ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  seq3p1 Unicode version

Theorem seq3p1 9872
Description: Value of the sequence builder function at a successor. (Contributed by Jim Kingdon, 30-Apr-2022.)
Hypotheses
Ref Expression
seq3p1.m  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
seq3p1.f  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
seq3p1.pl  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
Assertion
Ref Expression
seq3p1  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 ( N  + 
1 ) )  =  ( (  seq M
(  .+  ,  F
) `  N )  .+  ( F `  ( N  +  1 ) ) ) )
Distinct variable groups:    x,  .+ , y    x, F, y    x, M, y    x, N, y   
x, S, y    ph, x, y

Proof of Theorem seq3p1
Dummy variables  a  b  w  z  c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 seq3p1.m . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
2 eluzel2 9014 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
31, 2syl 14 . . . 4  |-  ( ph  ->  M  e.  ZZ )
4 fveq2 5299 . . . . . 6  |-  ( x  =  M  ->  ( F `  x )  =  ( F `  M ) )
54eleq1d 2156 . . . . 5  |-  ( x  =  M  ->  (
( F `  x
)  e.  S  <->  ( F `  M )  e.  S
) )
6 seq3p1.f . . . . . 6  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
76ralrimiva 2446 . . . . 5  |-  ( ph  ->  A. x  e.  (
ZZ>= `  M ) ( F `  x )  e.  S )
8 uzid 9023 . . . . . 6  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
93, 8syl 14 . . . . 5  |-  ( ph  ->  M  e.  ( ZZ>= `  M ) )
105, 7, 9rspcdva 2727 . . . 4  |-  ( ph  ->  ( F `  M
)  e.  S )
11 ssv 3046 . . . . 5  |-  S  C_  _V
1211a1i 9 . . . 4  |-  ( ph  ->  S  C_  _V )
13 seq3p1.pl . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
146, 13iseqovex 9858 . . . 4  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  M )  /\  y  e.  S
) )  ->  (
x ( z  e.  ( ZZ>= `  M ) ,  w  e.  S  |->  ( w  .+  ( F `  ( z  +  1 ) ) ) ) y )  e.  S )
15 iseqvalcbv 9860 . . . 4  |- frec ( ( a  e.  ( ZZ>= `  M ) ,  b  e.  _V  |->  <. (
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.  _V  |->  <.
( x  +  1 ) ,  ( x ( z  e.  (
ZZ>= `  M ) ,  w  e.  S  |->  ( w  .+  ( F `
 ( z  +  1 ) ) ) ) y ) >.
) ,  <. M , 
( F `  M
) >. )
163, 15, 6, 13seq3val 9862 . . . 4  |-  ( ph  ->  seq M (  .+  ,  F )  =  ran frec ( ( a  e.  (
ZZ>= `  M ) ,  b  e.  _V  |->  <.
( a  +  1 ) ,  ( a ( c  e.  (
ZZ>= `  M ) ,  d  e.  S  |->  ( d  .+  ( F `
 ( c  +  1 ) ) ) ) b ) >.
) ,  <. M , 
( F `  M
) >. ) )
173, 10, 12, 14, 15, 16frecuzrdgsuct 9819 . . 3  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  (  seq M (  .+  ,  F ) `  ( N  +  1 ) )  =  ( N ( z  e.  (
ZZ>= `  M ) ,  w  e.  S  |->  ( w  .+  ( F `
 ( z  +  1 ) ) ) ) (  seq M
(  .+  ,  F
) `  N )
) )
181, 17mpdan 412 . 2  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 ( N  + 
1 ) )  =  ( N ( z  e.  ( ZZ>= `  M
) ,  w  e.  S  |->  ( w  .+  ( F `  ( z  +  1 ) ) ) ) (  seq M (  .+  ,  F ) `  N
) ) )
19 eqid 2088 . . . . 5  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
2019, 3, 6, 13seqf 9868 . . . 4  |-  ( ph  ->  seq M (  .+  ,  F ) : (
ZZ>= `  M ) --> S )
2120, 1ffvelrnd 5429 . . 3  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 N )  e.  S )
22 fveq2 5299 . . . . . 6  |-  ( x  =  ( N  + 
1 )  ->  ( F `  x )  =  ( F `  ( N  +  1
) ) )
2322eleq1d 2156 . . . . 5  |-  ( x  =  ( N  + 
