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Theorem iseqss 9226
Description: Specifying a larger universe for  seq. As long as  F and  .+ are closed over  S, then any set which contains  S can be used as the last argument to  seq. This theorem does not allow  T to be a proper class, however. It also currently requires that  .+ be closed over  T (as well as  S). (Contributed by Jim Kingdon, 18-Aug-2021.)
Hypotheses
Ref Expression
iseqss.m  |-  ( ph  ->  M  e.  ZZ )
iseqss.t  |-  ( ph  ->  T  e.  V )
iseqss.ss  |-  ( ph  ->  S  C_  T )
iseqss.f  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
iseqss.pl  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
iseqss.plt  |-  ( (
ph  /\  ( x  e.  T  /\  y  e.  T ) )  -> 
( x  .+  y
)  e.  T )
Assertion
Ref Expression
iseqss  |-  ( ph  ->  seq M (  .+  ,  F ,  S )  =  seq M ( 
.+  ,  F ,  T ) )
Distinct variable groups:    x,  .+ , y    x, F, y    x, M, y    x, S, y   
x, T, y    ph, x, y
Allowed substitution hints:    V( x, y)

Proof of Theorem iseqss
Dummy variables  k  n  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iseqss.m . . 3  |-  ( ph  ->  M  e.  ZZ )
2 iseqss.t . . . 4  |-  ( ph  ->  T  e.  V )
3 iseqss.ss . . . 4  |-  ( ph  ->  S  C_  T )
42, 3ssexd 3897 . . 3  |-  ( ph  ->  S  e.  _V )
5 iseqss.f . . 3  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
6 iseqss.pl . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
71, 4, 5, 6iseqfn 9221 . 2  |-  ( ph  ->  seq M (  .+  ,  F ,  S )  Fn  ( ZZ>= `  M
) )
83sseld 2944 . . . . 5  |-  ( ph  ->  ( ( F `  x )  e.  S  ->  ( F `  x
)  e.  T ) )
98adantr 261 . . . 4  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( ( F `  x )  e.  S  ->  ( F `
 x )  e.  T ) )
105, 9mpd 13 . . 3  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  T
)
11 iseqss.plt . . 3  |-  ( (
ph  /\  ( x  e.  T  /\  y  e.  T ) )  -> 
( x  .+  y
)  e.  T )
121, 2, 10, 11iseqfn 9221 . 2  |-  ( ph  ->  seq M (  .+  ,  F ,  T )  Fn  ( ZZ>= `  M
) )
13 fveq2 5178 . . . . . 6  |-  ( w  =  M  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq M (  .+  ,  F ,  S ) `
 M ) )
14 fveq2 5178 . . . . . 6  |-  ( w  =  M  ->  (  seq M (  .+  ,  F ,  T ) `  w )  =  (  seq M (  .+  ,  F ,  T ) `
 M ) )
1513, 14eqeq12d 2054 . . . . 5  |-  ( w  =  M  ->  (
(  seq M (  .+  ,  F ,  S ) `
 w )  =  (  seq M ( 
.+  ,  F ,  T ) `  w
)  <->  (  seq M
(  .+  ,  F ,  S ) `  M
)  =  (  seq M (  .+  ,  F ,  T ) `  M ) ) )
1615imbi2d 219 . . . 4  |-  ( w  =  M  ->  (
( ph  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq M (  .+  ,  F ,  T ) `
 w ) )  <-> 
( ph  ->  (  seq M (  .+  ,  F ,  S ) `  M )  =  (  seq M (  .+  ,  F ,  T ) `
 M ) ) ) )
17 fveq2 5178 . . . . . 6  |-  ( w  =  k  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq M (  .+  ,  F ,  S ) `
 k ) )
18 fveq2 5178 . . . . . 6  |-  ( w  =  k  ->  (  seq M (  .+  ,  F ,  T ) `  w )  =  (  seq M (  .+  ,  F ,  T ) `
 k ) )
