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Theorem 4rexfrabdioph 35641
Description: Diophantine set builder for existential quantifier, explicit substitution, four variables. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
Hypotheses
Ref Expression
rexfrabdioph.1  |-  M  =  ( N  +  1 )
rexfrabdioph.2  |-  L  =  ( M  +  1 )
rexfrabdioph.3  |-  K  =  ( L  +  1 )
rexfrabdioph.4  |-  J  =  ( K  +  1 )
Assertion
Ref Expression
4rexfrabdioph  |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0 
^m  ( 1 ... J ) )  | 
[. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. [. ( t `  J )  /  y ]. ph }  e.  (Dioph `  J ) )  ->  { u  e.  ( NN0  ^m  ( 1 ... N ) )  |  E. v  e.  NN0  E. w  e.  NN0  E. x  e.  NN0  E. y  e. 
NN0  ph }  e.  (Dioph `  N ) )
Distinct variable groups:    u, t,
v, w, x, y, J    t, K, u, v, w, x, y   
t, L, u, v, w, x, y    t, M, u, v, w, x, y    t, N, u, v, w, x, y    ph, t
Allowed substitution hints:    ph( x, y, w, v, u)

Proof of Theorem 4rexfrabdioph
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 2sbcrex 35627 . . . . . . . 8  |-  ( [. ( a `  M
)  /  v ]. [. ( a `  L
)  /  w ]. E. x  e.  NN0  E. y  e.  NN0  ph  <->  E. x  e.  NN0  [. ( a `  M )  /  v ]. [. ( a `  L )  /  w ]. E. y  e.  NN0  ph )
2 2sbcrex 35627 . . . . . . . . 9  |-  ( [. ( a `  M
)  /  v ]. [. ( a `  L
)  /  w ]. E. y  e.  NN0  ph  <->  E. y  e.  NN0  [. (
a `  M )  /  v ]. [. (
a `  L )  /  w ]. ph )
32rexbii 2889 . . . . . . . 8  |-  ( E. x  e.  NN0  [. (
a `  M )  /  v ]. [. (
a `  L )  /  w ]. E. y  e.  NN0  ph  <->  E. x  e.  NN0  E. y  e.  NN0  [. (
a `  M )  /  v ]. [. (
a `  L )  /  w ]. ph )
41, 3bitri 253 . . . . . . 7  |-  ( [. ( a `  M
)  /  v ]. [. ( a `  L
)  /  w ]. E. x  e.  NN0  E. y  e.  NN0  ph  <->  E. x  e.  NN0  E. y  e. 
NN0  [. ( a `  M )  /  v ]. [. ( a `  L )  /  w ]. ph )
54sbcbii 3323 . . . . . 6  |-  ( [. ( a  |`  (
1 ... N ) )  /  u ]. [. (
a `  M )  /  v ]. [. (
a `  L )  /  w ]. E. x  e.  NN0  E. y  e. 
NN0  ph  <->  [. ( a  |`  ( 1 ... N
) )  /  u ]. E. x  e.  NN0  E. y  e.  NN0  [. (
a `  M )  /  v ]. [. (
a `  L )  /  w ]. ph )
6 sbc2rex 35630 . . . . . 6  |-  ( [. ( a  |`  (
1 ... N ) )  /  u ]. E. x  e.  NN0  E. y  e.  NN0  [. ( a `  M )  /  v ]. [. ( a `  L )  /  w ]. ph  <->  E. x  e.  NN0  E. y  e.  NN0  [. (
a  |`  ( 1 ... N ) )  /  u ]. [. ( a `
 M )  / 
v ]. [. ( a `
 L )  /  w ]. ph )
75, 6bitri 253 . . . . 5  |-  ( [. ( a  |`  (
1 ... N ) )  /  u ]. [. (
a `  M )  /  v ]. [. (
a `  L )  /  w ]. E. x  e.  NN0  E. y  e. 
