MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  seq1st Unicode version

Theorem seq1st 12949
Description: A sequence whose iteration function ignores the second argument is only affected by the first point of the initial value function. (Contributed by Mario Carneiro, 11-Feb-2015.)
Hypotheses
Ref Expression
algrf.1  |-  Z  =  ( ZZ>= `  M )
algrf.2  |-  R  =  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) )
Assertion
Ref Expression
seq1st  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  R  =  seq  M
( ( F  o.  1st ) ,  { <. M ,  A >. } ) )

Proof of Theorem seq1st
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 algrf.2 . 2  |-  R  =  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) )
2 seqfn 11222 . . . 4  |-  ( M  e.  ZZ  ->  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A }
) )  Fn  ( ZZ>=
`  M ) )
32adantr 451 . . 3  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) )  Fn  ( ZZ>= `  M
) )
4 seqfn 11222 . . . 4  |-  ( M  e.  ZZ  ->  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } )  Fn  ( ZZ>= `  M
) )
54adantr 451 . . 3  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } )  Fn  ( ZZ>=
`  M ) )
6 fveq2 5632 . . . . . . . 8  |-  ( y  =  M  ->  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  y )  =  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `
 M ) )
7 fveq2 5632 . . . . . . . 8  |-  ( y  =  M  ->  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  y
)  =  (  seq 
M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  M
) )
86, 7eqeq12d 2380 . . . . . . 7  |-  ( y  =  M  ->  (
(  seq  M (
( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  y )  =  (  seq  M
( ( F  o.  1st ) ,  { <. M ,  A >. } ) `
 y )  <->  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A }
) ) `  M
)  =  (  seq 
M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  M
) ) )
98imbi2d 307 . . . . . 6  |-  ( y  =  M  ->  (
( A  e.  V  ->  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  y )  =  (  seq  M
( ( F  o.  1st ) ,  { <. M ,  A >. } ) `
 y ) )  <-> 
( A  e.  V  ->  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  M )  =  (  seq  M
( ( F  o.  1st ) ,  { <. M ,  A >. } ) `
 M ) ) ) )
10 fveq2 5632 . . . . . . . 8  |-  ( y  =  x  ->  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  y )  =  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `
 x ) )
11 fveq2 5632 . . . . . . . 8  |-  ( y  =  x  ->  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  y
)  =  (  seq 
M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  x
) )
1210, 11eqeq12d 2380 . . . . . . 7  |-  ( y  =  x  ->  (
(  seq  M (
( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  y )  =  (  seq  M
( ( F  o.  1st ) ,  { <. M ,  A >. } ) `
 y )  <->  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A }
) ) `  x
)  =  (  seq 
M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  x
) ) )
1312imbi2d 307 . . . . . 6  |-  ( y  =  x  ->  (
( A  e.  V  ->  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  y )  =  (  seq  M
( ( F  o.  1st ) ,  { <. M ,  A >. } ) `
 y ) )  <-> 
( A  e.  V  ->  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  x )  =  (  seq  M
( ( F  o.  1st ) ,  { <. M ,  A >. } ) `
 x ) ) ) )
14 fveq2 5632 . . . . . . . 8  |-  ( y  =  ( x  + 
1 )  ->  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  y )  =  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `
 ( x  + 
1 ) ) )
15 fveq2 5632 . . . . . . . 8  |-  ( y  =  ( x  + 
1 )  ->  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  y
)  =  (  seq 
M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  (
x  +  1 ) ) )
1614, 15eqeq12d 2380 . . . . . . 7  |-  ( y  =  ( x  + 
1 )  ->  (
(  seq  M (
( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  y )  =  (  seq  M
( ( F  o.  1st ) ,  { <. M ,  A >. } ) `
 y )  <->  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A }
