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Theorem seq3z 10481
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 9992 . . 3  |-  ( K  e.  ( M ... N )  ->  N  e.  ( ZZ>= `  K )
)
31, 2syl 14 . 2  |-  ( ph  ->  N  e.  ( ZZ>= `  K ) )
4 fveqeq2 5516 . . . 4  |-  ( w  =  K  ->  (
(  seq M (  .+  ,  F ) `  w
)  =  Z  <->  (  seq M (  .+  ,  F ) `  K
)  =  Z ) )
54imbi2d 230 . . 3  |-  ( w  =  K  ->  (
( ph  ->  (  seq M (  .+  ,  F ) `  w
)  =  Z )  <-> 
( ph  ->  (  seq M (  .+  ,  F ) `  K
)  =  Z ) ) )
6 fveqeq2 5516 . . . 4  |-  ( w  =  k  ->  (
(  seq M (  .+  ,  F ) `  w
)  =  Z  <->  (  seq M (  .+  ,  F ) `  k
)  =  Z ) )
76imbi2d 230 . . 3  |-  ( w  =  k  ->  (
( ph  ->  (  seq M (  .+  ,  F ) `  w
)  =  Z )  <-> 
( ph  ->  (  seq M (  .+  ,  F ) `  k
)  =  Z ) ) )
8 fveqeq2 5516 . . . 4  |-  ( w  =  ( k  +  1 )  ->  (
(  seq M (  .+  ,  F ) `  w
)  =  Z  <->  (  seq M (  .+  ,  F ) `  (
k  +  1 ) )  =  Z ) )
98imbi2d 230 . . 3  |-  ( w  =  ( k  +  1 )  ->  (
( ph  ->  (  seq M (  .+  ,  F ) `  w
)  =  Z )  <-> 
( ph  ->  (  seq M (  .+  ,  F ) `  (
k  +  1 ) )  =  Z ) ) )
10 fveqeq2 5516 . . . 4  |-  ( w  =  N  ->  (
(  seq M (  .+  ,  F ) `  w
)  =  Z  <->  (  seq M (  .+  ,  F ) `  N
)  =  Z ) )
1110imbi2d 230 . . 3  |-  ( w  =  N  ->  (
( ph  ->  (  seq M (  .+  ,  F ) `  w
)  =  Z )  <-> 
( ph  ->  (  seq M (  .+  ,  F ) `  N
)  =  Z ) ) )
12 elfzuz 9991 . . . . . . . . . 10  |-  ( K  e.  ( M ... N )  ->  K  e.  ( ZZ>= `  M )
)
131, 12syl 14 . . . . . . . . 9  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
14 eluzelz 9510 . . . . . . . . 9  |-  ( K  e.  ( ZZ>= `  M
)  ->  K  e.  ZZ )
1513, 14syl 14 . . . . . . . 8  |-  ( ph  ->  K  e.  ZZ )
16 simpr 110 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  x  e.  ( ZZ>= `  K )
)
1713adantr 276 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  K  e.  ( ZZ>= `  M )
)
18 uztrn 9517 . . . . . . . . . 10  |-  ( ( x  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  M )
)  ->  x  e.  ( ZZ>= `  M )
)
1916, 17, 18syl2anc 411 . . . . . . . . 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 282 . . . . . . . 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 10430 . . . . . . 7  |-  ( ph  ->  (  seq K ( 
.+  ,  F ) `
 K )  =  ( F `  K
) )
24 seqz.7 . . . . . . 7  |-  ( ph  ->  ( F `  K
)  =  Z )
2523, 24eqtrd 2208 . . . . . 6  |-  ( ph  ->  (  seq K ( 
.+  ,  F ) `
 K )  =  Z )
26 seqeq1 10418 . . . . . . . 8  |-  ( K  =  M  ->  seq K (  .+  ,  F )  =  seq M (  .+  ,  F ) )
2726fveq1d 5509 . . . . . . 7  |-  ( K  =  M  ->  (  seq K (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  K
) )
2827eqeq1d 2184 . . . . . 6  |-  ( K  =  M  ->  (
(  seq K (  .+  ,  F ) `  K
)  =  Z  <->  (  seq M (  .+  ,  F ) `  K
)  =  Z ) )
2925, 28syl5ibcom 155 . . . . 5  |-  ( ph  ->  ( K  =  M  ->  (  seq M
(  .+  ,  F
) `  K )  =  Z ) )
30 eluzel2 9506 . . . . . . . . . 10  |-  ( K  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
3113, 30syl 14 . . . . . . . . 9  |-  ( ph  ->  M  e.  ZZ )
3231adantr 276 . . . . . . . 8  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  M  e.  ZZ )
