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Theorem seqz 11110
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
seqhomo.1  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
seqhomo.2  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  ( 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.6  |-  ( ph  ->  N  e.  V )
seqz.7  |-  ( ph  ->  ( F `  K
)  =  Z )
Assertion
Ref Expression
seqz  |-  ( 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
Allowed substitution hints:    V( x, y)

Proof of Theorem seqz
StepHypRef Expression
1 seqz.5 . . . 4  |-  ( ph  ->  K  e.  ( M ... N ) )
2 elfzuz 10810 . . . 4  |-  ( K  e.  ( M ... N )  ->  K  e.  ( ZZ>= `  M )
)
31, 2syl 15 . . 3  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
4 eluzelz 10254 . . . . . . . . 9  |-  ( K  e.  ( ZZ>= `  M
)  ->  K  e.  ZZ )
53, 4syl 15 . . . . . . . 8  |-  ( ph  ->  K  e.  ZZ )
6 seq1 11075 . . . . . . . 8  |-  ( K  e.  ZZ  ->  (  seq  K (  .+  ,  F ) `  K
)  =  ( F `
 K ) )
75, 6syl 15 . . . . . . 7  |-  ( ph  ->  (  seq  K ( 
.+  ,  F ) `
 K )  =  ( F `  K
) )
8 seqz.7 . . . . . . 7  |-  ( ph  ->  ( F `  K
)  =  Z )
97, 8eqtrd 2328 . . . . . 6  |-  ( ph  ->  (  seq  K ( 
.+  ,  F ) `
 K )  =  Z )
10 seqeq1 11065 . . . . . . . 8  |-  ( K  =  M  ->  seq  K (  .+  ,  F
)  =  seq  M
(  .+  ,  F
) )
1110fveq1d 5543 . . . . . . 7  |-  ( K  =  M  ->  (  seq  K (  .+  ,  F ) `  K
)  =  (  seq 
M (  .+  ,  F ) `  K
) )
1211eqeq1d 2304 . . . . . 6  |-  ( K  =  M  ->  (
(  seq  K (  .+  ,  F ) `  K )  =  Z  <-> 
(  seq  M (  .+  ,  F ) `  K )  =  Z ) )
139, 12syl5ibcom 211 . . . . 5  |-  ( ph  ->  ( K  =  M  ->  (  seq  M
(  .+  ,  F
) `  K )  =  Z ) )
14 eluzel2 10251 . . . . . . . . 9  |-  ( K  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
153, 14syl 15 . . . . . . . 8  |-  ( ph  ->  M  e.  ZZ )
16 seqm1 11079 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  K  e.  ( ZZ>= `  ( M  +  1
) ) )  -> 
(  seq  M (  .+  ,  F ) `  K )  =  ( (  seq  M ( 
.+  ,  F ) `
 ( K  - 
1 ) )  .+  ( F `  K ) ) )
1715, 16sylan 457 . . . . . . 7  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq  M (  .+  ,  F
) `  K )  =  ( (  seq 
M (  .+  ,  F ) `  ( K  -  1 ) )  .+  ( F `
 K ) ) )
188adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( F `  K )  =  Z )
1918oveq2d 5890 . . . . . . . 8  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( (  seq  M (  .+  ,  F ) `  ( K  -  1 ) )  .+  ( F `
 K ) )  =  ( (  seq 
M (  .+  ,  F ) `  ( K  -  1 ) )  .+  Z ) )
20 eluzp1m1 10267 . . . . . . . . . . 11  |-  ( ( M  e.  ZZ  /\  K  e.  ( ZZ>= `  ( M  +  1
) ) )  -> 
( K  -  1 )  e.  ( ZZ>= `  M ) )
2115, 20sylan 457 . . . . . . . . . 10  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( K  -  1 )  e.  ( ZZ>= `  M )
)
22 fzssp1 10850 . . . . . . . . . . . . . . 15  |-  ( M ... ( K  - 
1 ) )  C_  ( M ... ( ( K  -  1 )  +  1 ) )
235zcnd 10134 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  K  e.  CC )
24 ax-1cn 8811 . . . . . . . . . . . . . . . . 17  |-  1  e.  CC
25 npcan 9076 . . . . . . . . . . . . . . . . 17  |-  ( ( K  e.  CC  /\  1  e.  CC )  ->  ( ( K  - 
1 )  +  1 )  =  K )
2623, 24, 25sylancl 643 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( K  - 
1 )  +  1 )  =  K )
2726oveq2d 5890 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( M ... (
( K  -  1 )  +  1 ) )  =  ( M ... K ) )
2822, 27syl5sseq 3239 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( M ... ( K  -  1 ) )  C_  ( M ... K ) )
