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Theorem jm2.27b 27077
Description: Lemma for jm2.27 27079. 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 10374 . . . 4  |-  ( ph  ->  C  e.  ZZ )
6 rmxycomplete 26980 . . . 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 1184 . . 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 202 . 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 452 . . . 4  |-  ( (
ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  ->  ( ( F ^ 2 )  -  ( ( ( A ^ 2 )  - 
1 )  x.  ( E ^ 2 ) ) )  =  1 )
112adantr 452 . . . . 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 452 . . . . 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 10373 . . . . . 6  |-  ( ph  ->  E  e.  ZZ )
1615adantr 452 . . . . 5  |-  ( (
ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  ->  E  e.  ZZ )
17 rmxycomplete 26980 . . . . 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 1184 . . . 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 202 . . 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 707 . . . . 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 707 . . . . . 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 707 . . . . . 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 10373 . . . . . . 7  |-  ( ph  ->  H  e.  ZZ )
2827ad2antrr 707 . . . . . 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 26980 . . . . . 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 1184 . . . . 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 202 . . . 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 711 . . . . 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 711 . . . . 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 711 . . . . 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 711 . . . . 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 711 . . . . 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 711 . . . . 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 711 . . . . 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 711 . . . . 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 711 . . . . 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 711 . . . . 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 711 . . . . 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 711 . . . . 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 711 . . . . 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 711 . . . . 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 711 . . . . 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 711 . . . . 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 711 . . . . 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 711 . . . . 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 711 . . . . 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 711 . . . . 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 733 . . . . . 6  |-  ( (
ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  ->  p  e.  ZZ )
6261ad2antrr 707 . . . . 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 741 . . . . . 6  |-  ( (
ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  ->  D  =  ( A Xrm  p ) )
6463ad2antrr 707 . . . . 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 742 . . . . . 6  |-  ( (
ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  ->  C  =  ( A Yrm  p ) )
6665ad2antrr 707 . . . . 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 737 . . . . 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 733 . . . . . 6  |-  ( ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) )  ->  F  =  ( A Xrm  q ) )
6968ad2antlr 708 . . . . 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 734 . . . . . 6  |-  ( ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) )  ->  E  =  ( A Yrm  q ) )
7170ad2antlr 708 . . . . 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 733 . . . . 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 741 . . . . 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 742 . . . . 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 27076 . . . 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 2834 . . 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 2834 . 2  |-  ( (
ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  ->  C  =  ( A Yrm  B ) )
788, 77rexlimddv 2834 1  |-  ( ph  ->  C  =  ( A Yrm  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2706   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   1c1 8991    + caddc 8993    x. cmul 8995    <_ cle 9121    - cmin 9291   NNcn 10000   2c2 10049   NN0cn0 10221   ZZcz 10282   ZZ>=cuz 10488   ^cexp 11382    || cdivides 12852   Xrm crmx 26963   Yrm crmy 26964
This theorem is referenced by:  jm2.27  27079
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-omul 6729  df-er 6905  df-map 7020  df-pm 7021  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-fi 7416  df-sup 7446  df-oi 7479  df-card 7826  df-acn 7829  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-ioo 10920  df-ioc 10921  df-ico 10922  df-icc 10923  df-fz 11044  df-fzo 11136  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-fac 11567  df-bc 11594  df-hash 11619  df-shft 11882  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-limsup 12265  df-clim 12282  df-rlim 12283  df-sum 12480  df-ef 12670  df-sin 12672  df-cos 12673  df-pi 12675  df-dvds 12853  df-gcd 13007  df-prm 13080  df-numer 13127  df-denom 13128  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-starv 13544  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-hom 13553  df-cco 13554  df-rest 13650  df-topn 13651  df-topgen 13667  df-pt 13668  df-prds 13671  df-xrs 13726  df-0g 13727  df-gsum 13728  df-qtop 13733  df-imas 13734  df-xps 13736  df-mre 13811  df-mrc 13812  df-acs 13814  df-mnd 14690  df-submnd 14739  df-mulg 14815  df-cntz 15116  df-cmn 15414  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-mopn 16698  df-fbas 16699  df-fg 16700  df-cnfld 16704  df-top 16963  df-bases 16965  df-topon 16966  df-topsp 16967  df-cld 17083  df-ntr 17084  df-cls 17085  df-nei 17162  df-lp 17200  df-perf 17201  df-cn 17291  df-cnp 17292  df-haus 17379  df-tx 17594  df-hmeo 17787  df-fil 17878  df-fm 17970  df-flim 17971  df-flf 17972  df-xms 18350  df-ms 18351  df-tms 18352  df-cncf 18908  df-limc 19753  df-dv 19754  df-log 20454  df-squarenn 26904  df-pell1qr 26905  df-pell14qr 26906  df-pell1234qr 26907  df-pellfund 26908  df-rmx 26965  df-rmy 26966
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