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Theorem seq1st 14050
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 12077 . . . 4  |-  ( M  e.  ZZ  ->  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) )  Fn  ( ZZ>= `  M )
)
32adantr 465 . . 3  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) )  Fn  ( ZZ>= `  M
) )
4 seqfn 12077 . . . 4  |-  ( M  e.  ZZ  ->  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } )  Fn  ( ZZ>=
`  M ) )
54adantr 465 . . 3  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } )  Fn  ( ZZ>=
`  M ) )
6 fveq2 5859 . . . . . . . 8  |-  ( y  =  M  ->  (  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  y )  =  (  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `
 M ) )
7 fveq2 5859 . . . . . . . 8  |-  ( y  =  M  ->  (  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  y
)  =  (  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  M
) )
86, 7eqeq12d 2484 . . . . . . 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 5859 . . . . . . . 8  |-  ( y  =  x  ->  (  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  y )  =  (  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `
 x ) )
11 fveq2 5859 . . . . . . . 8  |-  ( y  =  x  ->  (  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  y
)  =  (  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  x
) )
1210, 11eqeq12d 2484 . . . . . . 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 5859 . . . . . . . 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 5859 . . . . . . . 8  |-  ( y  =  ( x  + 
1 )  ->  (  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  y
)  =  (  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  (
x  +  1 ) ) )
1614, 15eqeq12d 2484 . . . . . . 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 12078 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  (  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  M )  =  ( ( Z  X.  { A } ) `  M
) )
1918adantr 465 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  (  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  M )  =  ( ( Z  X.  { A }
) `  M )
)
20 seq1 12078 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  (  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  M
)  =  ( {
<. M ,  A >. } `
 M ) )
2120adantr 465 . . . . . . . . 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 11087 . . . . . . . . . . . 12  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
24 algrf.1 . . . . . . . . . . . 12  |-  Z  =  ( ZZ>= `  M )
2523, 24syl6eleqr 2561 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  M  e.  Z )
26 fvconst2g 6107 . . . . . . . . . . 11  |-  ( ( A  e.  V  /\  M  e.  Z )  ->  ( ( Z  X.  { A } ) `  M )  =  A )
2722, 25, 26syl2anr 478 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  ( ( Z  X.  { A } ) `  M )  =  A )
28 fvsng 6088 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  ( { <. M ,  A >. } `  M
)  =  A )
2927, 28eqtr4d 2506 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  ( ( Z  X.  { A } ) `  M )  =  ( { <. M ,  A >. } `  M ) )
3021, 29eqtr4d 2506 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  (  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  M )  =  ( ( Z  X.  { A } ) `  M
) )
3119, 30eqtr4d 2506 . . . . . . 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 5859 . . . . . . . . 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 12080 . . . . . . . . . . . 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 5869 . . . . . . . . . . . . 13  |-  (  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  x )  e.  _V
36 fvex 5869 . . . . . . . . . . . . 13  |-  ( ( Z  X.  { A } ) `  (
x  +  1 ) )  e.  _V
3735, 36algrflem 6884 . . . . . . . . . . . 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 2519 . . . . . . . . . . 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 12080 . . . . . . . . . . . 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 5869 . . . . . . . . . . . . 13  |-  (  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  x
)  e.  _V
41 fvex 5869 . . . . . . . . . . . . 13  |-  ( {
<. M ,  A >. } `
 ( x  + 
1 ) )  e. 
_V
4240, 41algrflem 6884 . . . . . . . . . . . 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 2519 . . . . . . . . . . 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 2484 . . . . . . . . . 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 466 . . . . . . . . 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 11130 . . . . 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 713 . . 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 5971 . 2  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) )  =  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) )
531, 52syl5eq 2515 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 1374    e. wcel 1762   {csn 4022   <.cop 4028    X. cxp 4992    o. ccom 4998    Fn wfn 5576   ` cfv 5581  (class class class)co 6277   1stc1st 6774   1c1 9484    + caddc 9486   ZZcz 10855   ZZ>=cuz 11073    seqcseq 12065
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-n0 10787  df-z 10856  df-uz 11074  df-seq 12066
This theorem is referenced by: (None)
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