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Theorem iseqsst 9578
Description: Specifying a larger universe for  seq. As long as  F and  .+ are closed over  S, then any class which contains  S can be used as the last argument to 
seq. (Contributed by Jim Kingdon, 28-Apr-2022.)
Hypotheses
Ref Expression
iseqsst.m  |-  ( ph  ->  M  e.  ZZ )
iseqsst.ss  |-  ( ph  ->  S  C_  T )
iseqsst.f  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
iseqsst.pl  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
Assertion
Ref Expression
iseqsst  |-  ( 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

Proof of Theorem iseqsst
Dummy variables  k  w  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2083 . . . 4  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
2 iseqsst.m . . . 4  |-  ( ph  ->  M  e.  ZZ )
3 iseqsst.f . . . 4  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
4 iseqsst.pl . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
51, 2, 3, 4iseqfcl 9571 . . 3  |-  ( ph  ->  seq M (  .+  ,  F ,  S ) : ( ZZ>= `  M
) --> S )
6 ffn 5098 . . 3  |-  (  seq M (  .+  ,  F ,  S ) : ( ZZ>= `  M
) --> S  ->  seq M (  .+  ,  F ,  S )  Fn  ( ZZ>= `  M )
)
75, 6syl 14 . 2  |-  ( ph  ->  seq M (  .+  ,  F ,  S )  Fn  ( ZZ>= `  M
) )
8 iseqsst.ss . . . 4  |-  ( ph  ->  S  C_  T )
91, 2, 3, 4, 8iseqfclt 9572 . . 3  |-  ( ph  ->  seq M (  .+  ,  F ,  T ) : ( ZZ>= `  M
) --> S )
10 ffn 5098 . . 3  |-  (  seq M (  .+  ,  F ,  T ) : ( ZZ>= `  M
) --> S  ->  seq M (  .+  ,  F ,  T )  Fn  ( ZZ>= `  M )
)
119, 10syl 14 . 2  |-  ( ph  ->  seq M (  .+  ,  F ,  T )  Fn  ( ZZ>= `  M
) )
12 fveq2 5230 . . . . . 6  |-  ( w  =  M  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq M (  .+  ,  F ,  S ) `
 M ) )
13 fveq2 5230 . . . . . 6  |-  ( w  =  M  ->  (  seq M (  .+  ,  F ,  T ) `  w )  =  (  seq M (  .+  ,  F ,  T ) `
 M ) )
1412, 13eqeq12d 2097 . . . . 5  |-  ( w  =  M  ->  (
(  seq M (  .+  ,  F ,  S ) `
 w )  =  (  seq M ( 
.+  ,  F ,  T ) `  w
)  <->  (  seq M
(  .+  ,  F ,  S ) `  M
)  =  (  seq M (  .+  ,  F ,  T ) `  M ) ) )
1514imbi2d 228 . . . 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 ) ) ) )
16 fveq2 5230 . . . . . 6  |-  ( w  =  k  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq M (  .+  ,  F ,  S ) `
 k ) )
17 fveq2 5230 . . . . . 6  |-  ( w  =  k  ->  (  seq M (  .+  ,  F ,  T ) `  w )  =  (  seq M (  .+  ,  F ,  T ) `
 k ) )
1816, 17eqeq12d 2097 . . . . 5  |-  ( w  =  k  ->  (
(  seq M (  .+  ,  F ,  S ) `
 w )  =  (  seq M ( 
.+  ,  F ,  T ) `  w
)  <->  (  seq M
(  .+  ,  F ,  S ) `  k
)  =  (  seq M (  .+  ,  F ,  T ) `  k ) ) )
1918imbi2d 228 . . . 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 ) ) ) )
20 fveq2 5230 . . . . . 6  |-  ( w  =  ( k  +  1 )  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq M (  .+  ,  F ,  S ) `
 ( k  +  1 ) ) )
21 fveq2 5230 . . . . . 6  |-  ( w  =  ( k  +  1 )  ->  (  seq M (  .+  ,  F ,  T ) `  w )  =  (  seq M (  .+  ,  F ,  T ) `
 ( k  +  1 ) ) )
2220, 21eqeq12d 2097 . . . . 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 ) ) ) )
2322imbi2d 228 . . . 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 ) ) ) ) )
24 fveq2 5230 . . . . . 6  |-  ( w  =  n  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq M (  .+  ,  F ,  S ) `
 n ) )
25 fveq2 5230 . . . . . 6  |-  ( w  =  n  ->  (  seq M (  .+  ,  F ,  T ) `  w )  =  (  seq M (  .+  ,  F ,  T ) `
 n ) )
2624, 25eqeq12d 2097 . . . . 5  |-  ( w  =  n  ->  (
(  seq M (  .+  ,  F ,  S ) `
 w )  =  (  seq M ( 