) ) `  (
x  +  1 ) )  =  (  seq 
M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  (
x  +  1 ) ) ) )
1716imbi2d 307 . . . . . 6  |-  ( y  =  ( x  + 
1 )  ->  (
( A  e.  V  ->  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  y )  =  (  seq  M
( ( F  o.  1st ) ,  { <. M ,  A >. } ) `
 y ) )  <-> 
( A  e.  V  ->  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  ( x  +  1 ) )  =  (  seq  M
( ( F  o.  1st ) ,  { <. M ,  A >. } ) `
 ( x  + 
1 ) ) ) ) )
18 seq1 11223 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  M )  =  ( ( Z  X.  { A } ) `  M
) )
1918adantr 451 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  M )  =  ( ( Z  X.  { A }
) `  M )
)
20 seq1 11223 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  M
)  =  ( {
<. M ,  A >. } `
 M ) )
2120adantr 451 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  M )  =  ( { <. M ,  A >. } `  M ) )
22 id 19 . . . . . . . . . . 11  |-  ( A  e.  V  ->  A  e.  V )
23 uzid 10393 . . . . . . . . . . . 12  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
24 algrf.1 . . . . . . . . . . . 12  |-  Z  =  ( ZZ>= `  M )
2523, 24syl6eleqr 2457 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  M  e.  Z )
26 fvconst2g 5845 . . . . . . . . . . 11  |-  ( ( A  e.  V  /\  M  e.  Z )  ->  ( ( Z  X.  { A } ) `  M )  =  A )
2722, 25, 26syl2anr 464 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  ( ( Z  X.  { A } ) `  M )  =  A )
28 fvsng 5827 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  ( { <. M ,  A >. } `  M
)  =  A )
2927, 28eqtr4d 2401 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  ( ( Z  X.  { A } ) `  M )  =  ( { <. M ,  A >. } `  M ) )
3021, 29eqtr4d 2401 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  M )  =  ( ( Z  X.  { A } ) `  M
) )
3119, 30eqtr4d 2401 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  M )  =  (  seq  M
( ( F  o.  1st ) ,  { <. M ,  A >. } ) `
 M ) )
3231ex 423 . . . . . 6  |-  ( M  e.  ZZ  ->  ( A  e.  V  ->  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `
 M )  =  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  M ) ) )
33 fveq2 5632 . . . . . . . . 9  |-  ( (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `
 x )  =  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  x )  ->  ( F `  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A }
) ) `  x
) )  =  ( F `  (  seq 
M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  x
) ) )
34 seqp1 11225 . . . . . . . . . . . 12  |-  ( x  e.  ( ZZ>= `  M
)  ->  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A }
) ) `  (
x  +  1 ) )  =  ( (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `
 x ) ( F  o.  1st )
( ( Z  X.  { A } ) `  ( x  +  1
) ) ) )
35 fvex 5646 . . . . . . . . . . . . 13  |-  (  seq 
M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  x )  e.  _V
36 fvex 5646 . . . . . . . . . . . . 13  |-  ( ( Z  X.  { A } ) `  (
x  +  1 ) )  e.  _V
3735, 36algrflem 6352 . . . . . . . . . . . 12  |-  ( (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `
 x ) ( F  o.  1st )
( ( Z  X.  { A } ) `  ( x  +  1
) ) )  =  ( F `  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  x ) )
3834, 37syl6eq 2414 . . . . . . . . . . 11  |-  ( x  e.  ( ZZ>= `  M
)  ->  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A }
) ) `  (
x  +  1 ) )  =  ( F `
 (  seq  M
( ( F  o.  1st ) ,  ( Z  X.  { A }
) ) `  x
) ) )
39 seqp1 11225 . . . . . . . . . . . 12  |-  ( x  e.  ( ZZ>= `  M
)  ->  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `
 ( x  + 
1 ) )  =  ( (  seq  M
( ( F  o.  1st ) ,  { <. M ,  A >. } ) `
 x ) ( F  o.  1st )
( { <. M ,  A >. } `  (
x  +  1 ) ) ) )
40 fvex 5646 . . . . . . . . . . . . 13  |-  (  seq 
M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  x
)  e.  _V
41 fvex 5646 . . . . . . . . . . . . 13  |-  ( {
<. M ,  A >. } `
 ( x  + 
1 ) )  e. 