1917, 18eqeq12d 2054 . . . . 5  |-  ( w  =  k  ->  (
(  seq M (  .+  ,  F ,  S ) `
 w )  =  (  seq M ( 
.+  ,  F ,  T ) `  w
)  <->  (  seq M
(  .+  ,  F ,  S ) `  k
)  =  (  seq M (  .+  ,  F ,  T ) `  k ) ) )
2019imbi2d 219 . . . 4  |-  ( w  =  k  ->  (
( ph  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq M (  .+  ,  F ,  T ) `
 w ) )  <-> 
( ph  ->  (  seq M (  .+  ,  F ,  S ) `  k )  =  (  seq M (  .+  ,  F ,  T ) `
 k ) ) ) )
21 fveq2 5178 . . . . . 6  |-  ( w  =  ( k  +  1 )  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq M (  .+  ,  F ,  S ) `
 ( k  +  1 ) ) )
22 fveq2 5178 . . . . . 6  |-  ( w  =  ( k  +  1 )  ->  (  seq M (  .+  ,  F ,  T ) `  w )  =  (  seq M (  .+  ,  F ,  T ) `
 ( k  +  1 ) ) )
2321, 22eqeq12d 2054 . . . . 5  |-  ( w  =  ( k  +  1 )  ->  (
(  seq M (  .+  ,  F ,  S ) `
 w )  =  (  seq M ( 
.+  ,  F ,  T ) `  w
)  <->  (  seq M
(  .+  ,  F ,  S ) `  (
k  +  1 ) )  =  (  seq M (  .+  ,  F ,  T ) `  ( k  +  1 ) ) ) )
2423imbi2d 219 . . . 4  |-  ( w  =  ( k  +  1 )  ->  (
( ph  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq M (  .+  ,  F ,  T ) `
 w ) )  <-> 
( ph  ->  (  seq M (  .+  ,  F ,  S ) `  ( k  +  1 ) )  =  (  seq M (  .+  ,  F ,  T ) `
 ( k  +  1 ) ) ) ) )
25 fveq2 5178 . . . . . 6  |-  ( w  =  n  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq M (  .+  ,  F ,  S ) `
 n ) )
26 fveq2 5178 . . . . . 6  |-  ( w  =  n  ->  (  seq M (  .+  ,  F ,  T ) `  w )  =  (  seq M (  .+  ,  F ,  T ) `
 n ) )
2725, 26eqeq12d 2054 . . . . 5  |-  ( w  =  n  ->  (
(  seq M (  .+  ,  F ,  S ) `
 w )  =  (  seq M ( 
.+  ,  F ,  T ) `  w
)  <->  (  seq M
(  .+  ,  F ,  S ) `  n
)  =  (  seq M (  .+  ,  F ,  T ) `  n ) ) )
2827imbi2d 219 . . . 4  |-  ( w  =  n  ->  (
( ph  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq M (  .+  ,  F ,  T ) `
 w ) )  <-> 
( ph  ->  (  seq M (  .+  ,  F ,  S ) `  n )  =  (  seq M (  .+  ,  F ,  T ) `
 n ) ) ) )
291, 4, 5, 6iseq1 9222 . . . . . 6  |-  ( ph  ->  (  seq M ( 
.+  ,  F ,  S ) `  M
)  =  ( F `
 M ) )
301, 2, 10, 11iseq1 9222 . . . . . 6  |-  ( ph  ->  (  seq M ( 
.+  ,  F ,  T ) `  M
)  =  ( F `
 M ) )
3129, 30eqtr4d 2075 . . . . 5  |-  ( ph  ->  (  seq M ( 
.+  ,  F ,  S ) `  M
)  =  (  seq M (  .+  ,  F ,  T ) `  M ) )
3231a1i 9 . . . 4  |-  ( M  e.  ZZ  ->  ( ph  ->  (  seq M
(  .+  ,  F ,  S ) `  M
)  =  (  seq M (  .+  ,  F ,  T ) `  M ) ) )
33 oveq1 5519 . . . . . . 7  |-  ( (  seq M (  .+  ,  F ,  S ) `
 k )  =  (  seq M ( 
.+  ,  F ,  T ) `  k
)  ->  ( (  seq M (  .+  ,  F ,  S ) `  k )  .+  ( F `  ( k  +  1 ) ) )  =  ( (  seq M (  .+  ,  F ,  T ) `
 k )  .+  ( F `  ( k  +  1 ) ) ) )
34 simpr 103 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  k  e.  ( ZZ>= `  M )
)
354adantr 261 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  S  e.  _V )
365adantlr 446 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
376adantlr 446 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
