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Theorem iseqz 9908
Description: If the operation  .+ has an absorbing element  Z (a.k.a. zero element), then any sequence containing a  Z evaluates to  Z. (Contributed by Mario Carneiro, 27-May-2014.)
Hypotheses
Ref Expression
iseqhomo.1  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
iseqhomo.2  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
iseqhomo.s  |-  ( ph  ->  S  e.  V )
iseqz.3  |-  ( (
ph  /\  x  e.  S )  ->  ( Z  .+  x )  =  Z )
iseqz.4  |-  ( (
ph  /\  x  e.  S )  ->  (
x  .+  Z )  =  Z )
iseqz.5  |-  ( ph  ->  K  e.  ( M ... N ) )
iseqz.6  |-  ( ph  ->  N  e.  V )
iseqz.7  |-  ( ph  ->  ( F `  K
)  =  Z )
Assertion
Ref Expression
iseqz  |-  ( ph  ->  (  seq M ( 
.+  ,  F ,  S ) `  N
)  =  Z )
Distinct variable groups:    x, y, F   
x, M, y    x, N, y    ph, x, y   
x, K, y    x,  .+ , y    x, S, y   
x, Z, y
Allowed substitution hints:    V( x, y)

Proof of Theorem iseqz
Dummy variables  k  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iseqz.5 . . 3  |-  ( ph  ->  K  e.  ( M ... N ) )
2 elfzuz3 9406 . . 3  |-  ( K  e.  ( M ... N )  ->  N  e.  ( ZZ>= `  K )
)
31, 2syl 14 . 2  |-  ( ph  ->  N  e.  ( ZZ>= `  K ) )
4 fveq2 5289 . . . . 5  |-  ( w  =  K  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq M (  .+  ,  F ,  S ) `
 K ) )
54eqeq1d 2096 . . . 4  |-  ( w  =  K  ->  (
(  seq M (  .+  ,  F ,  S ) `
 w )  =  Z  <->  (  seq M
(  .+  ,  F ,  S ) `  K
)  =  Z ) )
65imbi2d 228 . . 3  |-  ( w  =  K  ->  (
( ph  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  Z )  <->  ( ph  ->  (  seq M (  .+  ,  F ,  S ) `
 K )  =  Z ) ) )
7 fveq2 5289 . . . . 5  |-  ( w  =  k  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq M (  .+  ,  F ,  S ) `
 k ) )
87eqeq1d 2096 . . . 4  |-  ( w  =  k  ->  (
(  seq M (  .+  ,  F ,  S ) `
 w )  =  Z  <->  (  seq M
(  .+  ,  F ,  S ) `  k
)  =  Z ) )
98imbi2d 228 . . 3  |-  ( w  =  k  ->  (
( ph  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  Z )  <->  ( ph  ->  (  seq M (  .+  ,  F ,  S ) `
 k )  =  Z ) ) )
10 fveq2 5289 . . . . 5  |-  ( w  =  ( k  +  1 )  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq M (  .+  ,  F ,  S ) `
 ( k  +  1 ) ) )
1110eqeq1d 2096 . . . 4  |-  ( w  =  ( k  +  1 )  ->  (
(  seq M (  .+  ,  F ,  S ) `
 w )  =  Z  <->  (  seq M
(  .+  ,  F ,  S ) `  (
k  +  1 ) )  =  Z ) )
1211imbi2d 228 . . 3  |-  ( w  =  ( k  +  1 )  ->  (
( ph  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  Z )  <->  ( ph  ->  (  seq M (  .+  ,  F ,  S ) `
 ( k  +  1 ) )  =  Z ) ) )
13 fveq2 5289 . . . . 5  |-  ( w  =  N  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq M (  .+  ,  F ,  S ) `
 N ) )
1413eqeq1d 2096 . . . 4  |-  ( w  =  N  ->  (
(  seq M (  .+  ,  F ,  S ) `
 w )  =  Z  <->  (  seq M
(  .+  ,  F ,  S ) `  N
)  =  Z ) )
1514imbi2d 228 . . 3  |-  ( w  =  N  ->  (
( ph  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  Z )  <->  ( ph  ->  (  seq M (  .+  ,  F ,  S ) `
 N )  =  Z ) ) )
16 elfzuz 9405 . . . . . . . . . 10  |-  ( K  e.  ( M ... N )  ->  K  e.  ( ZZ>= `  M )
)
