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Theorem jm2.27b 30876
Description: Lemma for jm2.27 30878. 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 10977 . . . 4  |-  ( ph  ->  C  e.  ZZ )
6 rmxycomplete 30781 . . . 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 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 10976 . . . . . 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 30781 . . . . 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 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 10976 . . . . . . 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 30781 . . . . . 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 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 30875 . . . 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 2963 . . 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 2963 . 2  |-  ( (
ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  ->  C  =  ( A Yrm  B ) )
788, 77rexlimddv 2963 1  |-  ( ph  ->  C  =  ( A Yrm  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2818   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   1c1 9505    + caddc 9507    x. cmul 9509    <_ cle 9641    - cmin 9817   NNcn 10548   2c2 10597   NN0cn0 10807   ZZcz 10876   ZZ>=cuz 11094   ^cexp 12146    || cdivides 13864   Xrm crmx 30764   Yrm crmy 30765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583  ax-mulf 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-omul 7147  df-er 7323  df-map 7434  df-pm 7435  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-fi 7883  df-sup 7913  df-oi 7947  df-card 8332  df-acn 8335  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ioo 11545  df-ioc 11546  df-ico 11547  df-icc 11548  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-fac 12334  df-bc 12361  df-hash 12386  df-shft 12880  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-limsup 13274  df-clim 13291  df-rlim 13292  df-sum 13489  df-ef 13682  df-sin 13684  df-cos 13685  df-pi 13687  df-dvds 13865  df-gcd 14021  df-prm 14094  df-numer 14144  df-denom 14145  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-starv 14587  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-hom 14596  df-cco 14597  df-rest 14695  df-topn 14696  df-0g 14714  df-gsum 14715  df-topgen 14716  df-pt 14717  df-prds 14720  df-xrs 14774  df-qtop 14779  df-imas 14780  df-xps 14782  df-mre 14858  df-mrc 14859  df-acs 14861  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-mulg 15932  df-cntz 16227  df-cmn 16673  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-fbas 18286  df-fg 18287  df-cnfld 18291  df-top 19268  df-bases 19270  df-topon 19271  df-topsp 19272  df-cld 19388  df-ntr 19389  df-cls 19390  df-nei 19467  df-lp 19505  df-perf 19506  df-cn 19596  df-cnp 19597  df-haus 19684  df-tx 19931  df-hmeo 20124  df-fil 20215  df-fm 20307  df-flim 20308  df-flf 20309  df-xms 20691  df-ms 20692  df-tms 20693  df-cncf 21250  df-limc 22138  df-dv 22139  df-log 22810  df-squarenn 30705  df-pell1qr 30706  df-pell14qr 30707  df-pell1234qr 30708  df-pellfund 30709  df-rmx 30766  df-rmy 30767
This theorem is referenced by:  jm2.27  30878
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