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Theorem iseqss 9541
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). New proofs should use iseqsst 9542 which is the same without those two hypotheses. (Contributed by Jim Kingdon, 18-Aug-2021.) (New usage is discouraged.)
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 eqid 2082 . . . 4  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
2 iseqss.m . . . 4  |-  ( ph  ->  M  e.  ZZ )
3 iseqss.f . . . 4  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
4 iseqss.pl . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
51, 2, 3, 4iseqfcl 9535 . . 3  |-  ( ph  ->  seq M (  .+  ,  F ,  S ) : ( ZZ>= `  M
) --> S )
6 ffn 5077 . . 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 iseqss.ss . . . . . . 7  |-  ( ph  ->  S  C_  T )
98sseld 2999 . . . . . 6  |-  ( ph  ->  ( ( F `  x )  e.  S  ->  ( F `  x
)  e.  T ) )
109adantr 270 . . . . 5  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( ( F `  x )  e.  S  ->  ( F `
 x )  e.  T ) )
113, 10mpd 13 . . . 4  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  T
)
12 iseqss.plt . . . 4  |-  ( (
ph  /\  ( x  e.  T  /\  y  e.  T ) )  -> 
( x  .+  y
)  e.  T )
131, 2, 11, 12iseqfcl 9535 . . 3  |-  ( ph  ->  seq M (  .+  ,  F ,  T ) : ( ZZ>= `  M
) --> T )
14 ffn 5077 . . 3  |-  (  seq M (  .+  ,  F ,  T ) : ( ZZ>= `  M
) --> T  ->  seq M (  .+  ,  F ,  T )  Fn  ( ZZ>= `  M )
)
1513, 14syl 14 . 2  |-  ( ph  ->  seq M (  .+  ,  F ,  T )  Fn  ( ZZ>= `  M
) )
16 fveq2 5209 . . . . . 6  |-  ( w  =  M  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq M (  .+  ,  F ,  S ) `
 M ) )
17 fveq2 5209 . . . . . 6  |-  ( w  =  M  ->  (  seq M (  .+  ,  F ,  T ) `  w )  =  (  seq M (  .+  ,  F ,  T ) `
 M ) )
1816, 17eqeq12d 2096 . . . . 5  |-  ( w  =  M  ->  (
(  seq M (  .+  ,  F ,  S ) `
 w )  =  (  seq M ( 
.+  ,  F ,  T ) `  w
)  <->  (  seq M
(  .+  ,  F ,  S ) `  M
)  =  (  seq M (  .+  ,  F ,  T ) `  M ) ) )
1918imbi2d 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 ) ) ) )
20 fveq2 5209 . . . . . 6  |-  ( w  =  k  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq M (  .+  ,  F ,  S ) `
 k ) )
21 fveq2 5209 . . . . . 6  |-  ( w  =  k  ->  (  seq M (  .+  ,  F ,  T ) `  w )  =  (  seq M (  .+  ,  F ,  T ) `
 k ) )
2220, 21eqeq12d 2096 . . . . 5  |-  ( w  =  k  ->  (
(  seq M (  .+  ,  F ,  S ) `
 w )  =  (  seq M ( 
.+  ,  F ,  T ) `  w
)  <->  (  seq M
(  .+  ,  F ,  S ) `  k
)  =  (  seq M (  .+  ,  F ,  T ) `  k ) ) )
2322imbi2d 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 ) ) ) )
24 fveq2 5209 . . . . . 6  |-  ( w  =  ( k  +  1 )  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq M (  .+  ,  F ,  S ) `
 ( k  +  1 ) ) )
25 fveq2 5209 . . . . . 6  |-  ( w  =  ( k  +  1 )  ->  (  seq M (  .+  ,  F ,  T ) `  w )  =  (  seq M (  .+  ,  F ,  T ) `
 ( k  +  1 ) ) )
2624, 25eqeq12d 2096 . . . . 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 ) ) ) )
2726imbi2d 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 ) ) ) ) )
28 fveq2 5209 . . . . . 6  |-  ( w  =  n  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq M (  .+  ,  F ,  S ) `
 n ) )