33 simpr 110 . . . . . . . 8  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  K  e.  ( ZZ>= `  ( M  +  1 ) ) )
3420adantlr 477 . . . . . . . 8  |-  ( ( ( ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
3522adantlr 477 . . . . . . . 8  |-  ( ( ( ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
3632, 33, 34, 35seq3m1 10438 . . . . . . 7  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq M (  .+  ,  F ) `  K
)  =  ( (  seq M (  .+  ,  F ) `  ( K  -  1 ) )  .+  ( F `
 K ) ) )
3724adantr 276 . . . . . . . 8  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( F `  K )  =  Z )
3837oveq2d 5881 . . . . . . 7  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( (  seq M (  .+  ,  F ) `  ( K  -  1 ) )  .+  ( F `
 K ) )  =  ( (  seq M (  .+  ,  F ) `  ( K  -  1 ) )  .+  Z ) )
39 oveq1 5872 . . . . . . . . 9  |-  ( x  =  (  seq M
(  .+  ,  F
) `  ( K  -  1 ) )  ->  ( x  .+  Z )  =  ( (  seq M ( 
.+  ,  F ) `
 ( K  - 
1 ) )  .+  Z ) )
4039eqeq1d 2184 . . . . . . . 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 2548 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  S  ( x  .+  Z )  =  Z )
4342adantr 276 . . . . . . . 8  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  A. x  e.  S  ( x  .+  Z )  =  Z )
44 eqid 2175 . . . . . . . . . 10  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
4544, 32, 34, 35seqf 10431 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  seq M ( 
.+  ,  F ) : ( ZZ>= `  M
) --> S )
46 eluzp1m1 9524 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  K  e.  ( ZZ>= `  ( M  +  1
) ) )  -> 
( K  -  1 )  e.  ( ZZ>= `  M ) )
4731, 46sylan 283 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( K  -  1 )  e.  ( ZZ>= `  M )
)
4845, 47ffvelcdmd 5644 . . . . . . . 8  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq M (  .+  ,  F ) `  ( K  -  1 ) )  e.  S )
4940, 43, 48rspcdva 2844 . . . . . . 7  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( (  seq M (  .+  ,  F ) `  ( K  -  1 ) )  .+  Z )  =  Z )
5036, 38, 493eqtrd 2212 . . . . . 6  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq M (  .+  ,  F ) `  K
)  =  Z )
5150ex 115 . . . . 5  |-  ( ph  ->  ( K  e.  (
ZZ>= `  ( M  + 
1 ) )  -> 
(  seq M (  .+  ,  F ) `  K
)  =  Z ) )
52 uzp1 9534 . . . . . 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 718 . . . 4  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 K )  =  Z )
5554a1i 9 . . 3  |-  ( K  e.  ZZ  ->  ( ph  ->  (  seq M
(  .+  ,  F
) `  K )  =  Z ) )
56 simpr 110 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ZZ>= `  K )
)  ->  k  e.  ( ZZ>= `  K )
)
5713adantr 276 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ZZ>= `  K )
)  ->  K  e.  ( ZZ>= `  M )
)
58 uztrn 9517 . . . . . . . . . 10  |-  ( ( k  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  M )
)  ->  k  e.  ( ZZ>= `  M )
)
5956, 57, 58syl2anc 411 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ZZ>= `  K )
)  ->  k  e.  ( ZZ>= `  M )
)
6020adantlr 477 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  K )
)  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
6122adantlr 477 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  K )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