_V
4240, 41algrflem 6352 . . . . . . . . . . . 12  |-  ( (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  x
) ( F  o.  1st ) ( { <. M ,  A >. } `  ( x  +  1
) ) )  =  ( F `  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  x
) )
4339, 42syl6eq 2414 . . . . . . . . . . 11  |-  ( x  e.  ( ZZ>= `  M
)  ->  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `
 ( x  + 
1 ) )  =  ( F `  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  x
) ) )
4438, 43eqeq12d 2380 . . . . . . . . . 10  |-  ( x  e.  ( ZZ>= `  M
)  ->  ( (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  ( x  +  1
) )  =  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  (
x  +  1 ) )  <->  ( F `  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `
 x ) )  =  ( F `  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  x
) ) ) )
4544adantl 452 . . . . . . . . 9  |-  ( ( A  e.  V  /\  x  e.  ( ZZ>= `  M ) )  -> 
( (  seq  M
( ( F  o.  1st ) ,  ( Z  X.  { A }
) ) `  (
x  +  1 ) )  =  (  seq 
M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  (
x  +  1 ) )  <->  ( F `  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `
 x ) )  =  ( F `  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  x
) ) ) )
4633, 45syl5ibr 212 . . . . . . . 8  |-  ( ( A  e.  V  /\  x  e.  ( ZZ>= `  M ) )  -> 
( (  seq  M
( ( F  o.  1st ) ,  ( Z  X.  { A }
) ) `  x
)  =  (  seq 
M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  x
)  ->  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A }
) ) `  (
x  +  1 ) )  =  (  seq 
M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  (
x  +  1 ) ) ) )
4746expcom 424 . . . . . . 7  |-  ( x  e.  ( ZZ>= `  M
)  ->  ( A  e.  V  ->  ( (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `
 x )  =  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  x )  ->  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  ( x  +  1
) )  =  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  (
x  +  1 ) ) ) ) )
4847a2d 23 . . . . . 6  |-  ( x  e.  ( ZZ>= `  M
)  ->  ( ( A  e.  V  ->  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `
 x )  =  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  x ) )  -> 
( A  e.  V  ->  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  ( x  +  1 ) )  =  (  seq  M
( ( F  o.  1st ) ,  { <. M ,  A >. } ) `
 ( x  + 
1 ) ) ) ) )
499, 13, 17, 13, 32, 48uzind4 10427 . . . . 5  |-  ( x  e.  ( ZZ>= `  M
)  ->  ( A  e.  V  ->  (  seq 
M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  x )  =  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  x
) ) )
5049impcom 419 . . . 4  |-  ( ( A  e.  V  /\  x  e.  ( ZZ>= `  M ) )  -> 
(  seq  M (
( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  x )  =  (  seq  M
( ( F  o.  1st ) ,  { <. M ,  A >. } ) `
 x ) )
5150adantll 694 . . 3  |-  ( ( ( M  e.  ZZ  /\  A  e.  V )  /\  x  e.  (
ZZ>= `  M ) )  ->  (  seq  M
( ( F  o.  1st ) ,  ( Z  X.  { A }
) ) `  x
)  =  (  seq 
M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  x
) )
523, 5, 51eqfnfvd 5732 . 2  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) )  =  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) )
531, 52syl5eq 2410 1  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  R  =  seq  M
( ( F  o.  1st ) ,  { <. M ,  A >. } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1647    e. wcel 1715   {csn 3729   <.cop 3732    X. cxp 4790    o. ccom 4796    Fn wfn 5353   ` cfv 5358  (class class class)co 5981   1stc1st 6247   1c1 8885    + caddc 8887   ZZcz 10175   ZZ>=cuz 10381    seq cseq 11210
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-n0 10115  df-z 10176  df-uz 10382  df-seq 11211
  Copyright terms: Public domain W3C validator