1 )  ->  (
( F `  x
)  e.  S  <->  ( F `  ( N  +  1 ) )  e.  S
) )
24 peano2uz 9061 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  1 )  e.  ( ZZ>= `  M )
)
251, 24syl 14 . . . . 5  |-  ( ph  ->  ( N  +  1 )  e.  ( ZZ>= `  M ) )
2623, 7, 25rspcdva 2727 . . . 4  |-  ( ph  ->  ( F `  ( N  +  1 ) )  e.  S )
2713, 21, 26caovcld 5790 . . 3  |-  ( ph  ->  ( (  seq M
(  .+  ,  F
) `  N )  .+  ( F `  ( N  +  1 ) ) )  e.  S
)
28 fvoveq1 5667 . . . . 5  |-  ( z  =  N  ->  ( F `  ( z  +  1 ) )  =  ( F `  ( N  +  1
) ) )
2928oveq2d 5660 . . . 4  |-  ( z  =  N  ->  (
w  .+  ( F `  ( z  +  1 ) ) )  =  ( w  .+  ( F `  ( N  +  1 ) ) ) )
30 oveq1 5651 . . . 4  |-  ( w  =  (  seq M
(  .+  ,  F
) `  N )  ->  ( w  .+  ( F `  ( N  +  1 ) ) )  =  ( (  seq M (  .+  ,  F ) `  N
)  .+  ( F `  ( N  +  1 ) ) ) )
31 eqid 2088 . . . 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 5771 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  (  seq M (  .+  ,  F ) `  N
)  e.  S  /\  ( (  seq M
(  .+  ,  F
) `  N )  .+  ( F `  ( N  +  1 ) ) )  e.  S
)  ->  ( N
( z  e.  (
ZZ>= `  M ) ,  w  e.  S  |->  ( w  .+  ( F `
 ( z  +  1 ) ) ) ) (  seq M
(  .+  ,  F
) `  N )
)  =  ( (  seq M (  .+  ,  F ) `  N
)  .+  ( F `  ( N  +  1 ) ) ) )
331, 21, 27, 32syl3anc 1174 . 2  |-  ( ph  ->  ( N ( z  e.  ( ZZ>= `  M
) ,  w  e.  S  |->  ( w  .+  ( F `  ( z  +  1 ) ) ) ) (  seq M (  .+  ,  F ) `  N
) )  =  ( (  seq M ( 
.+  ,  F ) `
 N )  .+  ( F `  ( N  +  1 ) ) ) )
3418, 33eqtrd 2120 1  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 ( N  + 
1 ) )  =  ( (  seq M
(  .+  ,  F
) `  N )  .+  ( F `  ( N  +  1 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   _Vcvv 2619    C_ wss 2999   <.cop 3447   ` cfv 5010  (class class class)co 5644    |-> cmpt2 5646  freccfrec 6147   1c1 7341    + caddc 7343   ZZcz 8740   ZZ>=cuz 9009    seqcseq 9840
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-coll 3952  ax-sep 3955  ax-nul 3963  ax-pow 4007  ax-pr 4034  ax-un 4258  ax-setind 4351  ax-iinf 4401  ax-cnex 7426  ax-resscn 7427  ax-1cn 7428  ax-1re 7429  ax-icn 7430  ax-addcl 7431  ax-addrcl 7432  ax-mulcl 7433  ax-addcom 7435  ax-addass 7437  ax-distr 7439  ax-i2m1 7440  ax-0lt1 7441  ax-0id 7443  ax-rnegex 7444  ax-cnre 7446  ax-pre-ltirr 7447  ax-pre-ltwlin 7448  ax-pre-lttrn 7449  ax-pre-ltadd 7451
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-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-int 3687  df-iun 3730  df-br 3844  df-opab 3898  df-mpt 3899  df-tr 3935  df-id 4118  df-iord 4191  df-on 4193  df-ilim 4194  df-suc 4196  df-iom 4404  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-f1 5015  df-fo 5016  df-f1o 5017  df-fv 5018  df-riota 5600  df-ov 5647  df-oprab 5648  df-mpt2 5649  df-1st 5903  df-2nd 5904  df-recs 6062  df-frec 6148  df-pnf 7514  df-mnf 7515  df-xr 7516  df-ltxr 7517  df-le 7518  df-sub 7645  df-neg 7646  df-inn 8413  df-n0 8664  df-z 8741  df-uz 9010  df-iseq 9841  df-seq3 9842
This theorem is referenced by:  seq3clss  9875  seq3m1  9877  seq3fveq2  9880  seq3split  9895  seq3id2  9928  seq3homo  9932  ser3ge0  9940  exp3vallem  9944  expp1  9950  resqrexlemfp1  10430  climserle  10721  ege2le3  10948  efgt1p2  10972  efgt1p  10973
  Copyright terms: Public domain W3C validator