NN0  ph  <->  E. x  e.  NN0  E. y  e.  NN0  [. (
a  |`  ( 1 ... N ) )  /  u ]. [. ( a `
 M )  / 
v ]. [. ( a `
 L )  /  w ]. ph )
87a1i 11 . . . 4  |-  ( a  e.  ( NN0  ^m  ( 1 ... L
) )  ->  ( [. ( a  |`  (
1 ... N ) )  /  u ]. [. (
a `  M )  /  v ]. [. (
a `  L )  /  w ]. E. x  e.  NN0  E. y  e. 
NN0  ph  <->  E. x  e.  NN0  E. y  e.  NN0  [. (
a  |`  ( 1 ... N ) )  /  u ]. [. ( a `
 M )  / 
v ]. [. ( a `
 L )  /  w ]. ph ) )
98rabbiia 3033 . . 3  |-  { a  e.  ( NN0  ^m  ( 1 ... L
) )  |  [. ( a  |`  (
1 ... N ) )  /  u ]. [. (
a `  M )  /  v ]. [. (
a `  L )  /  w ]. E. x  e.  NN0  E. y  e. 
NN0  ph }  =  {
a  e.  ( NN0 
^m  ( 1 ... L ) )  |  E. x  e.  NN0  E. y  e.  NN0  [. (
a  |`  ( 1 ... N ) )  /  u ]. [. ( a `
 M )  / 
v ]. [. ( a `
 L )  /  w ]. ph }
10 rexfrabdioph.2 . . . . . . 7  |-  L  =  ( M  +  1 )
11 rexfrabdioph.1 . . . . . . . . 9  |-  M  =  ( N  +  1 )
12 nn0p1nn 10909 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  NN )
1311, 12syl5eqel 2533 . . . . . . . 8  |-  ( N  e.  NN0  ->  M  e.  NN )
1413peano2nnd 10626 . . . . . . 7  |-  ( N  e.  NN0  ->  ( M  +  1 )  e.  NN )
1510, 14syl5eqel 2533 . . . . . 6  |-  ( N  e.  NN0  ->  L  e.  NN )
1615nnnn0d 10925 . . . . 5  |-  ( N  e.  NN0  ->  L  e. 
NN0 )
1716adantr 467 . . . 4  |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0 
^m  ( 1 ... J ) )  | 
[. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. [. ( t `  J )  /  y ]. ph }  e.  (Dioph `  J ) )  ->  L  e.  NN0 )
18 sbcrot3 35634 . . . . . . . . . 10  |-  ( [. ( a  |`  (
1 ... N ) )  /  u ]. [. (
t `  K )  /  x ]. [. (
t `  J )  /  y ]. [. (
a `  M )  /  v ]. [. (
a `  L )  /  w ]. ph  <->  [. ( t `
 K )  /  x ]. [. ( t `
 J )  / 
y ]. [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. [. ( a `  L )  /  w ]. ph )
19 sbcrot3 35634 . . . . . . . . . . . . 13  |-  ( [. ( t `  J
)  /  y ]. [. ( a `  M
)  /  v ]. [. ( a `  L
)  /  w ]. ph  <->  [. ( a `  M
)  /  v ]. [. ( a `  L
)  /  w ]. [. ( t `  J
)  /  y ]. ph )
2019sbcbii 3323 . . . . . . . . . . . 12  |-  ( [. ( t `  K
)  /  x ]. [. ( t `  J
)  /  y ]. [. ( a `  M
)  /  v ]. [. ( a `  L
)  /  w ]. ph  <->  [. ( t `  K
)  /  x ]. [. ( a `  M
)  /  v ]. [. ( a `  L
)  /  w ]. [. ( t `  J
)  /  y ]. ph )
21 sbcrot3 35634 . . . . . . . . . . . 12  |-  ( [. ( t `  K
)  /  x ]. [. ( a `  M
)  /  v ]. [. ( a `  L
)  /  w ]. [. ( t `  J
)  /  y ]. ph  <->  [. ( a `  M
)  /  v ]. [. ( a `  L
)  /  w ]. [. ( t `  K
)  /  x ]. [. ( t `  J
)  /  y ]. ph )
2220, 21bitri 253 . . . . . . . . . . 11  |-  ( [. ( t `  K