171, 16syl 14 . . . . . . . . 9  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
18 eluzelz 8997 . . . . . . . . 9  |-  ( K  e.  ( ZZ>= `  M
)  ->  K  e.  ZZ )
1917, 18syl 14 . . . . . . . 8  |-  ( ph  ->  K  e.  ZZ )
20 simpr 108 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  x  e.  ( ZZ>= `  K )
)
2117adantr 270 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  K  e.  ( ZZ>= `  M )
)
22 uztrn 9004 . . . . . . . . . 10  |-  ( ( x  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  M )
)  ->  x  e.  ( ZZ>= `  M )
)
2320, 21, 22syl2anc 403 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  x  e.  ( ZZ>= `  M )
)
24 iseqhomo.2 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
2523, 24syldan 276 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  ( F `  x )  e.  S
)
26 iseqhomo.1 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
2719, 25, 26iseq1 9840 . . . . . . 7  |-  ( ph  ->  (  seq K ( 
.+  ,  F ,  S ) `  K
)  =  ( F `
 K ) )
28 iseqz.7 . . . . . . 7  |-  ( ph  ->  ( F `  K
)  =  Z )
2927, 28eqtrd 2120 . . . . . 6  |-  ( ph  ->  (  seq K ( 
.+  ,  F ,  S ) `  K
)  =  Z )
30 iseqeq1 9823 . . . . . . . 8  |-  ( K  =  M  ->  seq K (  .+  ,  F ,  S )  =  seq M (  .+  ,  F ,  S ) )
3130fveq1d 5291 . . . . . . 7  |-  ( K  =  M  ->  (  seq K (  .+  ,  F ,  S ) `  K )  =  (  seq M (  .+  ,  F ,  S ) `
 K ) )
3231eqeq1d 2096 . . . . . 6  |-  ( K  =  M  ->  (
(  seq K (  .+  ,  F ,  S ) `
 K )  =  Z  <->  (  seq M
(  .+  ,  F ,  S ) `  K
)  =  Z ) )
3329, 32syl5ibcom 153 . . . . 5  |-  ( ph  ->  ( K  =  M  ->  (  seq M
(  .+  ,  F ,  S ) `  K
)  =  Z ) )
34 eluzel2 8993 . . . . . . . . . 10  |-  ( K  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
3517, 34syl 14 . . . . . . . . 9  |-  ( ph  ->  M  e.  ZZ )
3635adantr 270 . . . . . . . 8  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  M  e.  ZZ )
37 simpr 108 . . . . . . . 8  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  K  e.  ( ZZ>= `  ( M  +  1 ) ) )
3824adantlr 461 . . . . . . . 8  |-  ( ( ( ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
3926adantlr 461 . . . . . . . 8  |-  ( ( ( ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
4036, 37, 38, 39iseqm1 9853 . . . . . . 7  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq M (  .+  ,  F ,  S ) `  K )  =  ( (  seq M ( 
.+  ,  F ,  S ) `  ( K  -  1 ) )  .+  ( F `
 K ) ) )
4128adantr 270 . . . . . . . 8  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( F `  K )  =  Z )
4241oveq2d 5650 . . . . . . 7  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( (  seq M (  .+  ,  F ,  S ) `  ( K  -  1 ) )  .+  ( F `  K )
)  =  ( (  seq M (  .+  ,  F ,  S ) `
 ( K  - 
1 ) )  .+  Z ) )
43 oveq1 5641 . . . . . . . . 9  |-  ( x  =  (  seq M
(  .+  ,  F ,  S ) `  ( K  -  1 ) )  ->  ( x  .+  Z )  =  ( (  seq M ( 
.+  ,  F ,  S ) `  ( K  -  1 ) )  .+  Z ) )
4443eqeq1d 2096 . . . . . . . 8  |-  ( x  =  (  seq M
(  .+  ,  F ,  S ) `  ( K  -  1 ) )  ->  ( (
x  .+  Z )  =  Z  <->  ( (  seq M (  .+  ,  F ,  S ) `  ( K  -  1 ) )  .+  Z
)  =  Z ) )
45 iseqz.4 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  S )  ->  (
x  .+  Z )  =  Z )
