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Theorem seq3z 10296
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.) (Revised by Jim Kingdon, 23-Apr-2023.)
Hypotheses
Ref Expression
seq3homo.1  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
seq3homo.2  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
seqz.3  |-  ( (
ph  /\  x  e.  S )  ->  ( Z  .+  x )  =  Z )
seqz.4  |-  ( (
ph  /\  x  e.  S )  ->  (
x  .+  Z )  =  Z )
seqz.5  |-  ( ph  ->  K  e.  ( M ... N ) )
seqz.7  |-  ( ph  ->  ( F `  K
)  =  Z )
Assertion
Ref Expression
seq3z  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 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

Proof of Theorem seq3z
Dummy variables  k  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 seqz.5 . . 3  |-  ( ph  ->  K  e.  ( M ... N ) )
2 elfzuz3 9815 . . 3  |-  ( K  e.  ( M ... N )  ->  N  e.  ( ZZ>= `  K )
)
31, 2syl 14 . 2  |-  ( ph  ->  N  e.  ( ZZ>= `  K ) )
4 fveqeq2 5430 . . . 4  |-  ( w  =  K  ->  (
(  seq M (  .+  ,  F ) `  w
)  =  Z  <->  (  seq M (  .+  ,  F ) `  K
)  =  Z ) )
54imbi2d 229 . . 3  |-  ( w  =  K  ->  (
( ph  ->  (  seq M (  .+  ,  F ) `  w
)  =  Z )  <-> 
( ph  ->  (  seq M (  .+  ,  F ) `  K
)  =  Z ) ) )
6 fveqeq2 5430 . . . 4  |-  ( w  =  k  ->  (
(  seq M (  .+  ,  F ) `  w
)  =  Z  <->  (  seq M (  .+  ,  F ) `  k
)  =  Z ) )
76imbi2d 229 . . 3  |-  ( w  =  k  ->  (
( ph  ->  (  seq M (  .+  ,  F ) `  w
)  =  Z )  <-> 
( ph  ->  (  seq M (  .+  ,  F ) `  k
)  =  Z ) ) )
8 fveqeq2 5430 . . . 4  |-  ( w  =  ( k  +  1 )  ->  (
(  seq M (  .+  ,  F ) `  w
)  =  Z  <->  (  seq M (  .+  ,  F ) `  (
k  +  1 ) )  =  Z ) )
98imbi2d 229 . . 3  |-  ( w  =  ( k  +  1 )  ->  (
( ph  ->  (  seq M (  .+  ,  F ) `  w
)  =  Z )  <-> 
( ph  ->  (  seq M (  .+  ,  F ) `  (
k  +  1 ) )  =  Z ) ) )
10 fveqeq2 5430 . . . 4  |-  ( w  =  N  ->  (
(  seq M (  .+  ,  F ) `  w
)  =  Z  <->  (  seq M (  .+  ,  F ) `  N
)  =  Z ) )
1110imbi2d 229 . . 3  |-  ( w  =  N  ->  (
( ph  ->  (  seq M (  .+  ,  F ) `  w
)  =  Z )  <-> 
( ph  ->  (  seq M (  .+  ,  F ) `  N
)  =  Z ) ) )
12 elfzuz 9814 . . . . . . . . . 10  |-  ( K  e.  ( M ... N )  ->  K  e.  ( ZZ>= `  M )
)
131, 12syl 14 . . . . . . . . 9  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
14 eluzelz 9347 . . . . . . . . 9  |-  ( K  e.  ( ZZ>= `  M
)  ->  K  e.  ZZ )
1513, 14syl 14 . . . . . . . 8  |-  ( ph  ->  K  e.  ZZ )
16 simpr 109 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  x  e.  ( ZZ>= `  K )
)
1713adantr 274 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  K  e.  ( ZZ>= `  M )
)
18 uztrn 9354 . . . . . . . . . 10  |-  ( ( x  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  M )
)  ->  x  e.  ( ZZ>= `  M )
)
1916, 17, 18syl2anc 408 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  x  e.  ( ZZ>= `  M )
)
20 seq3homo.2 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
2119, 20syldan 280 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  ( F `  x )  e.  S
)
22 seq3homo.1 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
2315, 21, 22seq3-1 10245 . . . . . . 7  |-  ( ph  ->  (  seq K ( 
.+  ,  F ) `
 K )  =  ( F `  K
) )
24 seqz.7 . . . . . . 7  |-  ( ph  ->  ( F `  K
)  =  Z )
2523, 24eqtrd 2172 . . . . . 6  |-  ( ph  ->  (  seq K ( 
.+  ,  F ) `
 K )  =  Z )
26 seqeq1 10233 . . . . . . . 8  |-  ( K  =  M  ->  seq K (  .+  ,  F )  =  seq M (  .+  ,  F ) )
2726fveq1d 5423 . . . . . . 7  |-  ( K  =  M  ->  (  seq K (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  K
) )
2827eqeq1d 2148 . . . . . 6  |-  ( K  =  M  ->  (
(  seq K (  .+  ,  F ) `  K
)  =  Z  <->  (  seq M (  .+  ,  F ) `  K
)  =  Z ) )
2925, 28syl5ibcom 154 . . . . 5  |-  ( ph  ->  ( K  =  M  ->  (  seq M
(  .+  ,  F
) `  K )  =  Z ) )
30 eluzel2 9343 . . . . . . . . . 10  |-  ( K  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
3113, 30syl 14 . . . . . . . . 9  |-  ( ph  ->  M  e.  ZZ )
