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Theorem seqz 12258
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 11794 . . . 4  |-  ( K  e.  ( M ... N )  ->  K  e.  ( ZZ>= `  M )
)
31, 2syl 17 . . 3  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
4 eluzelz 11168 . . . . . . . . 9  |-  ( K  e.  ( ZZ>= `  M
)  ->  K  e.  ZZ )
53, 4syl 17 . . . . . . . 8  |-  ( ph  ->  K  e.  ZZ )
6 seq1 12223 . . . . . . . 8  |-  ( K  e.  ZZ  ->  (  seq K (  .+  ,  F ) `  K
)  =  ( F `
 K ) )
75, 6syl 17 . . . . . . 7  |-  ( ph  ->  (  seq K ( 
.+  ,  F ) `
 K )  =  ( F `  K
) )
8 seqz.7 . . . . . . 7  |-  ( ph  ->  ( F `  K
)  =  Z )
97, 8eqtrd 2470 . . . . . 6  |-  ( ph  ->  (  seq K ( 
.+  ,  F ) `
 K )  =  Z )
10 seqeq1 12213 . . . . . . . 8  |-  ( K  =  M  ->  seq K (  .+  ,  F )  =  seq M (  .+  ,  F ) )
1110fveq1d 5883 . . . . . . 7  |-  ( K  =  M  ->  (  seq K (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  K
) )
1211eqeq1d 2431 . . . . . 6  |-  ( K  =  M  ->  (
(  seq K (  .+  ,  F ) `  K
)  =  Z  <->  (  seq M (  .+  ,  F ) `  K
)  =  Z ) )
139, 12syl5ibcom 223 . . . . 5  |-  ( ph  ->  ( K  =  M  ->  (  seq M
(  .+  ,  F
) `  K )  =  Z ) )
14 eluzel2 11164 . . . . . . . . 9  |-  ( K  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
153, 14syl 17 . . . . . . . 8  |-  ( ph  ->  M  e.  ZZ )
16 seqm1 12227 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  K  e.  ( ZZ>= `  ( M  +  1
) ) )  -> 
(  seq M (  .+  ,  F ) `  K
)  =  ( (  seq M (  .+  ,  F ) `  ( K  -  1 ) )  .+  ( F `
 K ) ) )
1715, 16sylan 473 . . . . . . 7  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq M (  .+  ,  F ) `  K
)  =  ( (  seq M (  .+  ,  F ) `  ( K  -  1 ) )  .+  ( F `
 K ) ) )
188adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( F `  K )  =  Z )
1918oveq2d 6321 . . . . . . . 8  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( (  seq M (  .+  ,  F ) `  ( K  -  1 ) )  .+  ( F `
 K ) )  =  ( (  seq M (  .+  ,  F ) `  ( K  -  1 ) )  .+  Z ) )
20 eluzp1m1 11182 . . . . . . . . . . 11  |-  ( ( M  e.  ZZ  /\  K  e.  ( ZZ>= `  ( M  +  1
) ) )  -> 
( K  -  1 )  e.  ( ZZ>= `  M ) )
2115, 20sylan 473 . . . . . . . . . 10  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( K  -  1 )  e.  ( ZZ>= `  M )
)
22 fzssp1 11839 . . . . . . . . . . . . . . 15  |-  ( M ... ( K  - 
1 ) )  C_  ( M ... ( ( K  -  1 )  +  1 ) )
235zcnd 11041 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  K  e.  CC )
24 ax-1cn 9596 . . . . . . . . . . . . . . . . 17  |-  1  e.  CC
25 npcan 9883 . . . . . . . . . . . . . . . . 17  |-  ( ( K  e.  CC  /\  1  e.  CC )  ->  ( ( K  - 
1 )  +  1 )  =  K )
2623, 24, 25sylancl 666 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( K  - 
1 )  +  1 )  =  K )
2726oveq2d 6321 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( M ... (
( K  -  1 )  +  1 ) )  =  ( M ... K ) )
2822, 27syl5sseq 3518 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( M ... ( K  -  1 ) )  C_  ( M ... K ) )