4645ralrimiva 2446 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  S  ( x  .+  Z )  =  Z )
4746adantr 270 . . . . . . . 8  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  A. x  e.  S  ( x  .+  Z )  =  Z )
48 eluzp1m1 9011 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  K  e.  ( ZZ>= `  ( M  +  1
) ) )  -> 
( K  -  1 )  e.  ( ZZ>= `  M ) )
4935, 48sylan 277 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( K  -  1 )  e.  ( ZZ>= `  M )
)
5049, 38, 39iseqcl 9846 . . . . . . . 8  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq M (  .+  ,  F ,  S ) `  ( K  -  1 ) )  e.  S
)
5144, 47, 50rspcdva 2727 . . . . . . 7  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( (  seq M (  .+  ,  F ,  S ) `  ( K  -  1 ) )  .+  Z
)  =  Z )
5240, 42, 513eqtrd 2124 . . . . . 6  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq M (  .+  ,  F ,  S ) `  K )  =  Z )
5352ex 113 . . . . 5  |-  ( ph  ->  ( K  e.  (
ZZ>= `  ( M  + 
1 ) )  -> 
(  seq M (  .+  ,  F ,  S ) `
 K )  =  Z ) )
54 uzp1 9021 . . . . . 6  |-  ( K  e.  ( ZZ>= `  M
)  ->  ( K  =  M  \/  K  e.  ( ZZ>= `  ( M  +  1 ) ) ) )
5517, 54syl 14 . . . . 5  |-  ( ph  ->  ( K  =  M  \/  K  e.  (
ZZ>= `  ( M  + 
1 ) ) ) )
5633, 53, 55mpjaod 673 . . . 4  |-  ( ph  ->  (  seq M ( 
.+  ,  F ,  S ) `  K
)  =  Z )
5756a1i 9 . . 3  |-  ( K  e.  ZZ  ->  ( ph  ->  (  seq M
(  .+  ,  F ,  S ) `  K
)  =  Z ) )
58 simpr 108 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ZZ>= `  K )
)  ->  k  e.  ( ZZ>= `  K )
)
5917adantr 270 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ZZ>= `  K )
)  ->  K  e.  ( ZZ>= `  M )
)
60 uztrn 9004 . . . . . . . . . 10  |-  ( ( k  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  M )
)  ->  k  e.  ( ZZ>= `  M )
)
6158, 59, 60syl2anc 403 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ZZ>= `  K )
)  ->  k  e.  ( ZZ>= `  M )
)
6224adantlr 461 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  K )
)  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
6326adantlr 461 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  K )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
6461, 62, 63iseqp1 9847 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  K )
)  ->  (  seq M (  .+  ,  F ,  S ) `  ( k  +  1 ) )  =  ( (  seq M ( 
.+  ,  F ,  S ) `  k
)  .+  ( F `  ( k  +  1 ) ) ) )
6564adantr 270 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  K )
)  /\  (  seq M (  .+  ,  F ,  S ) `  k )  =  Z )  ->  (  seq M (  .+  ,  F ,  S ) `  ( k  +  1 ) )  =  ( (  seq M ( 
.+  ,  F ,  S ) `  k
)  .+  ( F `  ( k  +  1 ) ) ) )
66 simpr 108 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  K )
)  /\  (  seq M (  .+  ,  F ,  S ) `  k )  =  Z )  ->  (  seq M (  .+  ,  F ,  S ) `  k )  =  Z )
6766oveq1d 5649 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  K )
)  /\  (  seq M (  .+  ,  F ,  S ) `  k )  =  Z )  ->  ( (  seq M (  .+  ,  F ,  S ) `  k )  .+  ( F `  ( k  +  1 ) ) )  =  ( Z 
.+  ( F `  ( k  +  1 ) ) ) )
68 oveq2 5642 . . . . . . . . . 10  |-  ( x  =  ( F `  ( k  +  1 ) )  ->  ( Z  .+  x )  =  ( Z  .+  ( F `  ( k  +  1 ) ) ) )