29 elfzuz3 10811 . . . . . . . . . . . . . . . 16  |-  ( K  e.  ( M ... N )  ->  N  e.  ( ZZ>= `  K )
)
301, 29syl 15 . . . . . . . . . . . . . . 15  |-  ( ph  ->  N  e.  ( ZZ>= `  K ) )
31 fzss2 10847 . . . . . . . . . . . . . . 15  |-  ( N  e.  ( ZZ>= `  K
)  ->  ( M ... K )  C_  ( M ... N ) )
3230, 31syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( M ... K
)  C_  ( M ... N ) )
3328, 32sstrd 3202 . . . . . . . . . . . . 13  |-  ( ph  ->  ( M ... ( K  -  1 ) )  C_  ( M ... N ) )
3433adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( M ... ( K  -  1 ) )  C_  ( M ... N ) )
3534sselda 3193 . . . . . . . . . . 11  |-  ( ( ( ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  /\  x  e.  ( M ... ( K  -  1 ) ) )  ->  x  e.  ( M ... N
) )
36 seqhomo.2 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  ( F `  x )  e.  S
)
3736adantlr 695 . . . . . . . . . . 11  |-  ( ( ( ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  /\  x  e.  ( M ... N
) )  ->  ( F `  x )  e.  S )
3835, 37syldan 456 . . . . . . . . . 10  |-  ( ( ( ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  /\  x  e.  ( M ... ( K  -  1 ) ) )  ->  ( F `  x )  e.  S )
39 seqhomo.1 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
4039adantlr 695 . . . . . . . . . 10  |-  ( ( ( ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
4121, 38, 40seqcl 11082 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq  M (  .+  ,  F
) `  ( K  -  1 ) )  e.  S )
42 seqz.4 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  S )  ->  (
x  .+  Z )  =  Z )
4342ralrimiva 2639 . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  S  ( x  .+  Z )  =  Z )
4443adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  A. x  e.  S  ( x  .+  Z )  =  Z )
45 oveq1 5881 . . . . . . . . . . 11  |-  ( x  =  (  seq  M
(  .+  ,  F
) `  ( K  -  1 ) )  ->  ( x  .+  Z )  =  ( (  seq  M ( 
.+  ,  F ) `
 ( K  - 
1 ) )  .+  Z ) )
4645eqeq1d 2304 . . . . . . . . . 10  |-  ( x  =  (  seq  M
(  .+  ,  F
) `  ( K  -  1 ) )  ->  ( ( x 
.+  Z )  =  Z  <->  ( (  seq 
M (  .+  ,  F ) `  ( K  -  1 ) )  .+  Z )  =  Z ) )
4746rspcv 2893 . . . . . . . . 9  |-  ( (  seq  M (  .+  ,  F ) `  ( K  -  1 ) )  e.  S  -> 
( A. x  e.  S  ( x  .+  Z )  =  Z  ->  ( (  seq 
M (  .+  ,  F ) `  ( K  -  1 ) )  .+  Z )  =  Z ) )
4841, 44, 47sylc 56 . . . . . . . 8  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( (  seq  M (  .+  ,  F ) `  ( K  -  1 ) )  .+  Z )  =  Z )
4919, 48eqtrd 2328 . . . . . . 7  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( (  seq  M (  .+  ,  F ) `  ( K  -  1 ) )  .+  ( F `
 K ) )  =  Z )
5017, 49eqtrd 2328 . . . . . 6  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq  M (  .+  ,  F
) `  K )  =  Z )
5150ex 423 . . . . 5  |-  ( ph  ->  ( K  e.  (
ZZ>= `  ( M  + 
1 ) )  -> 
(  seq  M (  .+  ,  F ) `  K )  =  Z ) )
52 uzp1 10277 . . . . . 6  |-  ( K  e.  ( ZZ>= `  M
)  ->  ( K  =  M  \/  K  e.  ( ZZ>= `  ( M  +  1 ) ) ) )
533, 52syl 15 . . . . 5  |-  ( ph  ->  ( K  =  M  \/  K  e.  (
ZZ>= `  ( M  + 
1 ) ) ) )
5413, 51, 53mpjaod 370 . . . 4  |-  ( ph  ->  (  seq  M ( 
.+  ,  F ) `
 K )  =  Z )
5554, 8eqtr4d 2331 . . 3  |-  ( ph  ->  (  seq  M ( 
.+  ,  F ) `
 K )  =  ( F `  K
) )
56 eqidd 2297 . . 3  |-  ( (
ph  /\  x  e.  ( ( K  + 
1 ) ... N
) )  ->  ( F `  x )  =  ( F `  x ) )
573, 55, 30, 56seqfveq2 11084 . 2  |-  ( ph  ->  (  seq  M ( 
.+  ,  F ) `
 N )  =  (  seq  K ( 
.+  ,  F ) `
 N ) )
58 fvex 5555 . . . . . 6  |-  ( F `
 K )  e. 
