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Theorem jm2.27b 27202
Description: Lemma for jm2.27 27204. 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 10132 . . . 4  |-  ( ph  ->  C  e.  ZZ )
6 rmxycomplete 27105 . . . 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 1182 . . 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 201 . 2  |-  ( ph  ->  E. p  e.  ZZ  ( D  =  ( A Xrm 
p )  /\  C  =  ( A Yrm  p ) ) )
9 jm2.27a12 . . . . . . 7  |-  ( ph  ->  ( ( F ^
2 )  -  (
( ( A ^
2 )  -  1 )  x.  ( E ^ 2 ) ) )  =  1 )
109adantr 451 . . . . . 6  |-  ( (
ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  ->  ( ( F ^ 2 )  -  ( ( ( A ^ 2 )  - 
1 )  x.  ( E ^ 2 ) ) )  =  1 )
112adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  ->  A  e.  ( ZZ>= `  2 )
)
12 jm2.27a6 . . . . . . . 8  |-  ( ph  ->  F  e.  NN0 )
1312adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  ->  F  e.  NN0 )
14 jm2.27a5 . . . . . . . . 9  |-  ( ph  ->  E  e.  NN0 )
1514nn0zd 10131 . . . . . . . 8  |-  ( ph  ->  E  e.  ZZ )
1615adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  ->  E  e.  ZZ )
17 rmxycomplete 27105 . . . . . . 7  |-  ( ( 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 1182 . . . . . 6  |-  ( (
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 201 . . . . 5  |-  ( (
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 . . . . . . . . . 10  |-  ( ph  ->  ( ( I ^
2 )  -  (
( ( G ^
2 )  -  1 )  x.  ( H ^ 2 ) ) )  =  1 )
2120ad2antrr 706 . . . . . . . . 9  |-  ( ( ( 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 . . . . . . . . . . 11  |-  ( ph  ->  G  e.  ( ZZ>= ` 
2 ) )
2322ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( 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 . . . . . . . . . . 11  |-  ( ph  ->  I  e.  NN0 )
2524ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( 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 . . . . . . . . . . . 12  |-  ( ph  ->  H  e.  NN0 )
2726nn0zd 10131 . . . . . . . . . . 11  |-  ( ph  ->  H  e.  ZZ )
2827ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( 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 27105 . . . . . . . . . 10  |-  ( ( 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 1182 . . . . . . . . 9  |-  ( ( ( 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 201 . . . . . . . 8  |-  ( ( ( 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 710 . . . . . . . . . . 11  |-  ( ( ( ( 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 . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  NN )
3433ad3antrrr 710 . . . . . . . . . . 11  |-  ( ( ( ( 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 710 . . . . . . . . . . 11  |-  ( ( ( ( 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 710 . . . . . . . . . . 11  |-  ( ( ( ( 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 710 . . . . . . . . . . 11  |-  ( ( ( ( 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 710 . . . . . . . . . . 11  |-  ( ( ( ( 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 . . . . . . . . . . . 12  |-  ( ph  ->  G  e.  NN0 )
4039ad3antrrr 710 . . . . . . . . . . 11  |-  ( ( ( ( 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 710 . . . . . . . . . . 11  |-  ( ( ( ( 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 710 . . . . . . . . . . 11  |-  ( ( ( ( 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 . . . . . . . . . . . 12  |-  ( ph  ->  J  e.  NN0 )
4443ad3antrrr 710 . . . . . . . . . . 11  |-  ( ( ( ( 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 710 . . . . . . . . . . 11  |-  ( ( ( ( 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 710 . . . . . . . . . . 11  |-  ( ( ( ( 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 710 . . . . . . . . . . 11  |-  ( ( ( ( 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 710 . . . . . . . . . . 11  |-  ( ( ( ( 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 . . . . . . . . . . . 12  |-  ( ph  ->  E  =  ( ( J  +  1 )  x.  ( 2  x.  ( C ^ 2 ) ) ) )
5049ad3antrrr 710 . . . . . . . . . . 11  |-  ( ( ( ( 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 . . . . . . . . . . . 12  |-  ( ph  ->  F  ||  ( G  -  A ) )
