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Theorem seq1st 13025
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 11298 . . . 4  |-  ( M  e.  ZZ  ->  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A }
) )  Fn  ( ZZ>=
`  M ) )
32adantr 452 . . 3  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) )  Fn  ( ZZ>= `  M
) )
4 seqfn 11298 . . . 4  |-  ( M  e.  ZZ  ->  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } )  Fn  ( ZZ>= `  M
) )
54adantr 452 . . 3  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } )  Fn  ( ZZ>=
`  M ) )
6 fveq2 5695 . . . . . . . 8  |-  ( y  =  M  ->  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  y )  =  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `
 M ) )
7 fveq2 5695 . . . . . . . 8  |-  ( y  =  M  ->  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  y
)  =  (  seq 
M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  M
) )
86, 7eqeq12d 2426 . . . . . . 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 308 . . . . . 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 5695 . . . . . . . 8  |-  ( y  =  x  ->  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  y )  =  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `
 x ) )
11 fveq2 5695 . . . . . . . 8  |-  ( y  =  x  ->  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  y
)  =  (  seq 
M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  x
) )
1210, 11eqeq12d 2426 . . . . . . 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 308 . . . . . 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 5695 . . . . . . . 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 5695 . . . . . . . 8  |-  ( y  =  ( x  + 
1 )  ->  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  y
)  =  (  seq 
M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  (
x  +  1 ) ) )
1614, 15eqeq12d 2426 . . . . . . 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 308 . . . . . 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 11299 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  M )  =  ( ( Z  X.  { A } ) `  M
) )
1918adantr 452 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  M )  =  ( ( Z  X.  { A }
) `  M )
)
20 seq1 11299 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  M
)  =  ( {
<. M ,  A >. } `
 M ) )
2120adantr 452 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  M )  =  ( { <. M ,  A >. } `  M ) )
22 id 20 . . . . . . . . . . 11  |-  ( A  e.  V  ->  A  e.  V )
23 uzid 10464 . . . . . . . . . . . 12  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
24 algrf.1 . . . . . . . . . . . 12  |-  Z  =  ( ZZ>= `  M )
2523, 24syl6eleqr 2503 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  M  e.  Z )
26 fvconst2g 5912 . . . . . . . . . . 11  |-  ( ( A  e.  V  /\  M  e.  Z )  ->  ( ( Z  X.  { A } ) `  M )  =  A )
2722, 25, 26syl2anr 465 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  ( ( Z  X.  { A } ) `  M )  =  A )
28 fvsng 5894 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  ( { <. M ,  A >. } `  M
)  =  A )
2927, 28eqtr4d 2447 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  ( ( Z  X.  { A } ) `  M )  =  ( { <. M ,  A >. } `  M ) )
3021, 29eqtr4d 2447 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  M )  =  ( ( Z  X.  { A } ) `  M
) )
3119, 30eqtr4d 2447 . . . . . . 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 424 . . . . . 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 5695 . . . . . . . . 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 11301 . . . . . . . . . . . 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 5709 . . . . . . . . . . . . 13  |-  (  seq 
M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  x )  e.  _V
36 fvex 5709 . . . . . . . . . . . . 13  |-  ( ( Z  X.  { A } ) `  (
x  +  1 ) )  e.  _V
3735, 36algrflem 6422 . . . . . . . . . . . 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 2460 . . . . . . . . . . 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 11301 . . . . . . . . . . . 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 5709 . . . . . . . . . . . . 13  |-  (  seq 
M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  x
)  e.  _V
41 fvex 5709 . . . . . . . . . . . . 13  |-  ( {
<. M ,  A >. } `
 ( x  + 
1 ) )  e. 
_V
4240, 41algrflem 6422 . . . . . . . . . . . 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 2460 . . . . . . . . . . 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 2426 . . . . . . . . . 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 453 . . . . . . . . 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 213 . . . . . . . 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 425 . . . . . . 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 24 . . . . . 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 10498 . . . . 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 420 . . . 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 695 . . 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 2456 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   {csn 3782   <.cop 3785    X. cxp 4843    o. ccom 4849    Fn wfn 5416   ` cfv 5421  (class class class)co 6048   1stc1st 6314   1c1 8955    + caddc 8957   ZZcz 10246   ZZ>=cuz 10452    seq cseq 11286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-n0 10186  df-z 10247  df-uz 10453  df-seq 11287
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