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Theorem seq1st 12741
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 11058 . . . 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 11058 . . . 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 5525 . . . . . . . 8  |-  ( y  =  M  ->  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  y )  =  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `
 M ) )
7 fveq2 5525 . . . . . . . 8  |-  ( y  =  M  ->  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  y
)  =  (  seq 
M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  M
) )
86, 7eqeq12d 2297 . . . . . . 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 5525 . . . . . . . 8  |-  ( y  =  x  ->  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  y )  =  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `
 x ) )
11 fveq2 5525 . . . . . . . 8  |-  ( y  =  x  ->  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  y
)  =  (  seq 
M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  x
) )
1210, 11eqeq12d 2297 . . . . . . 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 5525 . . . . . . . 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 5525 . . . . . . . 8  |-  ( y  =  ( x  + 
1 )  ->  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  y
)  =  (  seq 
M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  (
x  +  1 ) ) )
1614, 15eqeq12d 2297 . . . . . . 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 11059 . . . . . . . . 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 11059 . . . . . . . . . 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 10242 . . . . . . . . . . . 12  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
24 algrf.1 . . . . . . . . . . . 12  |-  Z  =  ( ZZ>= `  M )
2523, 24syl6eleqr 2374 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  M  e.  Z )
26 fvconst2g 5727 . . . . . . . . . . 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 5714 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  ( { <. M ,  A >. } `  M
)  =  A )
2927, 28eqtr4d 2318 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  ( ( Z  X.  { A } ) `  M )  =  ( { <. M ,  A >. } `  M ) )
3021, 29eqtr4d 2318 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  M )  =  ( ( Z  X.  { A } ) `  M
) )
3119, 30eqtr4d 2318 . . . . . . 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 5525 . . . . . . . . 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 11061 . . . . . . . . . . . 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 5539 . . . . . . . . . . . . 13  |-  (  seq 
M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  x )  e.  _V
36 fvex 5539 . . . . . . . . . . . . 13  |-  ( ( Z  X.  { A } ) `  (
x  +  1 ) )  e.  _V
3735, 36algrflem 6224 . . . . . . . . . . . 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 2331 . . . . . . . . . . 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 11061 . . . . . . . . . . . 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 5539 . . . . . . . . . . . . 13  |-  (  seq 
M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  x
)  e.  _V
41 fvex 5539 . . . . . . . . . . . . 13  |-  ( {
<. M ,  A >. } `
 ( x  + 
1 ) )  e. 
_V
4240, 41algrflem 6224 . . . . . . . . . . . 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 2331 . . . . . . . . . . 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 2297 . . . . . . . . . 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 10276 . . . . 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 5625 . 2  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) )  =  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) )
531, 52syl5eq 2327 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 1623    e. wcel 1684   {csn 3640   <.cop 3643    X. cxp 4687    o. ccom 4693    Fn wfn 5250   ` cfv 5255  (class class class)co 5858   1stc1st 6120   1c1 8738    + caddc 8740   ZZcz 10024   ZZ>=cuz 10230    seq cseq 11046
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-seq 11047
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