ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  iseqp1t Unicode version

Theorem iseqp1t 9609
Description: Value of the sequence builder function at a successor. (Contributed by Jim Kingdon, 30-Apr-2022.)
Hypotheses
Ref Expression
iseqp1t.m  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
iseqp1t.f  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
iseqp1t.pl  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
iseqp1t.t  |-  ( ph  ->  S  C_  T )
Assertion
Ref Expression
iseqp1t  |-  ( ph  ->  (  seq M ( 
.+  ,  F ,  T ) `  ( N  +  1 ) )  =  ( (  seq M (  .+  ,  F ,  T ) `
 N )  .+  ( F `  ( N  +  1 ) ) ) )
Distinct variable groups:    x,  .+ , y    x, F, y    x, M, y    x, N, y   
x, S, y    x, T, y    ph, x, y

Proof of Theorem iseqp1t
Dummy variables  a  b  w  z  c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iseqp1t.m . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
2 eluzel2 8775 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
31, 2syl 14 . . . 4  |-  ( ph  ->  M  e.  ZZ )
4 fveq2 5230 . . . . . 6  |-  ( x  =  M  ->  ( F `  x )  =  ( F `  M ) )
54eleq1d 2151 . . . . 5  |-  ( x  =  M  ->  (
( F `  x
)  e.  S  <->  ( F `  M )  e.  S
) )
6 iseqp1t.f . . . . . 6  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
76ralrimiva 2439 . . . . 5  |-  ( ph  ->  A. x  e.  (
ZZ>= `  M ) ( F `  x )  e.  S )
8 uzid 8784 . . . . . 6  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
93, 8syl 14 . . . . 5  |-  ( ph  ->  M  e.  ( ZZ>= `  M ) )
105, 7, 9rspcdva 2715 . . . 4  |-  ( ph  ->  ( F `  M
)  e.  S )
11 iseqp1t.t . . . 4  |-  ( ph  ->  S  C_  T )
12 iseqp1t.pl . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
136, 12iseqovex 9599 . . . 4  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  M )  /\  y  e.  S
) )  ->  (
x ( z  e.  ( ZZ>= `  M ) ,  w  e.  S  |->  ( w  .+  ( F `  ( z  +  1 ) ) ) ) y )  e.  S )
14 iseqvalcbv 9601 . . . 4  |- frec ( ( a  e.  ( ZZ>= `  M ) ,  b  e.  T  |->  <. (
a  +  1 ) ,  ( a ( c  e.  ( ZZ>= `  M ) ,  d  e.  S  |->  ( d 
.+  ( F `  ( c  +  1 ) ) ) ) b ) >. ) ,  <. M ,  ( F `  M )
>. )  = frec (
( x  e.  (
ZZ>= `  M ) ,  y  e.  T  |->  <.
( x  +  1 ) ,  ( x ( z  e.  (
ZZ>= `  M ) ,  w  e.  S  |->  ( w  .+  ( F `
 ( z  +  1 ) ) ) ) y ) >.
) ,  <. M , 
( F `  M
) >. )
153, 14, 6, 12, 11iseqvalt 9602 . . . 4  |-  ( ph  ->  seq M (  .+  ,  F ,  T )  =  ran frec ( (
a  e.  ( ZZ>= `  M ) ,  b  e.  T  |->  <. (
a  +  1 ) ,  ( a ( c  e.  ( ZZ>= `  M ) ,  d  e.  S  |->  ( d 
.+  ( F `  ( c  +  1 ) ) ) ) b ) >. ) ,  <. M ,  ( F `  M )
>. ) )
163, 10, 11, 13, 14, 15frecuzrdgsuct 9576 . . 3  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  (  seq M (  .+  ,  F ,  T ) `  ( N  +  1 ) )  =  ( N ( z  e.  ( ZZ>= `  M ) ,  w  e.  S  |->  ( w  .+  ( F `  ( z  +  1 ) ) ) ) (  seq M (  .+  ,  F ,  T ) `  N ) ) )
171, 16mpdan 412 . 2  |-  ( ph  ->  (  seq M ( 
.+  ,  F ,  T ) `  ( N  +  1 ) )  =  ( N ( z  e.  (
ZZ>= `  M ) ,  w  e.  S  |->  ( w  .+  ( F `
 ( z  +  1 ) ) ) ) (  seq M
(  .+  ,  F ,  T ) `  N
) ) )
18 eqid 2083 . . . . 5  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
1918, 3, 6, 12, 11iseqfclt 9606 . . . 4  |-  ( ph  ->  seq M (  .+  ,  F ,  T ) : ( ZZ>= `  M