)  /  x ]. [. ( t `  J
)  /  y ]. [. ( a `  M
)  /  v ]. [. ( a `  L
)  /  w ]. ph  <->  [. ( a `  M
)  /  v ]. [. ( a `  L
)  /  w ]. [. ( t `  K
)  /  x ]. [. ( t `  J
)  /  y ]. ph )
2322sbcbii 3323 . . . . . . . . . 10  |-  ( [. ( a  |`  (
1 ... N ) )  /  u ]. [. (
t `  K )  /  x ]. [. (
t `  J )  /  y ]. [. (
a `  M )  /  v ]. [. (
a `  L )  /  w ]. ph  <->  [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. [. ( a `  L )  /  w ]. [. ( t `  K )  /  x ]. [. ( t `  J )  /  y ]. ph )
2418, 23bitr3i 255 . . . . . . . . 9  |-  ( [. ( t `  K
)  /  x ]. [. ( t `  J
)  /  y ]. [. ( a  |`  (
1 ... N ) )  /  u ]. [. (
a `  M )  /  v ]. [. (
a `  L )  /  w ]. ph  <->  [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. [. ( a `  L )  /  w ]. [. ( t `  K )  /  x ]. [. ( t `  J )  /  y ]. ph )
2524sbcbii 3323 . . . . . . . 8  |-  ( [. ( t  |`  (
1 ... L ) )  /  a ]. [. (
t `  K )  /  x ]. [. (
t `  J )  /  y ]. [. (
a  |`  ( 1 ... N ) )  /  u ]. [. ( a `
 M )  / 
v ]. [. ( a `
 L )  /  w ]. ph  <->  [. ( t  |`  ( 1 ... L
) )  /  a ]. [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. [. ( a `  L )  /  w ]. [. ( t `  K )  /  x ]. [. ( t `  J )  /  y ]. ph )
26 reseq1 5099 . . . . . . . . . 10  |-  ( a  =  ( t  |`  ( 1 ... L
) )  ->  (
a  |`  ( 1 ... N ) )  =  ( ( t  |`  ( 1 ... L
) )  |`  (
1 ... N ) ) )
2726sbccomieg 35636 . . . . . . . . 9  |-  ( [. ( t  |`  (
1 ... L ) )  /  a ]. [. (
a  |`  ( 1 ... N ) )  /  u ]. [. ( a `
 M )  / 
v ]. [. ( a `
 L )  /  w ]. [. ( t `
 K )  /  x ]. [. ( t `
 J )  / 
y ]. ph  <->  [. ( ( t  |`  ( 1 ... L ) )  |`  ( 1 ... N
) )  /  u ]. [. ( t  |`  ( 1 ... L
) )  /  a ]. [. ( a `  M )  /  v ]. [. ( a `  L )  /  w ]. [. ( t `  K )  /  x ]. [. ( t `  J )  /  y ]. ph )
28 fzssp1 11841 . . . . . . . . . . . . 13  |-  ( 1 ... N )  C_  ( 1 ... ( N  +  1 ) )
2911oveq2i 6301 . . . . . . . . . . . . 13  |-  ( 1 ... M )  =  ( 1 ... ( N  +  1 ) )
3028, 29sseqtr4i 3465 . . . . . . . . . . . 12  |-  ( 1 ... N )  C_  ( 1 ... M
)
31 fzssp1 11841 . . . . . . . . . . . . 13  |-  ( 1 ... M )  C_  ( 1 ... ( M  +  1 ) )
3210oveq2i 6301 . . . . . . . . . . . . 13  |-  ( 1 ... L )  =  ( 1 ... ( M  +  1 ) )
3331, 32sseqtr4i 3465 . . . . . . . . . . . 12  |-  ( 1 ... M )  C_  ( 1 ... L
)
3430, 33sstri 3441 . . . . . . . . . . 11  |-  ( 1 ... N )  C_  ( 1 ... L
)
35 resabs1 5133 . . . . . . . . . . 11  |-  ( ( 1 ... N ) 
C_  ( 1 ... L )  ->  (
( t  |`  (
1 ... L ) )  |`  ( 1 ... N
) )  =  ( t  |`  ( 1 ... N ) ) )
36 dfsbcq 3269 . . . . . . . . . . 11  |-  ( ( ( t  |`  (
1 ... L ) )  |`  ( 1 ... N
