . . 4  |-  ( ( ( ( ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  /\  ( r  e.  ZZ  /\  ( I  =  ( G Xrm  r )  /\  H  =  ( G Yrm  r ) ) ) )  ->  C  =  ( A Yrm  B
) )
7631, 75rexlimddv 2897 . . 3  |-  ( ( ( ph  /\  (
p  e.  ZZ  /\  ( D  =  ( A Xrm 
p )  /\  C  =  ( A Yrm  p ) ) ) )  /\  ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) ) )  ->  C  =  ( A Yrm  B
) )
7719, 76rexlimddv 2897 . 2  |-  ( (
ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  ->  C  =  ( A Yrm  B ) )
788, 77rexlimddv 2897 1  |-  ( ph  ->  C  =  ( A Yrm  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1403    e. wcel 1840   E.wrex 2752   class class class wbr 4392   ` cfv 5523  (class class class)co 6232   1c1 9441    + caddc 9443    x. cmul 9445    <_ cle 9577    - cmin 9759   NNcn 10494   2c2 10544   NN0cn0 10754   ZZcz 10823   ZZ>=cuz 11043   ^cexp 12118    || cdvds 14085   Xrm crmx 35161   Yrm crmy 35162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-inf2 8009  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517  ax-pre-sup 9518  ax-addf 9519  ax-mulf 9520
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-fal 1409  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-iin 4271  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-isom 5532  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-of 6475  df-om 6637  df-1st 6736  df-2nd 6737  df-supp 6855  df-recs 6997  df-rdg 7031  df-1o 7085  df-2o 7086  df-oadd 7089  df-omul 7090  df-er 7266  df-map 7377  df-pm 7378  df-ixp 7426  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-fsupp 7782  df-fi 7823  df-sup 7853  df-oi 7887  df-card 8270  df-acn 8273  df-cda 8498  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-div 10166  df-nn 10495  df-2 10553  df-3 10554  df-4 10555  df-5 10556  df-6 10557  df-7 10558  df-8 10559  df-9 10560  df-10 10561  df-n0 10755  df-z 10824  df-dec 10938  df-uz 11044  df-q 11144  df-rp 11182  df-xneg 11287  df-xadd 11288  df-xmul 11289  df-ioo 11502  df-ioc 11503  df-ico 11504  df-icc 11505  df-fz 11642  df-fzo 11766  df-fl 11877  df-mod 11946  df-seq 12060  df-exp 12119  df-fac 12306  df-bc 12333  df-hash 12358  df-shft 12954  df-cj 12986  df-re 12987  df-im 12988  df-sqrt 13122  df-abs 13123  df-limsup 13348  df-clim 13365  df-rlim 13366  df-sum 13563  df-ef 13902  df-sin 13904  df-cos 13905  df-pi 13907  df-dvds 14086  df-gcd 14244  df-prm 14317  df-numer 14367  df-denom 14368  df-struct 14733  df-ndx 14734  df-slot 14735  df-base 14736  df-sets 14737  df-ress 14738  df-plusg 14812  df-mulr 14813  df-starv 14814  df-sca 14815  df-vsca 14816  df-ip 14817  df-tset 14818  df-ple 14819  df-ds 14821  df-unif 14822  df-hom 14823  df-cco 14824  df-rest 14927  df-topn 14928  df-0g 14946  df-gsum 14947  df-topgen 14948  df-pt 14949  df-prds 14952  df-xrs 15006  df-qtop 15011  df-imas 15012  df-xps 15014  df-mre 15090  df-mrc 15091  df-acs 15093  df-mgm 16086  df-sgrp 16125  df-mnd 16135  df-submnd 16181  df-mulg 16274  df-cntz 16569  df-cmn 17014  df-psmet 18621  df-xmet 18622  df-met 18623  df-bl 18624  df-mopn 18625  df-fbas 18626  df-fg 18627  df-cnfld 18631  df-top 19581  df-bases 19583  df-topon 19584  df-topsp 19585  df-cld 19702  df-ntr 19703  df-cls 19704  df-nei 19782  df-lp 19820  df-perf 19821  df-cn 19911  df-cnp 19912  df-haus 19999  df-tx 20245  df-hmeo 20438  df-fil 20529  df-fm 20621  df-flim 20622  df-flf 20623  df-xms 21005  df-ms 21006  df-tms 21007  df-cncf 21564  df-limc 22452  df-dv 22453  df-log 23126  df-squarenn 35102  df-pell1qr 35103  df-pell14qr 35104  df-pell1234qr 35105  df-pellfund 35106  df-rmx 35163  df-rmy 35164
This theorem is referenced by:  jm2.27  35276
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