6968eqeq1d 2096 . . . . . . . . 9  |-  ( x  =  ( F `  ( k  +  1 ) )  ->  (
( Z  .+  x
)  =  Z  <->  ( Z  .+  ( F `  (
k  +  1 ) ) )  =  Z ) )
70 iseqz.3 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  S )  ->  ( Z  .+  x )  =  Z )
7170ralrimiva 2446 . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  S  ( Z  .+  x )  =  Z )
7271adantr 270 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ZZ>= `  K )
)  ->  A. x  e.  S  ( Z  .+  x )  =  Z )
73 fveq2 5289 . . . . . . . . . . 11  |-  ( x  =  ( k  +  1 )  ->  ( F `  x )  =  ( F `  ( k  +  1 ) ) )
7473eleq1d 2156 . . . . . . . . . 10  |-  ( x  =  ( k  +  1 )  ->  (
( F `  x
)  e.  S  <->  ( F `  ( k  +  1 ) )  e.  S
) )
7524ralrimiva 2446 . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  (
ZZ>= `  M ) ( F `  x )  e.  S )
7675adantr 270 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ZZ>= `  K )
)  ->  A. x  e.  ( ZZ>= `  M )
( F `  x
)  e.  S )
77 peano2uz 9040 . . . . . . . . . . 11  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( k  +  1 )  e.  ( ZZ>= `  M )
)
7861, 77syl 14 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ZZ>= `  K )
)  ->  ( k  +  1 )  e.  ( ZZ>= `  M )
)
7974, 76, 78rspcdva 2727 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ZZ>= `  K )
)  ->  ( F `  ( k  +  1 ) )  e.  S
)
8069, 72, 79rspcdva 2727 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  K )
)  ->  ( Z  .+  ( F `  (
k  +  1 ) ) )  =  Z )
8180adantr 270 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  K )
)  /\  (  seq M (  .+  ,  F ,  S ) `  k )  =  Z )  ->  ( Z  .+  ( F `  (
k  +  1 ) ) )  =  Z )
8265, 67, 813eqtrd 2124 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  K )
)  /\  (  seq M (  .+  ,  F ,  S ) `  k )  =  Z )  ->  (  seq M (  .+  ,  F ,  S ) `  ( k  +  1 ) )  =  Z )
8382ex 113 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  K )
)  ->  ( (  seq M (  .+  ,  F ,  S ) `  k )  =  Z  ->  (  seq M
(  .+  ,  F ,  S ) `  (
k  +  1 ) )  =  Z ) )
8483expcom 114 . . . 4  |-  ( k  e.  ( ZZ>= `  K
)  ->  ( ph  ->  ( (  seq M
(  .+  ,  F ,  S ) `  k
)  =  Z  -> 
(  seq M (  .+  ,  F ,  S ) `
 ( k  +  1 ) )  =  Z ) ) )
8584a2d 26 . . 3  |-  ( k  e.  ( ZZ>= `  K
)  ->  ( ( ph  ->  (  seq M
(  .+  ,  F ,  S ) `  k
)  =  Z )  ->  ( ph  ->  (  seq M (  .+  ,  F ,  S ) `
 ( k  +  1 ) )  =  Z ) ) )
866, 9, 12, 15, 57, 85uzind4 9045 . 2  |-  ( N  e.  ( ZZ>= `  K
)  ->  ( ph  ->  (  seq M ( 
.+  ,  F ,  S ) `  N
)  =  Z ) )
873, 86mpcom 36 1  |-  ( ph  ->  (  seq M ( 
.+  ,  F ,  S ) `  N
)  =  Z )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    \/ wo 664    = wceq 1289    e. wcel 1438   A.wral 2359   ` cfv 5002  (class class class)co 5634   1c1 7330    + caddc 7332    - cmin 7632   ZZcz 8720   ZZ>=cuz 8988   ...cfz 9393    seqcseq4 9816
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-ltadd 7440
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-frec 6138  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-inn 8395  df-n0 8644  df-z 8721  df-uz 8989  df-fz 9394  df-iseq 9818
This theorem is referenced by:  ibcval5  10136
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