) )  =  ( t  |`  ( 1 ... N ) )  ->  ( [. (
( t  |`  (
1 ... L ) )  |`  ( 1 ... N
) )  /  u ]. [. ( t  |`  ( 1 ... L
) )  /  a ]. [. ( a `  M )  /  v ]. [. ( a `  L )  /  w ]. [. ( t `  K )  /  x ]. [. ( t `  J )  /  y ]. ph  <->  [. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t  |`  ( 1 ... L
) )  /  a ]. [. ( a `  M )  /  v ]. [. ( a `  L )  /  w ]. [. ( t `  K )  /  x ]. [. ( t `  J )  /  y ]. ph ) )
3734, 35, 36mp2b 10 . . . . . . . . . 10  |-  ( [. ( ( t  |`  ( 1 ... L
) )  |`  (
1 ... N ) )  /  u ]. [. (
t  |`  ( 1 ... L ) )  / 
a ]. [. ( a `
 M )  / 
v ]. [. ( a `
 L )  /  w ]. [. ( t `
 K )  /  x ]. [. ( t `
 J )  / 
y ]. ph  <->  [. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t  |`  ( 1 ... L
) )  /  a ]. [. ( a `  M )  /  v ]. [. ( a `  L )  /  w ]. [. ( t `  K )  /  x ]. [. ( t `  J )  /  y ]. ph )
38 fveq1 5864 . . . . . . . . . . . . 13  |-  ( a  =  ( t  |`  ( 1 ... L
) )  ->  (
a `  M )  =  ( ( t  |`  ( 1 ... L
) ) `  M
) )
3938sbccomieg 35636 . . . . . . . . . . . 12  |-  ( [. ( t  |`  (
1 ... L ) )  /  a ]. [. (
a `  M )  /  v ]. [. (
a `  L )  /  w ]. [. (
t `  K )  /  x ]. [. (
t `  J )  /  y ]. ph  <->  [. ( ( t  |`  ( 1 ... L ) ) `
 M )  / 
v ]. [. ( t  |`  ( 1 ... L
) )  /  a ]. [. ( a `  L )  /  w ]. [. ( t `  K )  /  x ]. [. ( t `  J )  /  y ]. ph )
40 elfz1end 11829 . . . . . . . . . . . . . . . 16  |-  ( M  e.  NN  <->  M  e.  ( 1 ... M
) )
4113, 40sylib 200 . . . . . . . . . . . . . . 15  |-  ( N  e.  NN0  ->  M  e.  ( 1 ... M
) )
4233, 41sseldi 3430 . . . . . . . . . . . . . 14  |-  ( N  e.  NN0  ->  M  e.  ( 1 ... L
) )
43 fvres 5879 . . . . . . . . . . . . . 14  |-  ( M  e.  ( 1 ... L )  ->  (
( t  |`  (
1 ... L ) ) `
 M )  =  ( t `  M
) )
44 dfsbcq 3269 . . . . . . . . . . . . . 14  |-  ( ( ( t  |`  (
1 ... L ) ) `
 M )  =  ( t `  M
)  ->  ( [. ( ( t  |`  ( 1 ... L
) ) `  M
)  /  v ]. [. ( t  |`  (
1 ... L ) )  /  a ]. [. (
a `  L )  /  w ]. [. (
t `  K )  /  x ]. [. (
t `  J )  /  y ]. ph  <->  [. ( t `
 M )  / 
v ]. [. ( t  |`  ( 1 ... L
) )  /  a ]. [. ( a `  L )  /  w ]. [. ( t `  K )  /  x ]. [. ( t `  J )  /  y ]. ph ) )
4542, 43, 443syl 18 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  ( [. ( ( t  |`  ( 1 ... L
) ) `  M
)  /  v ]. [. ( t  |`  (
1 ... L ) )  /  a ]. [. (
a `  L )  /  w ]. [. (
t `  K )  /  x ]. [. (
t `  J )  /  y ]. ph  <->  [. ( t `
 M )  / 
v ]. [. ( t  |`  ( 1 ... L
) )  /  a ]. [. ( a `  L )  /  w ]. [. ( t `  K )  /  x ]. [. ( t `  J )  /  y ]. ph ) )
46 vex 3048 . . . . . . . . . . . . . . . . 17  |-  t  e. 
