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Theorem seq1st 13742
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 11814 . . . 4  |-  ( M  e.  ZZ  ->  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) )  Fn  ( ZZ>= `  M )
)
32adantr 462 . . 3  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) )  Fn  ( ZZ>= `  M
) )
4 seqfn 11814 . . . 4  |-  ( M  e.  ZZ  ->  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } )  Fn  ( ZZ>=
`  M ) )
54adantr 462 . . 3  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } )  Fn  ( ZZ>=
`  M ) )
6 fveq2 5688 . . . . . . . 8  |-  ( y  =  M  ->  (  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  y )  =  (  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `
 M ) )
7 fveq2 5688 . . . . . . . 8  |-  ( y  =  M  ->  (  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  y
)  =  (  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  M
) )
86, 7eqeq12d 2455 . . . . . . 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 316 . . . . . 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 5688 . . . . . . . 8  |-  ( y  =  x  ->  (  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  y )  =  (  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `
 x ) )
11 fveq2 5688 . . . . . . . 8  |-  ( y  =  x  ->  (  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  y
)  =  (  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  x
) )
1210, 11eqeq12d 2455 . . . . . . 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 316 . . . . . 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 5688 . . . . . . . 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 5688 . . . . . . . 8  |-  ( y  =  ( x  + 
1 )  ->  (  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  y
)  =  (  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  (
x  +  1 ) ) )
1614, 15eqeq12d 2455 . . . . . . 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 316 . . . . . 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 11815 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  (  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  M )  =  ( ( Z  X.  { A } ) `  M
) )
1918adantr 462 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  (  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  M )  =  ( ( Z  X.  { A }
) `  M )
)
20 seq1 11815 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  (  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  M
)  =  ( {
<. M ,  A >. } `
 M ) )
2120adantr 462 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  (  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  M )  =  ( { <. M ,  A >. } `  M ) )
22 id 22 . . . . . . . . . . 11  |-  ( A  e.  V  ->  A  e.  V )
23 uzid 10871 . . . . . . . . . . . 12  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
24 algrf.1 . . . . . . . . . . . 12  |-  Z  =  ( ZZ>= `  M )
2523, 24syl6eleqr 2532 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  M  e.  Z )
26 fvconst2g 5928 . . . . . . . . . . 11  |-  ( ( A  e.  V  /\  M  e.  Z )  ->  ( ( Z  X.  { A } ) `  M )  =  A )
2722, 25, 26syl2anr 475 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  ( ( Z  X.  { A } ) `  M )  =  A )
28 fvsng 5909 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  ( { <. M ,  A >. } `  M
)  =  A )
2927, 28eqtr4d 2476 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  ( ( Z  X.  { A } ) `  M )  =  ( { <. M ,  A >. } `  M ) )
3021, 29eqtr4d 2476 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  (  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  M )  =  ( ( Z  X.  { A } ) `  M
) )
3119, 30eqtr4d 2476 . . . . . . 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 434 . . . . . 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 5688 . . . . . . . . 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 11817 . . . . . . . . . . . 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 5698 . . . . . . . . . . . . 13  |-  (  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  x )  e.  _V
36 fvex 5698 . . . . . . . . . . . . 13  |-  ( ( Z  X.  { A } ) `  (
x  +  1 ) )  e.  _V
3735, 36algrflem 6680 . . . . . . . . . . . 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 2489 . . . . . . . . . . 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 11817 . . . . . . . . . . . 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 5698 . . . . . . . . . . . . 13  |-  (  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  x
)  e.  _V
41 fvex 5698 . . . . . . . . . . . . 13  |-  ( {
<. M ,  A >. } `
 ( x  + 
1 ) )  e. 
_V
4240, 41algrflem 6680 . . . . . . . . . . . 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 2489 . . . . . . . . . . 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 2455 . . . . . . . . . 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 463 . . . . . . . . 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 221 . . . . . . . 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 435 . . . . . . 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 26 . . . . . 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 10908 . . . . 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 430 . . . 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 708 . . 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 5797 . 2  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) )  =  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) )
531, 52syl5eq 2485 1  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  R  =  seq M
( ( F  o.  1st ) ,  { <. M ,  A >. } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   {csn 3874   <.cop 3880    X. cxp 4834    o. ccom 4840    Fn wfn 5410   ` cfv 5415  (class class class)co 6090   1stc1st 6574   1c1 9279    + caddc 9281   ZZcz 10642   ZZ>=cuz 10857    seqcseq 11802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-seq 11803
This theorem is referenced by: (None)
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