29 elfzuz3 11795 . . . . . . . . . . . . . . . 16  |-  ( K  e.  ( M ... N )  ->  N  e.  ( ZZ>= `  K )
)
301, 29syl 17 . . . . . . . . . . . . . . 15  |-  ( ph  ->  N  e.  ( ZZ>= `  K ) )
31 fzss2 11836 . . . . . . . . . . . . . . 15  |-  ( N  e.  ( ZZ>= `  K
)  ->  ( M ... K )  C_  ( M ... N ) )
3230, 31syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( M ... K
)  C_  ( M ... N ) )
3328, 32sstrd 3480 . . . . . . . . . . . . 13  |-  ( ph  ->  ( M ... ( K  -  1 ) )  C_  ( M ... N ) )
3433adantr 466 . . . . . . . . . . . 12  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( M ... ( K  -  1 ) )  C_  ( M ... N ) )
3534sselda 3470 . . . . . . . . . . 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 719 . . . . . . . . . . 11  |-  ( ( ( ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  /\  x  e.  ( M ... N
) )  ->  ( F `  x )  e.  S )
3835, 37syldan 472 . . . . . . . . . 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 719 . . . . . . . . . 10  |-  ( ( ( ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
4121, 38, 40seqcl 12230 . . . . . . . . 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 2846 . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  S  ( x  .+  Z )  =  Z )
4443adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  A. x  e.  S  ( x  .+  Z )  =  Z )
45 oveq1 6312 . . . . . . . . . . 11  |-  ( x  =  (  seq M
(  .+  ,  F
) `  ( K  -  1 ) )  ->  ( x  .+  Z )  =  ( (  seq M ( 
.+  ,  F ) `
 ( K  - 
1 ) )  .+  Z ) )
4645eqeq1d 2431 . . . . . . . . . 10  |-  ( x  =  (  seq M
(  .+  ,  F
) `  ( K  -  1 ) )  ->  ( ( x 
.+  Z )  =  Z  <->  ( (  seq M (  .+  ,  F ) `  ( K  -  1 ) )  .+  Z )  =  Z ) )
4746rspcv 3184 . . . . . . . . 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 62 . . . . . . . 8  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( (  seq M (  .+  ,  F ) `  ( K  -  1 ) )  .+  Z )  =  Z )
4919, 48eqtrd 2470 . . . . . . 7  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( (  seq M (  .+  ,  F ) `  ( K  -  1 ) )  .+  ( F `
 K ) )  =  Z )
5017, 49eqtrd 2470 . . . . . 6  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq M (  .+  ,  F ) `  K
)  =  Z )
5150ex 435 . . . . 5  |-  ( ph  ->  ( K  e.  (
ZZ>= `  ( M  + 
1 ) )  -> 
(  seq M (  .+  ,  F ) `  K
)  =  Z ) )
52 uzp1 11192 . . . . . 6  |-  ( K  e.  ( ZZ>= `  M
)  ->  ( K  =  M  \/  K  e.  ( ZZ>= `  ( M  +  1 ) ) ) )
533, 52syl 17 . . . . 5  |-  ( ph  ->  ( K  =  M  \/  K  e.  (
ZZ>= `  ( M  + 
1 ) ) ) )
5413, 51, 53mpjaod 382 . . . 4  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 K )  =  Z )
5554, 8eqtr4d 2473 . . 3  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 K )  =  ( F `  K
) )
56 eqidd 2430 . . 3  |-  ( (
ph  /\  x  e.  ( ( K  + 
1 ) ... N
) )  ->  ( F `  x )  =  ( F `  x ) )
573, 55, 30, 56seqfveq2 12232 . 2  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 N )  =  (  seq K ( 
.+  ,  F ) `
 N ) )
58 fvex 5891 . . . . . 6  |-  ( F `
 K )  e. 