_V
4746resex 5148 . . . . . . . . . . . . . . . 16  |-  ( t  |`  ( 1 ... L
) )  e.  _V
48 fveq1 5864 . . . . . . . . . . . . . . . . 17  |-  ( a  =  ( t  |`  ( 1 ... L
) )  ->  (
a `  L )  =  ( ( t  |`  ( 1 ... L
) ) `  L
) )
4948sbcco3g 3788 . . . . . . . . . . . . . . . 16  |-  ( ( t  |`  ( 1 ... L ) )  e.  _V  ->  ( [. ( t  |`  (
1 ... L ) )  /  a ]. [. (
a `  L )  /  w ]. [. (
t `  K )  /  x ]. [. (
t `  J )  /  y ]. ph  <->  [. ( ( t  |`  ( 1 ... L ) ) `
 L )  /  w ]. [. ( t `
 K )  /  x ]. [. ( t `
 J )  / 
y ]. ph ) )
5047, 49ax-mp 5 . . . . . . . . . . . . . . 15  |-  ( [. ( t  |`  (
1 ... L ) )  /  a ]. [. (
a `  L )  /  w ]. [. (
t `  K )  /  x ]. [. (
t `  J )  /  y ]. ph  <->  [. ( ( t  |`  ( 1 ... L ) ) `
 L )  /  w ]. [. ( t `
 K )  /  x ]. [. ( t `
 J )  / 
y ]. ph )
51 elfz1end 11829 . . . . . . . . . . . . . . . . 17  |-  ( L  e.  NN  <->  L  e.  ( 1 ... L
) )
5215, 51sylib 200 . . . . . . . . . . . . . . . 16  |-  ( N  e.  NN0  ->  L  e.  ( 1 ... L
) )
53 fvres 5879 . . . . . . . . . . . . . . . 16  |-  ( L  e.  ( 1 ... L )  ->  (
( t  |`  (
1 ... L ) ) `
 L )  =  ( t `  L
) )
54 dfsbcq 3269 . . . . . . . . . . . . . . . 16  |-  ( ( ( t  |`  (
1 ... L ) ) `
 L )  =  ( t `  L
)  ->  ( [. ( ( t  |`  ( 1 ... L
) ) `  L
)  /  w ]. [. ( t `  K
)  /  x ]. [. ( t `  J
)  /  y ]. ph  <->  [. ( t `  L
)  /  w ]. [. ( t `  K
)  /  x ]. [. ( t `  J
)  /  y ]. ph ) )
5552, 53, 543syl 18 . . . . . . . . . . . . . . 15  |-  ( N  e.  NN0  ->  ( [. ( ( t  |`  ( 1 ... L
) ) `  L
)  /  w ]. [. ( t `  K
)  /  x ]. [. ( t `  J
)  /  y ]. ph  <->  [. ( t `  L
)  /  w ]. [. ( t `  K
)  /  x ]. [. ( t `  J
)  /  y ]. ph ) )
5650, 55syl5bb 261 . . . . . . . . . . . . . 14  |-  ( N  e.  NN0  ->  ( [. ( t  |`  (
1 ... L ) )  /  a ]. [. (
a `  L )  /  w ]. [. (
t `  K )  /  x ]. [. (
t `  J )  /  y ]. ph  <->  [. ( t `
 L )  /  w ]. [. ( t `
 K )  /  x ]. [. ( t `
 J )  / 
y ]. ph ) )
5756sbcbidv 3322 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  ( [. ( t `  M
)  /  v ]. [. ( t  |`  (
1 ... L ) )  /  a ]. [. (
a `  L )  /  w ]. [. (
t `  K )  /  x ]. [. (
t `  J )  /  y ]. ph  <->  [. ( t `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. [. ( t `
 K )  /  x ]. [. ( t `
 J )  / 
y ]. ph ) )
5845, 57bitrd 257 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  ( [. ( ( t  |`  ( 1 ... L
) ) `  M
)  /  v ]. [. ( t  |`  (
1 ... L ) )  /  a ]. [. (
a `  L )  /  w ]. [. (
t `  K )  /  x ]. [. (
t `  J )  /  y ]. ph  <->  [. ( t `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. [. ( t `
 K )  /  x ]. [. ( t `
 J )  / 
y ]. ph ) )
5939, 58syl5bb 261 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  ( [. ( t  |`  (
