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Theorem iseqovex 9529
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 2083 . . 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 5558 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  ( ZZ>= `  M )  /\  y  e.  S ) )  /\  ( z  =  x  /\  w  =  y ) )  ->  (
z  +  1 )  =  ( x  + 
1 ) )
54fveq2d 5213 . . . 4  |-  ( ( ( ph  /\  (
x  e.  ( ZZ>= `  M )  /\  y  e.  S ) )  /\  ( z  =  x  /\  w  =  y ) )  ->  ( F `  ( z  +  1 ) )  =  ( F `  ( x  +  1
) ) )
62, 5oveq12d 5561 . . 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 5684 . . . . 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 peano2uz 8752 . . . . . 6  |-  ( x  e.  ( ZZ>= `  M
)  ->  ( x  +  1 )  e.  ( ZZ>= `  M )
)
137, 12syl 14 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  M )  /\  y  e.  S
) )  ->  (
x  +  1 )  e.  ( ZZ>= `  M
) )
14 iseqovex.f . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
1514ralrimiva 2435 . . . . . . 7  |-  ( ph  ->  A. x  e.  (
ZZ>= `  M ) ( F `  x )  e.  S )
16 fveq2 5209 . . . . . . . . 9  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
1716eleq1d 2148 . . . . . . . 8  |-  ( x  =  z  ->  (
( F `  x
)  e.  S  <->  ( F `  z )  e.  S
) )
1817cbvralv 2578 . . . . . . 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 fveq2 5209 . . . . . . 7  |-  ( z  =  ( x  + 
1 )  ->  ( F `  z )  =  ( F `  ( x  +  1
) ) )
2221eleq1d 2148 . . . . . 6  |-  ( z  =  ( x  + 
1 )  ->  (
( F `  z
)  e.  S  <->  ( F `  ( x  +  1 ) )  e.  S
) )
2322rspcv 2698 . . . . 5  |-  ( ( x  +  1 )  e.  ( ZZ>= `  M
)  ->  ( A. z  e.  ( ZZ>= `  M ) ( F `
 z )  e.  S  ->  ( F `  ( x  +  1 ) )  e.  S
) )
2413, 20, 23sylc 61 . . . 4  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  M )  /\  y  e.  S
) )  ->  ( F `  ( x  +  1 ) )  e.  S )
2511, 8, 24caovcld 5685 . . 3  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  M )  /\  y  e.  S
) )  ->  (
y  .+  ( F `  ( x  +  1 ) ) )  e.  S )
261, 6, 7, 8, 25ovmpt2d 5659 . 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 ) ) ) )
2726, 25eqeltrd 2156 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 1285    e. wcel 1434   A.wral 2349   ` cfv 4932  (class class class)co 5543    |-> cmpt2 5545   1c1 7044    + caddc 7046   ZZ>=cuz 8700
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-addass 7140  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-ltadd 7154
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-inn 8107  df-n0 8356  df-z 8433  df-uz 8701
This theorem is referenced by:  iseqvalt  9532  iseq1  9533  iseq1t  9534  iseqfcl  9535  iseqcl  9537  iseqp1  9538  iseqp1t  9539
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