ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  iseqovex Unicode version

Theorem iseqovex 10222
Description: Closure of a function used in proving sequence builder theorems. This can be thought of as a lemma for the small number of sequence builder theorems which need it. (Contributed by Jim Kingdon, 31-May-2020.)
Hypotheses
Ref Expression
iseqovex.f  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
iseqovex.pl  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
Assertion
Ref Expression
iseqovex  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  M )  /\  y  e.  S
) )  ->  (
x ( z  e.  ( ZZ>= `  M ) ,  w  e.  S  |->  ( w  .+  ( F `  ( z  +  1 ) ) ) ) y )  e.  S )
Distinct variable groups:    w, F, x, y, z    w,  .+ , x, y, z    w, S, x, y, z    ph, w, x, y, z    w, M, x, z
Allowed substitution hint:    M( y)

Proof of Theorem iseqovex
StepHypRef Expression
1 eqidd 2138 . . 3  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  M )  /\  y  e.  S
) )  ->  (
z  e.  ( ZZ>= `  M ) ,  w  e.  S  |->  ( w 
.+  ( F `  ( z  +  1 ) ) ) )  =  ( z  e.  ( ZZ>= `  M ) ,  w  e.  S  |->  ( w  .+  ( F `  ( z  +  1 ) ) ) ) )
2 simprr 521 . . . 4  |-  ( ( ( ph  /\  (
x  e.  ( ZZ>= `  M )  /\  y  e.  S ) )  /\  ( z  =  x  /\  w  =  y ) )  ->  w  =  y )
3 simprl 520 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( ZZ>= `  M )  /\  y  e.  S ) )  /\  ( z  =  x  /\  w  =  y ) )  ->  z  =  x )
43oveq1d 5782 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  ( ZZ>= `  M )  /\  y  e.  S ) )  /\  ( z  =  x  /\  w  =  y ) )  ->  (
z  +  1 )  =  ( x  + 
1 ) )
54fveq2d 5418 . . . 4  |-  ( ( ( ph  /\  (
x  e.  ( ZZ>= `  M )  /\  y  e.  S ) )  /\  ( z  =  x  /\  w  =  y ) )  ->  ( F `  ( z  +  1 ) )  =  ( F `  ( x  +  1
) ) )
62, 5oveq12d 5785 . . 3  |-  ( ( ( ph  /\  (
x  e.  ( ZZ>= `  M )  /\  y  e.  S ) )  /\  ( z  =  x  /\  w  =  y ) )  ->  (
w  .+  ( F `  ( z  +  1 ) ) )  =  ( y  .+  ( F `  ( x  +  1 ) ) ) )
7 simprl 520 . . 3  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  M )  /\  y  e.  S
) )  ->  x  e.  ( ZZ>= `  M )
)
8 simprr 521 . . 3  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  M )  /\  y  e.  S
) )  ->  y  e.  S )
9 iseqovex.pl . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
109caovclg 5916 . . . . 5  |-  ( (
ph  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( z  .+  w
)  e.  S )
1110adantlr 468 . . . 4  |-  ( ( ( ph  /\  (
x  e.  ( ZZ>= `  M )  /\  y  e.  S ) )  /\  ( z  e.  S  /\  w  e.  S
) )  ->  (
z  .+  w )  e.  S )
12 fveq2 5414 . . . . . 6  |-  ( z  =  ( x  + 
1 )  ->  ( F `  z )  =  ( F `  ( x  +  1
) ) )
1312eleq1d 2206 . . . . 5  |-  ( z  =  ( x  + 
1 )  ->  (
( F `  z
)  e.  S  <->  ( F `  ( x  +  1 ) )  e.  S
) )
14 iseqovex.f . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
1514ralrimiva 2503 . . . . . . 7  |-  ( ph  ->  A. x  e.  (
ZZ>= `  M ) ( F `  x )  e.  S )
16 fveq2 5414 . . . . . . . . 9  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
1716eleq1d 2206 . . . . . . . 8  |-  ( x  =  z  ->  (
( F `  x
)  e.  S  <->  ( F `  z )  e.  S
) )
1817cbvralv 2652 . . . . . . 7  |-  ( A. x  e.  ( ZZ>= `  M ) ( F `
 x )  e.  S  <->  A. z  e.  (
ZZ>= `  M ) ( F `  z )  e.  S )
1915, 18sylib 121 . . . . . 6  |-  ( ph  ->  A. z  e.  (
ZZ>= `  M ) ( F `  z )  e.  S )
2019adantr 274 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  M )  /\  y  e.  S
) )  ->  A. z  e.  ( ZZ>= `  M )
( F `  z
)  e.  S )
21 peano2uz 9371 . . . . . 6  |-  ( x  e.  ( ZZ>= `  M
)  ->  ( x  +  1 )  e.  ( ZZ>= `  M )
)
227, 21syl 14 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  M )  /\  y  e.  S
) )  ->  (
x  +  1 )  e.  ( ZZ>= `  M
) )
2313, 20, 22rspcdva 2789 . . . 4  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  M )  /\  y  e.  S
) )  ->  ( F `  ( x  +  1 ) )  e.  S )
2411, 8, 23caovcld 5917 . . 3  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  M )  /\  y  e.  S
) )  ->  (
y  .+  ( F `  ( x  +  1 ) ) )  e.  S )
251, 6, 7, 8, 24ovmpod 5891 . 2  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  M )  /\  y  e.  S
) )  ->  (
x ( z  e.  ( ZZ>= `  M ) ,  w  e.  S  |->  ( w  .+  ( F `  ( z  +  1 ) ) ) ) y )  =  ( y  .+  ( F `  ( x  +  1 ) ) ) )
2625, 24eqeltrd 2214 1  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  M )  /\  y  e.  S
) )  ->  (
x ( z  e.  ( ZZ>= `  M ) ,  w  e.  S  |->  ( w  .+  ( F `  ( z  +  1 ) ) ) ) y )  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   A.wral 2414   ` cfv 5118  (class class class)co 5767    e. cmpo 5769   1c1 7614    + caddc 7616   ZZ>=cuz 9319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-0id 7721  ax-rnegex 7722  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-ltadd 7729
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-inn 8714  df-n0 8971  df-z 9048  df-uz 9320
This theorem is referenced by:  seq3val  10224  seq3-1  10226  seq3p1  10228
  Copyright terms: Public domain W3C validator