3231adantr 274 . . . . . . . 8  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  M  e.  ZZ )
33 simpr 109 . . . . . . . 8  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  K  e.  ( ZZ>= `  ( M  +  1 ) ) )
3420adantlr 468 . . . . . . . 8  |-  ( ( ( ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
3522adantlr 468 . . . . . . . 8  |-  ( ( ( ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
3632, 33, 34, 35seq3m1 10253 . . . . . . 7  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq M (  .+  ,  F ) `  K
)  =  ( (  seq M (  .+  ,  F ) `  ( K  -  1 ) )  .+  ( F `
 K ) ) )
3724adantr 274 . . . . . . . 8  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( F `  K )  =  Z )
3837oveq2d 5790 . . . . . . 7  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( (  seq M (  .+  ,  F ) `  ( K  -  1 ) )  .+  ( F `
 K ) )  =  ( (  seq M (  .+  ,  F ) `  ( K  -  1 ) )  .+  Z ) )
39 oveq1 5781 . . . . . . . . 9  |-  ( x  =  (  seq M
(  .+  ,  F
) `  ( K  -  1 ) )  ->  ( x  .+  Z )  =  ( (  seq M ( 
.+  ,  F ) `
 ( K  - 
1 ) )  .+  Z ) )
4039eqeq1d 2148 . . . . . . . 8  |-  ( x  =  (  seq M
(  .+  ,  F
) `  ( K  -  1 ) )  ->  ( ( x 
.+  Z )  =  Z  <->  ( (  seq M (  .+  ,  F ) `  ( K  -  1 ) )  .+  Z )  =  Z ) )
41 seqz.4 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  S )  ->  (
x  .+  Z )  =  Z )
4241ralrimiva 2505 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  S  ( x  .+  Z )  =  Z )
4342adantr 274 . . . . . . . 8  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  A. x  e.  S  ( x  .+  Z )  =  Z )
44 eqid 2139 . . . . . . . . . 10  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
4544, 32, 34, 35seqf 10246 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  seq M ( 
.+  ,  F ) : ( ZZ>= `  M
) --> S )
46 eluzp1m1 9361 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  K  e.  ( ZZ>= `  ( M  +  1
) ) )  -> 
( K  -  1 )  e.  ( ZZ>= `  M ) )
4731, 46sylan 281 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( K  -  1 )  e.  ( ZZ>= `  M )
)
4845, 47ffvelrnd 5556 . . . . . . . 8  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq M (  .+  ,  F ) `  ( K  -  1 ) )  e.  S )
4940, 43, 48rspcdva 2794 . . . . . . 7  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( (  seq M (  .+  ,  F ) `  ( K  -  1 ) )  .+  Z )  =  Z )
5036, 38, 493eqtrd 2176 . . . . . 6  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq M (  .+  ,  F ) `  K
)  =  Z )
5150ex 114 . . . . 5  |-  ( ph  ->  ( K  e.  (
ZZ>= `  ( M  + 
1 ) )  -> 
(  seq M (  .+  ,  F ) `  K
)  =  Z ) )
52 uzp1 9371 . . . . . 6  |-  ( K  e.  ( ZZ>= `  M
)  ->  ( K  =  M  \/  K  e.  ( ZZ>= `  ( M  +  1 ) ) ) )
5313, 52syl 14 . . . . 5  |-  ( ph  ->  ( K  =  M  \/  K  e.  (
ZZ>= `  ( M  + 
1 ) ) ) )
5429, 51, 53mpjaod 707 . . . 4  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 K )  =  Z )
5554a1i 9 . . 3  |-  ( K  e.  ZZ  ->  ( ph  ->  (  seq M
(  .+  ,  F
) `  K )  =  Z ) )
56 simpr 109 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ZZ>= `  K )
)  ->  k  e.  ( ZZ>= `  K )
)
5713adantr 274 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ZZ>= `  K )
)  ->  K  e.  ( ZZ>= `  M )
)
58 uztrn 9354 . . . . . . . . . 10  |-  ( ( k  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  M )
)  ->  k  e.  ( ZZ>= `  M )
)
5956, 57, 58syl2anc 408 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ZZ>= `  K )
)  ->  k  e.  ( ZZ>= `  M )
)
6020adantlr 468 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  K )
)  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
6122adantlr 468 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  K )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
6259, 60, 61seq3p1 10247 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  K )
)  ->  (  seq M (  .+  ,  F ) `  (
k  +  1 ) )  =  ( (  seq M (  .+  ,  F ) `  k
)  .+  ( F `  ( k  +  1 ) ) ) )