1 ... L ) )  /  a ]. [. (
a `  M )  /  v ]. [. (
a `  L )  /  w ]. [. (
t `  K )  /  x ]. [. (
t `  J )  /  y ]. ph  <->  [. ( t `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. [. ( t `
 K )  /  x ]. [. ( t `
 J )  / 
y ]. ph ) )
6059sbcbidv 3322 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  ( [. ( t  |`  (
1 ... N ) )  /  u ]. [. (
t  |`  ( 1 ... L ) )  / 
a ]. [. ( a `
 M )  / 
v ]. [. ( a `
 L )  /  w ]. [. ( t `
 K )  /  x ]. [. ( t `
 J )  / 
y ]. ph  <->  [. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. [. ( t `  J )  /  y ]. ph ) )
6137, 60syl5bb 261 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( [. ( ( t  |`  ( 1 ... L
) )  |`  (
1 ... N ) )  /  u ]. [. (
t  |`  ( 1 ... L ) )  / 
a ]. [. ( a `
 M )  / 
v ]. [. ( a `
 L )  /  w ]. [. ( t `
 K )  /  x ]. [. ( t `
 J )  / 
y ]. ph  <->  [. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. [. ( t `  J )  /  y ]. ph ) )
6227, 61syl5bb 261 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( [. ( t  |`  (
1 ... L ) )  /  a ]. [. (
a  |`  ( 1 ... N ) )  /  u ]. [. ( a `
 M )  / 
v ]. [. ( a `
 L )  /  w ]. [. ( t `
 K )  /  x ]. [. ( t `
 J )  / 
y ]. ph  <->  [. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. [. ( t `  J )  /  y ]. ph ) )
6325, 62syl5bb 261 . . . . . . 7  |-  ( N  e.  NN0  ->  ( [. ( t  |`  (
1 ... L ) )  /  a ]. [. (
t `  K )  /  x ]. [. (
t `  J )  /  y ]. [. (
a  |`  ( 1 ... N ) )  /  u ]. [. ( a `
 M )  / 
v ]. [. ( a `
 L )  /  w ]. ph  <->  [. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. [. ( t `  J )  /  y ]. ph ) )
6463rabbidv 3036 . . . . . 6  |-  ( N  e.  NN0  ->  { t  e.  ( NN0  ^m  ( 1 ... J
) )  |  [. ( t  |`  (
1 ... L ) )  /  a ]. [. (
t `  K )  /  x ]. [. (
t `  J )  /  y ]. [. (
a  |`  ( 1 ... N ) )  /  u ]. [. ( a `
 M )  / 
v ]. [. ( a `
 L )  /  w ]. ph }  =  { t  e.  ( NN0  ^m  ( 1 ... J ) )  |  [. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. [. ( t `  J )  /  y ]. ph } )
6564eleq1d 2513 . . . . 5  |-  ( N  e.  NN0  ->  ( { t  e.  ( NN0 
^m  ( 1 ... J ) )  | 
[. ( t  |`  ( 1 ... L
) )  /  a ]. [. ( t `  K )  /  x ]. [. ( t `  J )  /  y ]. [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. [. ( a `  L )  /  w ]. ph }  e.  (Dioph `  J )  <->  { t  e.  ( NN0  ^m  (
1 ... J ) )  |  [. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. [. ( t `  J )  /  y ]. ph }  e.  (Dioph `  J ) ) )
6665biimpar 488 . . . 4  |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0 
^m  ( 1 ... J ) )  | 
[. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. [. ( t `  J )  /  y ]. ph }  e.  (Dioph `  J ) )  ->  { t  e.  ( NN0  ^m  ( 1 ... J ) )  |  [. ( t  |`  ( 1 ... L
