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Theorem iseqovex 9866
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 2089 . . 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 499 . . . 4  |-  ( ( ( ph  /\  (
x  e.  ( ZZ>= `  M )  /\  y  e.  S ) )  /\  ( z  =  x  /\  w  =  y ) )  ->  w  =  y )
3 simprl 498 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( ZZ>= `  M )  /\  y  e.  S ) )  /\  ( z  =  x  /\  w  =  y ) )  ->  z  =  x )
43oveq1d 5667 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  ( ZZ>= `  M )  /\  y  e.  S ) )  /\  ( z  =  x  /\  w  =  y ) )  ->  (
z  +  1 )  =  ( x  + 
1 ) )
54fveq2d 5309 . . . 4  |-  ( ( ( ph  /\  (
x  e.  ( ZZ>= `  M )  /\  y  e.  S ) )  /\  ( z  =  x  /\  w  =  y ) )  ->  ( F `  ( z  +  1 ) )  =  ( F `  ( x  +  1
) ) )
62, 5oveq12d 5670 . . 3  |-  ( ( ( ph  /\  (
x  e.  ( ZZ>= `  M )  /\  y  e.  S ) )  /\  ( z  =  x  /\  w  =  y ) )  ->  (
w  .+  ( F `  ( z  +  1 ) ) )  =  ( y  .+  ( F `  ( x  +  1 ) ) ) )
7 simprl 498 . . 3  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  M )  /\  y  e.  S
) )  ->  x  e.  ( ZZ>= `  M )
)
8 simprr 499 . . 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 5797 . . . . 5  |-  ( (
ph  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( z  .+  w
)  e.  S )
1110adantlr 461 . . . 4  |-  ( ( ( ph  /\  (
x  e.  ( ZZ>= `  M )  /\  y  e.  S ) )  /\  ( z  e.  S  /\  w  e.  S
) )  ->  (
z  .+  w )  e.  S )
12 fveq2 5305 . . . . . 6  |-  ( z  =  ( x  + 
1 )  ->  ( F `  z )  =  ( F `  ( x  +  1
) ) )
1312eleq1d 2156 . . . . 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 2446 . . . . . . 7  |-  ( ph  ->  A. x  e.  (
ZZ>= `  M ) ( F `  x )  e.  S )
16 fveq2 5305 . . . . . . . . 9  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
1716eleq1d 2156 . . . . . . . 8  |-  ( x  =  z  ->  (
( F `  x
)  e.  S  <->  ( F `  z )  e.  S
) )
1817cbvralv 2590 . . . . . . 7  |-  ( A. x  e.  ( ZZ>= `  M ) ( F `
 x )  e.  S  <->  A. z  e.  (
ZZ>= `  M ) ( F `  z )  e.  S )
1915, 18sylib 120 . . . . . 6  |-  ( ph  ->  A. z  e.  (
ZZ>= `  M ) ( F `  z )  e.  S )
2019adantr 270 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  M )  /\  y  e.  S
) )  ->  A. z  e.  ( ZZ>= `  M )
( F `  z
)  e.  S )
21 peano2uz 9069 . . . . . 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 2727 . . . 4  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  M )  /\  y  e.  S
) )  ->  ( F `  ( x  +  1 ) )  e.  S )
2411, 8, 23caovcld 5798 . . 3  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  M )  /\  y  e.  S
) )  ->  (
y  .+  ( F `  ( x  +  1 ) ) )  e.  S )
251, 6, 7, 8, 24ovmpt2d 5772 . 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 2164 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 102    = wceq 1289    e. wcel 1438   A.wral 2359   ` cfv 5015  (class class class)co 5652    |-> cmpt2 5654   1c1 7349    + caddc 7351   ZZ>=cuz 9017
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7434  ax-resscn 7435  ax-1cn 7436  ax-1re 7437  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-addcom 7443  ax-addass 7445  ax-distr 7447  ax-i2m1 7448  ax-0lt1 7449  ax-0id 7451  ax-rnegex 7452  ax-cnre 7454  ax-pre-ltirr 7455  ax-pre-ltwlin 7456  ax-pre-lttrn 7457  ax-pre-ltadd 7459
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525  df-le 7526  df-sub 7653  df-neg 7654  df-inn 8421  df-n0 8672  df-z 8749  df-uz 9018
This theorem is referenced by:  iseqvalt  9869  seq3val  9870  iseq1  9871  iseq1t  9872  seq3-1  9873  iseqfcl  9874  iseqcl  9877  iseqp1  9878  iseqp1t  9879  seq3p1  9880
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