6259, 60, 61seq3p1 10432 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  K )
)  ->  (  seq M (  .+  ,  F ) `  (
k  +  1 ) )  =  ( (  seq M (  .+  ,  F ) `  k
)  .+  ( F `  ( k  +  1 ) ) ) )
6362adantr 276 . . . . . . 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 110 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  K )
)  /\  (  seq M (  .+  ,  F ) `  k
)  =  Z )  ->  (  seq M
(  .+  ,  F
) `  k )  =  Z )
6564oveq1d 5880 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  K )
)  /\  (  seq M (  .+  ,  F ) `  k
)  =  Z )  ->  ( (  seq M (  .+  ,  F ) `  k
)  .+  ( F `  ( k  +  1 ) ) )  =  ( Z  .+  ( F `  ( k  +  1 ) ) ) )
66 oveq2 5873 . . . . . . . . . 10  |-  ( x  =  ( F `  ( k  +  1 ) )  ->  ( Z  .+  x )  =  ( Z  .+  ( F `  ( k  +  1 ) ) ) )
6766eqeq1d 2184 . . . . . . . . 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 2548 . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  S  ( Z  .+  x )  =  Z )
7069adantr 276 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ZZ>= `  K )
)  ->  A. x  e.  S  ( Z  .+  x )  =  Z )
71 fveq2 5507 . . . . . . . . . . 11  |-  ( x  =  ( k  +  1 )  ->  ( F `  x )  =  ( F `  ( k  +  1 ) ) )
7271eleq1d 2244 . . . . . . . . . 10  |-  ( x  =  ( k  +  1 )  ->  (
( F `  x
)  e.  S  <->  ( F `  ( k  +  1 ) )  e.  S
) )
7320ralrimiva 2548 . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  (
ZZ>= `  M ) ( F `  x )  e.  S )
7473adantr 276 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ZZ>= `  K )
)  ->  A. x  e.  ( ZZ>= `  M )
( F `  x
)  e.  S )
75 peano2uz 9556 . . . . . . . . . . 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 2844 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ZZ>= `  K )
)  ->  ( F `  ( k  +  1 ) )  e.  S
)
7867, 70, 77rspcdva 2844 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ZZ>= `  K )
)  ->  ( Z  .+  ( F `  (
k  +  1 ) ) )  =  Z )
7978adantr 276 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  K )
)  /\  (  seq M (  .+  ,  F ) `  k
)  =  Z )  ->  ( Z  .+  ( F `  ( k  +  1 ) ) )  =  Z )
8063, 65, 793eqtrd 2212 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  K )
)  /\  (  seq M (  .+  ,  F ) `  k
)  =  Z )  ->  (  seq M
(  .+  ,  F
) `  ( k  +  1 ) )  =  Z )
8180ex 115 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  K )
)  ->  ( (  seq M (  .+  ,  F ) `  k
)  =  Z  -> 
(  seq M (  .+  ,  F ) `  (
k  +  1 ) )  =  Z ) )
8281expcom 116 . . . 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 9561 . 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 104    \/ wo 708    = wceq 1353    e. wcel 2146   A.wral 2453   ` cfv 5208  (class class class)co 5865   1c1 7787    + caddc 7789    - cmin 8102   ZZcz 9226   ZZ>=cuz 9501   ...cfz 9979    seqcseq 10415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-addcom 7886  ax-addass 7888  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-0id 7894  ax-rnegex 7895  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-ltadd 7902
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-inn 8893  df-n0 9150  df-z 9227  df-uz 9502  df-fz 9980  df-seqfrec 10416
This theorem is referenced by:  bcval5  10711  lgsne0  14019
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