) )  /  a ]. [. ( t `  K )  /  x ]. [. ( t `  J )  /  y ]. [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. [. ( a `  L )  /  w ]. ph }  e.  (Dioph `  J ) )
67 rexfrabdioph.3 . . . . 5  |-  K  =  ( L  +  1 )
68 rexfrabdioph.4 . . . . 5  |-  J  =  ( K  +  1 )
6967, 682rexfrabdioph 35639 . . . 4  |-  ( ( L  e.  NN0  /\  { t  e.  ( NN0 
^m  ( 1 ... J ) )  | 
[. ( t  |`  ( 1 ... L
) )  /  a ]. [. ( t `  K )  /  x ]. [. ( t `  J )  /  y ]. [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. [. ( a `  L )  /  w ]. ph }  e.  (Dioph `  J ) )  ->  { a  e.  ( NN0  ^m  ( 1 ... L ) )  |  E. x  e. 
NN0  E. y  e.  NN0  [. ( a  |`  (
1 ... N ) )  /  u ]. [. (
a `  M )  /  v ]. [. (
a `  L )  /  w ]. ph }  e.  (Dioph `  L )
)
7017, 66, 69syl2anc 667 . . 3  |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0 
^m  ( 1 ... J ) )  | 
[. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. [. ( t `  J )  /  y ]. ph }  e.  (Dioph `  J ) )  ->  { a  e.  ( NN0  ^m  ( 1 ... L ) )  |  E. x  e. 
NN0  E. y  e.  NN0  [. ( a  |`  (
1 ... N ) )  /  u ]. [. (
a `  M )  /  v ]. [. (
a `  L )  /  w ]. ph }  e.  (Dioph `  L )
)
719, 70syl5eqel 2533 . 2  |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0 
^m  ( 1 ... J ) )  | 
[. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. [. ( t `  J )  /  y ]. ph }  e.  (Dioph `  J ) )  ->  { a  e.  ( NN0  ^m  ( 1 ... L ) )  |  [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. [. ( a `  L )  /  w ]. E. x  e.  NN0  E. y  e.  NN0  ph }  e.  (Dioph `  L )
)
7211, 102rexfrabdioph 35639 . 2  |-  ( ( N  e.  NN0  /\  { a  e.  ( NN0 
^m  ( 1 ... L ) )  | 
[. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. [. ( a `  L )  /  w ]. E. x  e.  NN0  E. y  e.  NN0  ph }  e.  (Dioph `  L )
)  ->  { u  e.  ( NN0  ^m  (
1 ... N ) )  |  E. v  e. 
NN0  E. w  e.  NN0  E. x  e.  NN0  E. y  e.  NN0  ph }  e.  (Dioph `  N ) )
7371, 72syldan 473 1  |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0 
^m  ( 1 ... J ) )  | 
[. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. [. ( t `  J )  /  y ]. ph }  e.  (Dioph `  J ) )  ->  { u  e.  ( NN0  ^m  ( 1 ... N ) )  |  E. v  e.  NN0  E. w  e.  NN0  E. x  e.  NN0  E. y  e. 
NN0  ph }  e.  (Dioph `  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887   E.wrex 2738   {crab 2741   _Vcvv 3045   [.wsbc 3267    C_ wss 3404    |` cres 4836   ` cfv 5582  (class class class)co 6290    ^m cmap 7472   1c1 9540    + caddc 9542   NNcn 10609   NN0cn0 10869   ...cfz 11784  Diophcdioph 35597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-hash 12516  df-mzpcl 35565  df-mzp 35566  df-dioph 35598
This theorem is referenced by:  6rexfrabdioph  35642
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