) --> S )
2019, 1ffvelrnd 5356 . . 3  |-  ( ph  ->  (  seq M ( 
.+  ,  F ,  T ) `  N
)  e.  S )
21 fveq2 5230 . . . . . 6  |-  ( x  =  ( N  + 
1 )  ->  ( F `  x )  =  ( F `  ( N  +  1
) ) )
2221eleq1d 2151 . . . . 5  |-  ( x  =  ( N  + 
1 )  ->  (
( F `  x
)  e.  S  <->  ( F `  ( N  +  1 ) )  e.  S
) )
23 peano2uz 8822 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  1 )  e.  ( ZZ>= `  M )
)
241, 23syl 14 . . . . 5  |-  ( ph  ->  ( N  +  1 )  e.  ( ZZ>= `  M ) )
2522, 7, 24rspcdva 2715 . . . 4  |-  ( ph  ->  ( F `  ( N  +  1 ) )  e.  S )
2612, 20, 25caovcld 5706 . . 3  |-  ( ph  ->  ( (  seq M
(  .+  ,  F ,  T ) `  N
)  .+  ( F `  ( N  +  1 ) ) )  e.  S )
27 oveq1 5571 . . . . . 6  |-  ( z  =  N  ->  (
z  +  1 )  =  ( N  + 
1 ) )
2827fveq2d 5234 . . . . 5  |-  ( z  =  N  ->  ( F `  ( z  +  1 ) )  =  ( F `  ( N  +  1
) ) )
2928oveq2d 5580 . . . 4  |-  ( z  =  N  ->  (
w  .+  ( F `  ( z  +  1 ) ) )  =  ( w  .+  ( F `  ( N  +  1 ) ) ) )
30 oveq1 5571 . . . 4  |-  ( w  =  (  seq M
(  .+  ,  F ,  T ) `  N
)  ->  ( w  .+  ( F `  ( N  +  1 ) ) )  =  ( (  seq M ( 
.+  ,  F ,  T ) `  N
)  .+  ( F `  ( N  +  1 ) ) ) )
31 eqid 2083 . . . 4  |-  ( z  e.  ( ZZ>= `  M
) ,  w  e.  S  |->  ( w  .+  ( F `  ( z  +  1 ) ) ) )  =  ( z  e.  ( ZZ>= `  M ) ,  w  e.  S  |->  ( w 
.+  ( F `  ( z  +  1 ) ) ) )
3229, 30, 31ovmpt2g 5687 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  (  seq M (  .+  ,  F ,  T ) `  N )  e.  S  /\  ( (  seq M
(  .+  ,  F ,  T ) `  N
)  .+  ( F `  ( N  +  1 ) ) )  e.  S )  ->  ( N ( z  e.  ( ZZ>= `  M ) ,  w  e.  S  |->  ( w  .+  ( F `  ( z  +  1 ) ) ) ) (  seq M (  .+  ,  F ,  T ) `  N ) )  =  ( (  seq M
(  .+  ,  F ,  T ) `  N
)  .+  ( F `  ( N  +  1 ) ) ) )
331, 20, 26, 32syl3anc 1170 . 2  |-  ( ph  ->  ( N ( z  e.  ( ZZ>= `  M
) ,  w  e.  S  |->  ( w  .+  ( F `  ( z  +  1 ) ) ) ) (  seq M (  .+  ,  F ,  T ) `  N ) )  =  ( (  seq M
(  .+  ,  F ,  T ) `  N
)  .+  ( F `  ( N  +  1 ) ) ) )
3417, 33eqtrd 2115 1  |-  ( ph  ->  (  seq M ( 
.+  ,  F ,  T ) `  ( N  +  1 ) )  =  ( (  seq M (  .+  ,  F ,  T ) `
 N )  .+  ( F `  ( N  +  1 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434    C_ wss 2982   <.cop 3419   ` cfv 4952  (class class class)co 5564    |-> cmpt2 5566  freccfrec 6060   1c1 7114    + caddc 7116   ZZcz 8502   ZZ>=cuz 8770    seqcseq 9591
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 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7199  ax-resscn 7200  ax-1cn 7201  ax-1re 7202  ax-icn 7203  ax-addcl 7204  ax-addrcl 7205  ax-mulcl 7206  ax-addcom 7208  ax-addass 7210  ax-distr 7212  ax-i2m1 7213  ax-0lt1 7214  ax-0id 7216  ax-rnegex 7217  ax-cnre 7219  ax-pre-ltirr 7220  ax-pre-ltwlin 7221  ax-pre-lttrn 7222  ax-pre-ltadd 7224
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 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5520  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-frec 6061  df-pnf 7287  df-mnf 7288  df-xr 7289  df-ltxr 7290  df-le 7291  df-sub 7418  df-neg 7419  df-inn 8177  df-n0 8426  df-z 8503  df-uz 8771  df-iseq 9592
This theorem is referenced by:  iseqsst  9612
  Copyright terms: Public domain W3C validator