_V
5958elsnc 3676 . . . . 5  |-  ( ( F `  K )  e.  { Z }  <->  ( F `  K )  =  Z )
608, 59sylibr 203 . . . 4  |-  ( ph  ->  ( F `  K
)  e.  { Z } )
61 simprl 732 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  { Z }  /\  y  e.  S )
)  ->  x  e.  { Z } )
62 elsn 3668 . . . . . . . 8  |-  ( x  e.  { Z }  <->  x  =  Z )
6361, 62sylib 188 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  { Z }  /\  y  e.  S )
)  ->  x  =  Z )
6463oveq1d 5889 . . . . . 6  |-  ( (
ph  /\  ( x  e.  { Z }  /\  y  e.  S )
)  ->  ( x  .+  y )  =  ( Z  .+  y ) )
65 simprr 733 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  { Z }  /\  y  e.  S )
)  ->  y  e.  S )
66 seqz.3 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  S )  ->  ( Z  .+  x )  =  Z )
6766ralrimiva 2639 . . . . . . . 8  |-  ( ph  ->  A. x  e.  S  ( Z  .+  x )  =  Z )
6867adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  { Z }  /\  y  e.  S )
)  ->  A. x  e.  S  ( Z  .+  x )  =  Z )
69 oveq2 5882 . . . . . . . . 9  |-  ( x  =  y  ->  ( Z  .+  x )  =  ( Z  .+  y
) )
7069eqeq1d 2304 . . . . . . . 8  |-  ( x  =  y  ->  (
( Z  .+  x
)  =  Z  <->  ( Z  .+  y )  =  Z ) )
7170rspcv 2893 . . . . . . 7  |-  ( y  e.  S  ->  ( A. x  e.  S  ( Z  .+  x )  =  Z  ->  ( Z  .+  y )  =  Z ) )
7265, 68, 71sylc 56 . . . . . 6  |-  ( (
ph  /\  ( x  e.  { Z }  /\  y  e.  S )
)  ->  ( Z  .+  y )  =  Z )
7364, 72eqtrd 2328 . . . . 5  |-  ( (
ph  /\  ( x  e.  { Z }  /\  y  e.  S )
)  ->  ( x  .+  y )  =  Z )
74 ovex 5899 . . . . . 6  |-  ( x 
.+  y )  e. 
_V
7574elsnc 3676 . . . . 5  |-  ( ( x  .+  y )  e.  { Z }  <->  ( x  .+  y )  =  Z )
7673, 75sylibr 203 . . . 4  |-  ( (
ph  /\  ( x  e.  { Z }  /\  y  e.  S )
)  ->  ( x  .+  y )  e.  { Z } )
77 peano2uz 10288 . . . . . . . 8  |-  ( K  e.  ( ZZ>= `  M
)  ->  ( K  +  1 )  e.  ( ZZ>= `  M )
)
783, 77syl 15 . . . . . . 7  |-  ( ph  ->  ( K  +  1 )  e.  ( ZZ>= `  M ) )
79 fzss1 10846 . . . . . . 7  |-  ( ( K  +  1 )  e.  ( ZZ>= `  M
)  ->  ( ( K  +  1 ) ... N )  C_  ( M ... N ) )
8078, 79syl 15 . . . . . 6  |-  ( ph  ->  ( ( K  + 
1 ) ... N
)  C_  ( M ... N ) )
8180sselda 3193 . . . . 5  |-  ( (
ph  /\  x  e.  ( ( K  + 
1 ) ... N
) )  ->  x  e.  ( M ... N
) )
8281, 36syldan 456 . . . 4  |-  ( (
ph  /\  x  e.  ( ( K  + 
1 ) ... N
) )  ->  ( F `  x )  e.  S )
8360, 76, 30, 82seqcl2 11080 . . 3  |-  ( ph  ->  (  seq  K ( 
.+  ,  F ) `
 N )  e. 
{ Z } )
84 elsni 3677 . . 3  |-  ( (  seq  K (  .+  ,  F ) `  N
)  e.  { Z }  ->  (  seq  K
(  .+  ,  F
) `  N )  =  Z )
8583, 84syl 15 . 2  |-  ( ph  ->  (  seq  K ( 
.+  ,  F ) `
 N )  =  Z )
8657, 85eqtrd 2328 1  |-  ( ph  ->  (  seq  M ( 
.+  ,  F ) `
 N )  =  Z )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   {csn 3653   ` cfv 5271  (class class class)co 5874   CCcc 8751   1c1 8754    + caddc 8756    - cmin 9053   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798    seq cseq 11062
This theorem is referenced by:  bcval5  11346  elqaalem2  19716  lgsne0  20588
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-seq 11063
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