29 fveq2 5209 . . . . . 6  |-  ( w  =  n  ->  (  seq M (  .+  ,  F ,  T ) `  w )  =  (  seq M (  .+  ,  F ,  T ) `
 n ) )
3028, 29eqeq12d 2096 . . . . 5  |-  ( w  =  n  ->  (
(  seq M (  .+  ,  F ,  S ) `
 w )  =  (  seq M ( 
.+  ,  F ,  T ) `  w
)  <->  (  seq M
(  .+  ,  F ,  S ) `  n
)  =  (  seq M (  .+  ,  F ,  T ) `  n ) ) )
3130imbi2d 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 ) ) ) )
322, 3, 4iseq1 9533 . . . . . 6  |-  ( ph  ->  (  seq M ( 
.+  ,  F ,  S ) `  M
)  =  ( F `
 M ) )
332, 11, 12iseq1 9533 . . . . . 6  |-  ( ph  ->  (  seq M ( 
.+  ,  F ,  T ) `  M
)  =  ( F `
 M ) )
3432, 33eqtr4d 2117 . . . . 5  |-  ( ph  ->  (  seq M ( 
.+  ,  F ,  S ) `  M
)  =  (  seq M (  .+  ,  F ,  T ) `  M ) )
3534a1i 9 . . . 4  |-  ( M  e.  ZZ  ->  ( ph  ->  (  seq M
(  .+  ,  F ,  S ) `  M
)  =  (  seq M (  .+  ,  F ,  T ) `  M ) ) )
36 oveq1 5550 . . . . . . 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 ) ) ) )
37 simpr 108 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  k  e.  ( ZZ>= `  M )
)
383adantlr 461 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
394adantlr 461 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
4037, 38, 39iseqp1 9538 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  (  seq M (  .+  ,  F ,  S ) `  ( k  +  1 ) )  =  ( (  seq M ( 
.+  ,  F ,  S ) `  k
)  .+  ( F `  ( k  +  1 ) ) ) )
4111adantlr 461 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  T
)
4212adantlr 461 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  ( x  e.  T  /\  y  e.  T ) )  -> 
( x  .+  y
)  e.  T )
4337, 41, 42iseqp1 9538 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  (  seq M (  .+  ,  F ,  T ) `  ( k  +  1 ) )  =  ( (  seq M ( 
.+  ,  F ,  T ) `  k
)  .+  ( F `  ( k  +  1 ) ) ) )
4440, 43eqeq12d 2096 . . . . . . 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 ) ) ) ) )
4536, 44syl5ibr 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 ) ) ) )
4645expcom 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 ) ) ) ) )
4746a2d 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 ) ) ) ) )
4819, 23, 27, 31, 35, 47uzind4 8757 . . 3  |-  ( n  e.  ( ZZ>= `  M
)  ->  ( ph  ->  (  seq M ( 
.+  ,  F ,  S ) `  n
)  =  (  seq M (  .+  ,  F ,  T ) `  n ) ) )
4948impcom 123 . 2  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  (  seq M (  .+  ,  F ,  S ) `  n )  =  (  seq M (  .+  ,  F ,  T ) `
 n ) )
507, 15, 49eqfnfvd 5300 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 2974    Fn wfn 4927   -->wf 4928   ` cfv 4932  (class class class)co 5543   1c1 7044    + caddc 7046   ZZcz 8432   ZZ>=cuz 8700    seqcseq 9521
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-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337  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-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-iord 4129  df-on 4131  df-ilim 4132  df-suc 4134  df-iom 4340  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-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-frec 6040  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  df-iseq 9522
This theorem is referenced by:  serige0  9570  serile  9571  iserile  10318  climserile  10321
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