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Theorem seq1st 13067
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 11340 . . . 4  |-  ( M  e.  ZZ  ->  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A }
) )  Fn  ( ZZ>=
`  M ) )
32adantr 453 . . 3  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) )  Fn  ( ZZ>= `  M
) )
4 seqfn 11340 . . . 4  |-  ( M  e.  ZZ  ->  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } )  Fn  ( ZZ>= `  M
) )
54adantr 453 . . 3  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } )  Fn  ( ZZ>=
`  M ) )
6 fveq2 5731 . . . . . . . 8  |-  ( y  =  M  ->  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  y )  =  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `
 M ) )
7 fveq2 5731 . . . . . . . 8  |-  ( y  =  M  ->  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  y
)  =  (  seq 
M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  M
) )
86, 7eqeq12d 2452 . . . . . . 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 309 . . . . . 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 5731 . . . . . . . 8  |-  ( y  =  x  ->  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  y )  =  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `
 x ) )
11 fveq2 5731 . . . . . . . 8  |-  ( y  =  x  ->  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  y
)  =  (  seq 
M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  x
) )
1210, 11eqeq12d 2452 . . . . . . 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 309 . . . . . 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 5731 . . . . . . . 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 5731 . . . . . . . 8  |-  ( y  =  ( x  + 
1 )  ->  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  y
)  =  (  seq 
M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  (
x  +  1 ) ) )
1614, 15eqeq12d 2452 . . . . . . 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 309 . . . . . 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 11341 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  M )  =  ( ( Z  X.  { A } ) `  M
) )
1918adantr 453 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  (  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  M )  =  ( ( Z  X.  { A }
) `  M )
)
20 seq1 11341 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  M
)  =  ( {
<. M ,  A >. } `
 M ) )
2120adantr 453 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  M )  =  ( { <. M ,  A >. } `  M ) )
22 id 21 . . . . . . . . . . 11  |-  ( A  e.  V  ->  A  e.  V )
23 uzid 10505 . . . . . . . . . . . 12  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
24 algrf.1 . . . . . . . . . . . 12  |-  Z  =  ( ZZ>= `  M )
2523, 24syl6eleqr 2529 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  M  e.  Z )
26 fvconst2g 5948 . . . . . . . . . . 11  |-  ( ( A  e.  V  /\  M  e.  Z )  ->  ( ( Z  X.  { A } ) `  M )  =  A )
2722, 25, 26syl2anr 466 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  ( ( Z  X.  { A } ) `  M )  =  A )
28 fvsng 5930 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  ( { <. M ,  A >. } `  M
)  =  A )
2927, 28eqtr4d 2473 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  ( ( Z  X.  { A } ) `  M )  =  ( { <. M ,  A >. } `  M ) )
3021, 29eqtr4d 2473 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  (  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  M )  =  ( ( Z  X.  { A } ) `  M
) )
3119, 30eqtr4d 2473 . . . . . . 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 425 . . . . . 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 5731 . . . . . . . . 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 11343 . . . . . . . . . . . 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 5745 . . . . . . . . . . . . 13  |-  (  seq 
M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  x )  e.  _V
36 fvex 5745 . . . . . . . . . . . . 13  |-  ( ( Z  X.  { A } ) `  (
x  +  1 ) )  e.  _V
3735, 36algrflem 6458 . . . . . . . . . . . 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 2486 . . . . . . . . . . 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 11343 . . . . . . . . . . . 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 5745 . . . . . . . . . . . . 13  |-  (  seq 
M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  x
)  e.  _V
41 fvex 5745 . . . . . . . . . . . . 13  |-  ( {
<. M ,  A >. } `
 ( x  + 
1 ) )  e. 
_V
4240, 41algrflem 6458 . . . . . . . . . . . 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 2486 . . . . . . . . . . 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 2452 . . . . . . . . . 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 454 . . . . . . . . 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 214 . . . . . . . 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 426 . . . . . . 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 25 . . . . . 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 10539 . . . . 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 421 . . . 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 696 . . 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 5833 . 2  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) )  =  seq  M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) )
531, 52syl5eq 2482 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   {csn 3816   <.cop 3819    X. cxp 4879    o. ccom 4885    Fn wfn 5452   ` cfv 5457  (class class class)co 6084   1stc1st 6350   1c1 8996    + caddc 8998   ZZcz 10287   ZZ>=cuz 10493    seq cseq 11328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-n0 10227  df-z 10288  df-uz 10494  df-seq 11329
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