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Theorem seq1st 14072
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 12093 . . . 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 12093 . . . 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 5852 . . . . . . . 8  |-  ( y  =  M  ->  (  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  y )  =  (  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `
 M ) )
7 fveq2 5852 . . . . . . . 8  |-  ( y  =  M  ->  (  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  y
)  =  (  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  M
) )
86, 7eqeq12d 2463 . . . . . . 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 5852 . . . . . . . 8  |-  ( y  =  x  ->  (  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  y )  =  (  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `
 x ) )
11 fveq2 5852 . . . . . . . 8  |-  ( y  =  x  ->  (  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  y
)  =  (  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  x
) )
1210, 11eqeq12d 2463 . . . . . . 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 5852 . . . . . . . 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 5852 . . . . . . . 8  |-  ( y  =  ( x  + 
1 )  ->  (  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  y
)  =  (  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  (
x  +  1 ) ) )
1614, 15eqeq12d 2463 . . . . . . 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 12094 . . . . . . . . 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 12094 . . . . . . . . . 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 11099 . . . . . . . . . . . 12  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
24 algrf.1 . . . . . . . . . . . 12  |-  Z  =  ( ZZ>= `  M )
2523, 24syl6eleqr 2540 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  M  e.  Z )
26 fvconst2g 6105 . . . . . . . . . . 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 6086 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  ( { <. M ,  A >. } `  M
)  =  A )
2927, 28eqtr4d 2485 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  ( ( Z  X.  { A } ) `  M )  =  ( { <. M ,  A >. } `  M ) )
3021, 29eqtr4d 2485 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  (  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  M )  =  ( ( Z  X.  { A } ) `  M
) )
3119, 30eqtr4d 2485 . . . . . . 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 5852 . . . . . . . . 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 12096 . . . . . . . . . . . 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 5862 . . . . . . . . . . . . 13  |-  (  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) `  x )  e.  _V
36 fvex 5862 . . . . . . . . . . . . 13  |-  ( ( Z  X.  { A } ) `  (
x  +  1 ) )  e.  _V
3735, 36algrflem 6890 . . . . . . . . . . . 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 2498 . . . . . . . . . . 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 12096 . . . . . . . . . . . 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 5862 . . . . . . . . . . . . 13  |-  (  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) `  x
)  e.  _V
41 fvex 5862 . . . . . . . . . . . . 13  |-  ( {
<. M ,  A >. } `
 ( x  + 
1 ) )  e. 
_V
4240, 41algrflem 6890 . . . . . . . . . . . 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 2498 . . . . . . . . . . 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 2463 . . . . . . . . . 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 11143 . . . . 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 5965 . 2  |-  ( ( M  e.  ZZ  /\  A  e.  V )  ->  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) )  =  seq M ( ( F  o.  1st ) ,  { <. M ,  A >. } ) )
531, 52syl5eq 2494 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 1381    e. wcel 1802   {csn 4010   <.cop 4016    X. cxp 4983    o. ccom 4989    Fn wfn 5569   ` cfv 5574  (class class class)co 6277   1stc1st 6779   1c1 9491    + caddc 9493   ZZcz 10865   ZZ>=cuz 11085    seqcseq 12081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-n0 10797  df-z 10866  df-uz 11086  df-seq 12082
This theorem is referenced by: (None)
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