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Theorem jm2.27b 29360
Description: Lemma for jm2.27 29362. 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 10751 . . . 4  |-  ( ph  ->  C  e.  ZZ )
6 rmxycomplete 29263 . . . 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 1218 . . 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 465 . . . 4  |-  ( (
ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  ->  ( ( F ^ 2 )  -  ( ( ( A ^ 2 )  - 
1 )  x.  ( E ^ 2 ) ) )  =  1 )
112adantr 465 . . . . 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 465 . . . . 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 10750 . . . . . 6  |-  ( ph  ->  E  e.  ZZ )
1615adantr 465 . . . . 5  |-  ( (
ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  ->  E  e.  ZZ )
17 rmxycomplete 29263 . . . . 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 1218 . . . 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 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 ) ) ) )  -> 
( ( I ^
2 )  -  (
( ( G ^
2 )  -  1 )  x.  ( H ^ 2 ) ) )  =  1 )
22 jm2.27a13 . . . . . . 7  |-  ( ph  ->  G  e.  ( ZZ>= ` 
2 ) )
2322ad2antrr 725 . . . . . 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 725 . . . . . 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 10750 . . . . . . 7  |-  ( ph  ->  H  e.  ZZ )
2827ad2antrr 725 . . . . . 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 29263 . . . . . 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 1218 . . . . 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 729 . . . . 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 729 . . . . 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 729 . . . . 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 729 . . . . 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 729 . . . . 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 729 . . . . 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 729 . . . . 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 729 . . . . 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 729 . . . . 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 729 . . . . 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 729 . . . . 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 729 . . . . 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 729 . . . . 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 729 . . . . 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 729 . . . . 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 729 . . . . 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 729 . . . . 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 729 . . . . 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 729 . . . . 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 729 . . . . 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 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 ) ) ) )  ->  p  e.  ZZ )
63 simprrl 763 . . . . . 6  |-  ( (
ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  ->  D  =  ( A Xrm  p ) )
6463ad2antrr 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 ) ) ) )  ->  D  =  ( A Xrm  p
) )
65 simprrr 764 . . . . . 6  |-  ( (
ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  ->  C  =  ( A Yrm  p ) )
6665ad2antrr 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 ) ) ) )  ->  C  =  ( A Yrm  p
) )
67 simplrl 759 . . . . 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 726 . . . . 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 726 . . . . 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 763 . . . . 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 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 ) ) ) )  ->  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 29359 . . . 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 2850 . . 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 2850 . 2  |-  ( (
ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  ->  C  =  ( A Yrm  B ) )
788, 77rexlimddv 2850 1  |-  ( ph  ->  C  =  ( A Yrm  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2721   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   1c1 9288    + caddc 9290    x. cmul 9292    <_ cle 9424    - cmin 9600   NNcn 10327   2c2 10376   NN0cn0 10584   ZZcz 10651   ZZ>=cuz 10866   ^cexp 11870    || cdivides 13540   Xrm crmx 29246   Yrm crmy 29247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365  ax-addf 9366  ax-mulf 9367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-omul 6930  df-er 7106  df-map 7221  df-pm 7222  df-ixp 7269  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fsupp 7626  df-fi 7666  df-sup 7696  df-oi 7729  df-card 8114  df-acn 8117  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-q 10959  df-rp 10997  df-xneg 11094  df-xadd 11095  df-xmul 11096  df-ioo 11309  df-ioc 11310  df-ico 11311  df-icc 11312  df-fz 11443  df-fzo 11554  df-fl 11647  df-mod 11714  df-seq 11812  df-exp 11871  df-fac 12057  df-bc 12084  df-hash 12109  df-shft 12561  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-limsup 12954  df-clim 12971  df-rlim 12972  df-sum 13169  df-ef 13358  df-sin 13360  df-cos 13361  df-pi 13363  df-dvds 13541  df-gcd 13696  df-prm 13769  df-numer 13818  df-denom 13819  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-starv 14258  df-sca 14259  df-vsca 14260  df-ip 14261  df-tset 14262  df-ple 14263  df-ds 14265  df-unif 14266  df-hom 14267  df-cco 14268  df-rest 14366  df-topn 14367  df-0g 14385  df-gsum 14386  df-topgen 14387  df-pt 14388  df-prds 14391  df-xrs 14445  df-qtop 14450  df-imas 14451  df-xps 14453  df-mre 14529  df-mrc 14530  df-acs 14532  df-mnd 15420  df-submnd 15470  df-mulg 15553  df-cntz 15840  df-cmn 16284  df-psmet 17814  df-xmet 17815  df-met 17816  df-bl 17817  df-mopn 17818  df-fbas 17819  df-fg 17820  df-cnfld 17824  df-top 18508  df-bases 18510  df-topon 18511  df-topsp 18512  df-cld 18628  df-ntr 18629  df-cls 18630  df-nei 18707  df-lp 18745  df-perf 18746  df-cn 18836  df-cnp 18837  df-haus 18924  df-tx 19140  df-hmeo 19333  df-fil 19424  df-fm 19516  df-flim 19517  df-flf 19518  df-xms 19900  df-ms 19901  df-tms 19902  df-cncf 20459  df-limc 21346  df-dv 21347  df-log 22013  df-squarenn 29187  df-pell1qr 29188  df-pell14qr 29189  df-pell1234qr 29190  df-pellfund 29191  df-rmx 29248  df-rmy 29249
This theorem is referenced by:  jm2.27  29362
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