6362adantr 274 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  K )
)  /\  (  seq M (  .+  ,  F ) `  k
)  =  Z )  ->  (  seq M
(  .+  ,  F
) `  ( k  +  1 ) )  =  ( (  seq M (  .+  ,  F ) `  k
)  .+  ( F `  ( k  +  1 ) ) ) )
64 simpr 109 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  K )
)  /\  (  seq M (  .+  ,  F ) `  k
)  =  Z )  ->  (  seq M
(  .+  ,  F
) `  k )  =  Z )
6564oveq1d 5789 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  K )
)  /\  (  seq M (  .+  ,  F ) `  k
)  =  Z )  ->  ( (  seq M (  .+  ,  F ) `  k
)  .+  ( F `  ( k  +  1 ) ) )  =  ( Z  .+  ( F `  ( k  +  1 ) ) ) )
66 oveq2 5782 . . . . . . . . . 10  |-  ( x  =  ( F `  ( k  +  1 ) )  ->  ( Z  .+  x )  =  ( Z  .+  ( F `  ( k  +  1 ) ) ) )
6766eqeq1d 2148 . . . . . . . . 9  |-  ( x  =  ( F `  ( k  +  1 ) )  ->  (
( Z  .+  x
)  =  Z  <->  ( Z  .+  ( F `  (
k  +  1 ) ) )  =  Z ) )
68 seqz.3 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  S )  ->  ( Z  .+  x )  =  Z )
6968ralrimiva 2505 . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  S  ( Z  .+  x )  =  Z )
7069adantr 274 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ZZ>= `  K )
)  ->  A. x  e.  S  ( Z  .+  x )  =  Z )
71 fveq2 5421 . . . . . . . . . . 11  |-  ( x  =  ( k  +  1 )  ->  ( F `  x )  =  ( F `  ( k  +  1 ) ) )
7271eleq1d 2208 . . . . . . . . . 10  |-  ( x  =  ( k  +  1 )  ->  (
( F `  x
)  e.  S  <->  ( F `  ( k  +  1 ) )  e.  S
) )
7320ralrimiva 2505 . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  (
ZZ>= `  M ) ( F `  x )  e.  S )
7473adantr 274 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ZZ>= `  K )
)  ->  A. x  e.  ( ZZ>= `  M )
( F `  x
)  e.  S )
75 peano2uz 9390 . . . . . . . . . . 11  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( k  +  1 )  e.  ( ZZ>= `  M )
)
7659, 75syl 14 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ZZ>= `  K )
)  ->  ( k  +  1 )  e.  ( ZZ>= `  M )
)
7772, 74, 76rspcdva 2794 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ZZ>= `  K )
)  ->  ( F `  ( k  +  1 ) )  e.  S
)
7867, 70, 77rspcdva 2794 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  K )
)  ->  ( Z  .+  ( F `  (
k  +  1 ) ) )  =  Z )
7978adantr 274 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  K )
)  /\  (  seq M (  .+  ,  F ) `  k
)  =  Z )  ->  ( Z  .+  ( F `  ( k  +  1 ) ) )  =  Z )
8063, 65, 793eqtrd 2176 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  K )
)  /\  (  seq M (  .+  ,  F ) `  k
)  =  Z )  ->  (  seq M
(  .+  ,  F
) `  ( k  +  1 ) )  =  Z )
8180ex 114 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  K )
)  ->  ( (  seq M (  .+  ,  F ) `  k
)  =  Z  -> 
(  seq M (  .+  ,  F ) `  (
k  +  1 ) )  =  Z ) )
8281expcom 115 . . . 4  |-  ( k  e.  ( ZZ>= `  K
)  ->  ( ph  ->  ( (  seq M
(  .+  ,  F
) `  k )  =  Z  ->  (  seq M (  .+  ,  F ) `  (
k  +  1 ) )  =  Z ) ) )
8382a2d 26 . . 3  |-  ( k  e.  ( ZZ>= `  K
)  ->  ( ( ph  ->  (  seq M
(  .+  ,  F
) `  k )  =  Z )  ->  ( ph  ->  (  seq M
(  .+  ,  F
) `  ( k  +  1 ) )  =  Z ) ) )
845, 7, 9, 11, 55, 83uzind4 9395 . 2  |-  ( N  e.  ( ZZ>= `  K
)  ->  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 N )  =  Z ) )
853, 84mpcom 36 1  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 N )  =  Z )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 697    = wceq 1331    e. wcel 1480   A.wral 2416   ` cfv 5123  (class class class)co 5774   1c1 7633    + caddc 7635    - cmin 7945   ZZcz 9066   ZZ>=cuz 9338   ...cfz 9802    seqcseq 10230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-addcom 7732  ax-addass 7734  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-0id 7740  ax-rnegex 7741  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-ltadd 7748
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-inn 8733  df-n0 8990  df-z 9067  df-uz 9339  df-fz 9803  df-seqfrec 10231
This theorem is referenced by:  bcval5  10521
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