3834, 35, 36, 37iseqp1 9225 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  (  seq M (  .+  ,  F ,  S ) `  ( k  +  1 ) )  =  ( (  seq M ( 
.+  ,  F ,  S ) `  k
)  .+  ( F `  ( k  +  1 ) ) ) )
392adantr 261 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  T  e.  V )
4010adantlr 446 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  T
)
4111adantlr 446 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  ( x  e.  T  /\  y  e.  T ) )  -> 
( x  .+  y
)  e.  T )
4234, 39, 40, 41iseqp1 9225 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  (  seq M (  .+  ,  F ,  T ) `  ( k  +  1 ) )  =  ( (  seq M ( 
.+  ,  F ,  T ) `  k
)  .+  ( F `  ( k  +  1 ) ) ) )
4338, 42eqeq12d 2054 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (  seq M (  .+  ,  F ,  S ) `  ( k  +  1 ) )  =  (  seq M (  .+  ,  F ,  T ) `
 ( k  +  1 ) )  <->  ( (  seq M (  .+  ,  F ,  S ) `  k )  .+  ( F `  ( k  +  1 ) ) )  =  ( (  seq M (  .+  ,  F ,  T ) `
 k )  .+  ( F `  ( k  +  1 ) ) ) ) )
4433, 43syl5ibr 145 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (  seq M (  .+  ,  F ,  S ) `  k )  =  (  seq M (  .+  ,  F ,  T ) `
 k )  -> 
(  seq M (  .+  ,  F ,  S ) `
 ( k  +  1 ) )  =  (  seq M ( 
.+  ,  F ,  T ) `  (
k  +  1 ) ) ) )
4544expcom 109 . . . . 5  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( ph  ->  ( (  seq M
(  .+  ,  F ,  S ) `  k
)  =  (  seq M (  .+  ,  F ,  T ) `  k )  ->  (  seq M (  .+  ,  F ,  S ) `  ( k  +  1 ) )  =  (  seq M (  .+  ,  F ,  T ) `
 ( k  +  1 ) ) ) ) )
4645a2d 23 . . . 4  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( ( ph  ->  (  seq M
(  .+  ,  F ,  S ) `  k
)  =  (  seq M (  .+  ,  F ,  T ) `  k ) )  -> 
( ph  ->  (  seq M (  .+  ,  F ,  S ) `  ( k  +  1 ) )  =  (  seq M (  .+  ,  F ,  T ) `
 ( k  +  1 ) ) ) ) )
4716, 20, 24, 28, 32, 46uzind4 8531 . . 3  |-  ( n  e.  ( ZZ>= `  M
)  ->  ( ph  ->  (  seq M ( 
.+  ,  F ,  S ) `  n
)  =  (  seq M (  .+  ,  F ,  T ) `  n ) ) )
4847impcom 116 . 2  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  (  seq M (  .+  ,  F ,  S ) `  n )  =  (  seq M (  .+  ,  F ,  T ) `
 n ) )
497, 12, 48eqfnfvd 5268 1  |-  ( ph  ->  seq M (  .+  ,  F ,  S )  =  seq M ( 
.+  ,  F ,  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243    e. wcel 1393   _Vcvv 2557    C_ wss 2917   ` cfv 4902  (class class class)co 5512   1c1 6890    + caddc 6892   ZZcz 8245   ZZ>=cuz 8473    seqcseq 9211
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-pre-ltadd 7000
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-frec 5978  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816  df-1r 6817  df-0 6896  df-1 6897  df-r 6899  df-lt 6902  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-sub 7184  df-neg 7185  df-inn 7915  df-n0 8182  df-z 8246  df-uz 8474  df-iseq 9212
This theorem is referenced by:  serige0  9252  serile  9253  iserile  9862  climserile  9865
  Copyright terms: Public domain W3C validator