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Theorem seq3z 10497
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 10008 . . 3  |-  ( K  e.  ( M ... N )  ->  N  e.  ( ZZ>= `  K )
)
31, 2syl 14 . 2  |-  ( ph  ->  N  e.  ( ZZ>= `  K ) )
4 fveqeq2 5520 . . . 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 5520 . . . 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 5520 . . . 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 5520 . . . 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 10007 . . . . . . . . . 10  |-  ( K  e.  ( M ... N )  ->  K  e.  ( ZZ>= `  M )
)
131, 12syl 14 . . . . . . . . 9  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
14 eluzelz 9526 . . . . . . . . 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 9533 . . . . . . . . . 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 10446 . . . . . . 7  |-  ( ph  ->  (  seq K ( 
.+  ,  F ) `
 K )  =  ( F `  K
) )
24 seqz.7 . . . . . . 7  |-  ( ph  ->  ( F `  K
)  =  Z )
2523, 24eqtrd 2210 . . . . . 6  |-  ( ph  ->  (  seq K ( 
.+  ,  F ) `
 K )  =  Z )
26 seqeq1 10434 . . . . . . . 8  |-  ( K  =  M  ->  seq K (  .+  ,  F )  =  seq M (  .+  ,  F ) )
2726fveq1d 5513 . . . . . . 7  |-  ( K  =  M  ->  (  seq K (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  K
) )
2827eqeq1d 2186 . . . . . 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 9522 . . . . . . . . . 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 10454 . . . . . . 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 5885 . . . . . . 7  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( (  seq M (  .+  ,  F ) `  ( K  -  1 ) )  .+  ( F `
 K ) )  =  ( (  seq M (  .+  ,  F ) `  ( K  -  1 ) )  .+  Z ) )
39 oveq1 5876 . . . . . . . . 9  |-  ( x  =  (  seq M
(  .+  ,  F
) `  ( K  -  1 ) )  ->  ( x  .+  Z )  =  ( (  seq M ( 
.+  ,  F ) `
 ( K  - 
1 ) )  .+  Z ) )
4039eqeq1d 2186 . . . . . . . 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 2550 . . . . . . . . 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 2177 . . . . . . . . . 10  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
4544, 32, 34, 35seqf 10447 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  seq M ( 
.+  ,  F ) : ( ZZ>= `  M
) --> S )
46 eluzp1m1 9540 . . . . . . . . . 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 5648 . . . . . . . 8  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq M (  .+  ,  F ) `  ( K  -  1 ) )  e.  S )
4940, 43, 48rspcdva 2846 . . . . . . 7  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( (  seq M (  .+  ,  F ) `  ( K  -  1 ) )  .+  Z )  =  Z )
5036, 38, 493eqtrd 2214 . . . . . 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 9550 . . . . . 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 9533 . . . . . . . . . 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 10448 . . . . . . . 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 5884 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  K )
)  /\  (  seq M (  .+  ,  F ) `  k
)  =  Z )  ->  ( (  seq M (  .+  ,  F ) `  k
)  .+  ( F `  ( k  +  1 ) ) )  =  ( Z  .+  ( F `  ( k  +  1 ) ) ) )
66 oveq2 5877 . . . . . . . . . 10  |-  ( x  =  ( F `  ( k  +  1 ) )  ->  ( Z  .+  x )  =  ( Z  .+  ( F `  ( k  +  1 ) ) ) )
6766eqeq1d 2186 . . . . . . . . 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 2550 . . . . . . . . . 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 5511 . . . . . . . . . . 11  |-  ( x  =  ( k  +  1 )  ->  ( F `  x )  =  ( F `  ( k  +  1 ) ) )
7271eleq1d 2246 . . . . . . . . . 10  |-  ( x  =  ( k  +  1 )  ->  (
( F `  x
)  e.  S  <->  ( F `  ( k  +  1 ) )  e.  S
) )
7320ralrimiva 2550 . . . . . . . . . . 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 9572 . . . . . . . . . . 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 2846 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ZZ>= `  K )
)  ->  ( F `  ( k  +  1 ) )  e.  S
)
7867, 70, 77rspcdva 2846 . . . . . . . 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 2214 . . . . . 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 9577 . 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 2148   A.wral 2455   ` cfv 5212  (class class class)co 5869   1c1 7803    + caddc 7805    - cmin 8118   ZZcz 9242   ZZ>=cuz 9517   ...cfz 9995    seqcseq 10431
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-ltadd 7918
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-inn 8909  df-n0 9166  df-z 9243  df-uz 9518  df-fz 9996  df-seqfrec 10432
This theorem is referenced by:  bcval5  10727  lgsne0  14106
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