.+  ,  F ,  T ) `  w
)  <->  (  seq M
(  .+  ,  F ,  S ) `  n
)  =  (  seq M (  .+  ,  F ,  T ) `  n ) ) )
2726imbi2d 228 . . . 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 ) ) ) )
282, 3, 4iseq1 9569 . . . . . 6  |-  ( ph  ->  (  seq M ( 
.+  ,  F ,  S ) `  M
)  =  ( F `
 M ) )
292, 3, 4, 8iseq1t 9570 . . . . . 6  |-  ( ph  ->  (  seq M ( 
.+  ,  F ,  T ) `  M
)  =  ( F `
 M ) )
3028, 29eqtr4d 2118 . . . . 5  |-  ( ph  ->  (  seq M ( 
.+  ,  F ,  S ) `  M
)  =  (  seq M (  .+  ,  F ,  T ) `  M ) )
3130a1i 9 . . . 4  |-  ( M  e.  ZZ  ->  ( ph  ->  (  seq M
(  .+  ,  F ,  S ) `  M
)  =  (  seq M (  .+  ,  F ,  T ) `  M ) ) )
32 oveq1 5571 . . . . . . 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 ) ) ) )
33 simpr 108 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  k  e.  ( ZZ>= `  M )
)
343adantlr 461 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
354adantlr 461 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
3633, 34, 35iseqp1 9574 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  (  seq M (  .+  ,  F ,  S ) `  ( k  +  1 ) )  =  ( (  seq M ( 
.+  ,  F ,  S ) `  k
)  .+  ( F `  ( k  +  1 ) ) ) )
378adantr 270 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  S  C_  T
)
3833, 34, 35, 37iseqp1t 9575 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  (  seq M (  .+  ,  F ,  T ) `  ( k  +  1 ) )  =  ( (  seq M ( 
.+  ,  F ,  T ) `  k
)  .+  ( F `  ( k  +  1 ) ) ) )
3936, 38eqeq12d 2097 . . . . . . 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 ) ) ) ) )
4032, 39syl5ibr 154 . . . . . 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 ) ) ) )
4140expcom 114 . . . . 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 ) ) ) ) )
4241a2d 26 . . . 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 ) ) ) ) )
4315, 19, 23, 27, 31, 42uzind4 8793 . . 3  |-  ( n  e.  ( ZZ>= `  M
)  ->  ( ph  ->  (  seq M ( 
.+  ,  F ,  S ) `  n
)  =  (  seq M (  .+  ,  F ,  T ) `  n ) ) )
4443impcom 123 . 2  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  (  seq M (  .+  ,  F ,  S ) `  n )  =  (  seq M (  .+  ,  F ,  T ) `
 n ) )
457, 11, 44eqfnfvd 5321 1  |-  ( ph  ->  seq M (  .+  ,  F ,  S )  =  seq M ( 
.+  ,  F ,  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434    C_ wss 2983    Fn wfn 4948   -->wf 4949   ` cfv 4953  (class class class)co 5564   1c1 7080    + caddc 7082   ZZcz 8468   ZZ>=cuz 8736    seqcseq 9557
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 3914  ax-sep 3917  ax-nul 3925  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-iinf 4358  ax-cnex 7165  ax-resscn 7166  ax-1cn 7167  ax-1re 7168  ax-icn 7169  ax-addcl 7170  ax-addrcl 7171  ax-mulcl 7172  ax-addcom 7174  ax-addass 7176  ax-distr 7178  ax-i2m1 7179  ax-0lt1 7180  ax-0id 7182  ax-rnegex 7183  ax-cnre 7185  ax-pre-ltirr 7186  ax-pre-ltwlin 7187  ax-pre-lttrn 7188  ax-pre-ltadd 7190
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 2826  df-csb 2919  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-int 3658  df-iun 3701  df-br 3807  df-opab 3861  df-mpt 3862  df-tr 3897  df-id 4077  df-iord 4150  df-on 4152  df-ilim 4153  df-suc 4155  df-iom 4361  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-f1 4958  df-fo 4959  df-f1o 4960  df-fv 4961  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 7253  df-mnf 7254  df-xr 7255  df-ltxr 7256  df-le 7257  df-sub 7384  df-neg 7385  df-inn 8143  df-n0 8392  df-z 8469  df-uz 8737  df-iseq 9558
This theorem is referenced by: (None)
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