5251ad3antrrr 710 . . . . . . . . . . 11  |-  ( ( ( ( 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 . . . . . . . . . . . 12  |-  ( ph  ->  ( 2  x.  C
)  ||  ( G  -  1 ) )
5453ad3antrrr 710 . . . . . . . . . . 11  |-  ( ( ( ( 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 . . . . . . . . . . . 12  |-  ( ph  ->  F  ||  ( H  -  C ) )
5655ad3antrrr 710 . . . . . . . . . . 11  |-  ( ( ( ( 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 . . . . . . . . . . . 12  |-  ( ph  ->  ( 2  x.  C
)  ||  ( H  -  B ) )
5857ad3antrrr 710 . . . . . . . . . . 11  |-  ( ( ( ( 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 . . . . . . . . . . . 12  |-  ( ph  ->  B  <_  C )
6059ad3antrrr 710 . . . . . . . . . . 11  |-  ( ( ( ( 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 732 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  ->  p  e.  ZZ )
6261ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ( 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 740 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  ->  D  =  ( A Xrm  p ) )
6463ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ( 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 741 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  ->  C  =  ( A Yrm  p ) )
6665ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ( 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 736 . . . . . . . . . . 11  |-  ( ( ( ( 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 732 . . . . . . . . . . . 12  |-  ( ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) )  ->  F  =  ( A Xrm  q ) )
6968ad2antlr 707 . . . . . . . . . . 11  |-  ( ( ( ( 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 733 . . . . . . . . . . . 12  |-  ( ( q  e.  ZZ  /\  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) ) )  ->  E  =  ( A Yrm  q ) )
7170ad2antlr 707 . . . . . . . . . . 11  |-  ( ( ( ( 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 732 . . . . . . . . . . 11  |-  ( ( ( ( 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 740 . . . . . . . . . . 11  |-  ( ( ( ( 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 741 . . . . . . . . . . 11  |-  ( ( ( ( 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 27201 . . . . . . . . . 10  |-  ( ( ( ( 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
) )
7675exp32 588 . . . . . . . . 9  |-  ( ( ( 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 ) ) ) )
7776rexlimdv 2679 . . . . . . . 8  |-  ( ( ( 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 ) )  ->  C  =  ( A Yrm  B ) ) )
7831, 77mpd 14 . . . . . . 7  |-  ( ( ( 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
) )
7978exp32 588 . . . . . 6  |-  ( (
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 ) ) ) )
8079rexlimdv 2679 . . . . 5  |-  ( (
ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  ->  ( E. q  e.  ZZ  ( F  =  ( A Xrm  q )  /\  E  =  ( A Yrm  q ) )  ->  C  =  ( A Yrm  B ) ) )
8119, 80mpd 14 . . . 4  |-  ( (
ph  /\  ( p  e.  ZZ  /\  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p ) ) ) )  ->  C  =  ( A Yrm  B ) )
8281exp32 588 . . 3  |-  ( ph  ->  ( p  e.  ZZ  ->  ( ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p
) )  ->  C  =  ( A Yrm  B ) ) ) )
8382rexlimdv 2679 . 2  |-  ( ph  ->  ( E. p  e.  ZZ  ( D  =  ( A Xrm  p )  /\  C  =  ( A Yrm  p
) )  ->  C  =  ( A Yrm  B ) ) )
848, 83mpd 14 1  |-  ( ph  ->  C  =  ( A Yrm  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   1c1 8754    + caddc 8756    x. cmul 8758    <_ cle 8884    - cmin 9053   NNcn 9762   2c2 9811   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   ^cexp 11120    || cdivides 12547   Xrm crmx 27088   Yrm crmy 27089
This theorem is referenced by:  jm2.27  27204
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-omul 6500  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-acn 7591  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-sin 12367  df-cos 12368  df-pi 12370  df-dvds 12548  df-gcd 12702  df-prm 12775  df-numer 12822  df-denom 12823  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233  df-log 19930  df-squarenn 27029  df-pell1qr 27030  df-pell14qr 27031  df-pell1234qr 27032  df-pellfund 27033  df-rmx 27090  df-rmy 27091
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