_V
5958elsnc 4026 . . . . 5  |-  ( ( F `  K )  e.  { Z }  <->  ( F `  K )  =  Z )
608, 59sylibr 215 . . . 4  |-  ( ph  ->  ( F `  K
)  e.  { Z } )
61 simprl 762 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  { Z }  /\  y  e.  S )
)  ->  x  e.  { Z } )
62 elsn 4016 . . . . . . . 8  |-  ( x  e.  { Z }  <->  x  =  Z )
6361, 62sylib 199 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  { Z }  /\  y  e.  S )
)  ->  x  =  Z )
6463oveq1d 6320 . . . . . 6  |-  ( (
ph  /\  ( x  e.  { Z }  /\  y  e.  S )
)  ->  ( x  .+  y )  =  ( Z  .+  y ) )
65 simprr 764 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  { Z }  /\  y  e.  S )
)  ->  y  e.  S )
66 seqz.3 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  S )  ->  ( Z  .+  x )  =  Z )
6766ralrimiva 2846 . . . . . . . 8  |-  ( ph  ->  A. x  e.  S  ( Z  .+  x )  =  Z )
6867adantr 466 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  { Z }  /\  y  e.  S )
)  ->  A. x  e.  S  ( Z  .+  x )  =  Z )
69 oveq2 6313 . . . . . . . . 9  |-  ( x  =  y  ->  ( Z  .+  x )  =  ( Z  .+  y
) )
7069eqeq1d 2431 . . . . . . . 8  |-  ( x  =  y  ->  (
( Z  .+  x
)  =  Z  <->  ( Z  .+  y )  =  Z ) )
7170rspcv 3184 . . . . . . 7  |-  ( y  e.  S  ->  ( A. x  e.  S  ( Z  .+  x )  =  Z  ->  ( Z  .+  y )  =  Z ) )
7265, 68, 71sylc 62 . . . . . 6  |-  ( (
ph  /\  ( x  e.  { Z }  /\  y  e.  S )
)  ->  ( Z  .+  y )  =  Z )
7364, 72eqtrd 2470 . . . . 5  |-  ( (
ph  /\  ( x  e.  { Z }  /\  y  e.  S )
)  ->  ( x  .+  y )  =  Z )
74 ovex 6333 . . . . . 6  |-  ( x 
.+  y )  e. 
_V
7574elsnc 4026 . . . . 5  |-  ( ( x  .+  y )  e.  { Z }  <->  ( x  .+  y )  =  Z )
7673, 75sylibr 215 . . . 4  |-  ( (
ph  /\  ( x  e.  { Z }  /\  y  e.  S )
)  ->  ( x  .+  y )  e.  { Z } )
77 peano2uz 11212 . . . . . . . 8  |-  ( K  e.  ( ZZ>= `  M
)  ->  ( K  +  1 )  e.  ( ZZ>= `  M )
)
783, 77syl 17 . . . . . . 7  |-  ( ph  ->  ( K  +  1 )  e.  ( ZZ>= `  M ) )
79 fzss1 11835 . . . . . . 7  |-  ( ( K  +  1 )  e.  ( ZZ>= `  M
)  ->  ( ( K  +  1 ) ... N )  C_  ( M ... N ) )
8078, 79syl 17 . . . . . 6  |-  ( ph  ->  ( ( K  + 
1 ) ... N
)  C_  ( M ... N ) )
8180sselda 3470 . . . . 5  |-  ( (
ph  /\  x  e.  ( ( K  + 
1 ) ... N
) )  ->  x  e.  ( M ... N
) )
8281, 36syldan 472 . . . 4  |-  ( (
ph  /\  x  e.  ( ( K  + 
1 ) ... N
) )  ->  ( F `  x )  e.  S )
8360, 76, 30, 82seqcl2 12228 . . 3  |-  ( ph  ->  (  seq K ( 
.+  ,  F ) `
 N )  e. 
{ Z } )
84 elsni 4027 . . 3  |-  ( (  seq K (  .+  ,  F ) `  N
)  e.  { Z }  ->  (  seq K
(  .+  ,  F
) `  N )  =  Z )
8583, 84syl 17 . 2  |-  ( ph  ->  (  seq K ( 
.+  ,  F ) `
 N )  =  Z )
8657, 85eqtrd 2470 1  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 N )  =  Z )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782    C_ wss 3442   {csn 4002   ` cfv 5601  (class class class)co 6305   CCcc 9536   1c1 9539    + caddc 9541    - cmin 9859   ZZcz 10937   ZZ>=cuz 11159   ...cfz 11782    seqcseq 12210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-seq 12211
This theorem is referenced by:  bcval5  12500  elqaalem2  23141  lgsne0  24124
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