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Theorem seqz 12124
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 11685 . . . 4  |-  ( K  e.  ( M ... N )  ->  K  e.  ( ZZ>= `  M )
)
31, 2syl 16 . . 3  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
4 eluzelz 11092 . . . . . . . . 9  |-  ( K  e.  ( ZZ>= `  M
)  ->  K  e.  ZZ )
53, 4syl 16 . . . . . . . 8  |-  ( ph  ->  K  e.  ZZ )
6 seq1 12089 . . . . . . . 8  |-  ( K  e.  ZZ  ->  (  seq K (  .+  ,  F ) `  K
)  =  ( F `
 K ) )
75, 6syl 16 . . . . . . 7  |-  ( ph  ->  (  seq K ( 
.+  ,  F ) `
 K )  =  ( F `  K
) )
8 seqz.7 . . . . . . 7  |-  ( ph  ->  ( F `  K
)  =  Z )
97, 8eqtrd 2508 . . . . . 6  |-  ( ph  ->  (  seq K ( 
.+  ,  F ) `
 K )  =  Z )
10 seqeq1 12079 . . . . . . . 8  |-  ( K  =  M  ->  seq K (  .+  ,  F )  =  seq M (  .+  ,  F ) )
1110fveq1d 5868 . . . . . . 7  |-  ( K  =  M  ->  (  seq K (  .+  ,  F ) `  K
)  =  (  seq M (  .+  ,  F ) `  K
) )
1211eqeq1d 2469 . . . . . 6  |-  ( K  =  M  ->  (
(  seq K (  .+  ,  F ) `  K
)  =  Z  <->  (  seq M (  .+  ,  F ) `  K
)  =  Z ) )
139, 12syl5ibcom 220 . . . . 5  |-  ( ph  ->  ( K  =  M  ->  (  seq M
(  .+  ,  F
) `  K )  =  Z ) )
14 eluzel2 11088 . . . . . . . . 9  |-  ( K  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
153, 14syl 16 . . . . . . . 8  |-  ( ph  ->  M  e.  ZZ )
16 seqm1 12093 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  K  e.  ( ZZ>= `  ( M  +  1
) ) )  -> 
(  seq M (  .+  ,  F ) `  K
)  =  ( (  seq M (  .+  ,  F ) `  ( K  -  1 ) )  .+  ( F `
 K ) ) )
1715, 16sylan 471 . . . . . . 7  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq M (  .+  ,  F ) `  K
)  =  ( (  seq M (  .+  ,  F ) `  ( K  -  1 ) )  .+  ( F `
 K ) ) )
188adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( F `  K )  =  Z )
1918oveq2d 6301 . . . . . . . 8  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( (  seq M (  .+  ,  F ) `  ( K  -  1 ) )  .+  ( F `
 K ) )  =  ( (  seq M (  .+  ,  F ) `  ( K  -  1 ) )  .+  Z ) )
20 eluzp1m1 11106 . . . . . . . . . . 11  |-  ( ( M  e.  ZZ  /\  K  e.  ( ZZ>= `  ( M  +  1
) ) )  -> 
( K  -  1 )  e.  ( ZZ>= `  M ) )
2115, 20sylan 471 . . . . . . . . . 10  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( K  -  1 )  e.  ( ZZ>= `  M )
)
22 fzssp1 11727 . . . . . . . . . . . . . . 15  |-  ( M ... ( K  - 
1 ) )  C_  ( M ... ( ( K  -  1 )  +  1 ) )
235zcnd 10968 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  K  e.  CC )
24 ax-1cn 9551 . . . . . . . . . . . . . . . . 17  |-  1  e.  CC
25 npcan 9830 . . . . . . . . . . . . . . . . 17  |-  ( ( K  e.  CC  /\  1  e.  CC )  ->  ( ( K  - 
1 )  +  1 )  =  K )
2623, 24, 25sylancl 662 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( K  - 
1 )  +  1 )  =  K )
2726oveq2d 6301 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( M ... (
( K  -  1 )  +  1 ) )  =  ( M ... K ) )
2822, 27syl5sseq 3552 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( M ... ( K  -  1 ) )  C_  ( M ... K ) )
29 elfzuz3 11686 . . . . . . . . . . . . . . . 16  |-  ( K  e.  ( M ... N )  ->  N  e.  ( ZZ>= `  K )
)
301, 29syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  N  e.  ( ZZ>= `  K ) )
31 fzss2 11724 . . . . . . . . . . . . . . 15  |-  ( N  e.  ( ZZ>= `  K
)  ->  ( M ... K )  C_  ( M ... N ) )
3230, 31syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( M ... K
)  C_  ( M ... N ) )
3328, 32sstrd 3514 . . . . . . . . . . . . 13  |-  ( ph  ->  ( M ... ( K  -  1 ) )  C_  ( M ... N ) )
3433adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( M ... ( K  -  1 ) )  C_  ( M ... N ) )
3534sselda 3504 . . . . . . . . . . 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 714 . . . . . . . . . . 11  |-  ( ( ( ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  /\  x  e.  ( M ... N
) )  ->  ( F `  x )  e.  S )
3835, 37syldan 470 . . . . . . . . . 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 714 . . . . . . . . . 10  |-  ( ( ( ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
4121, 38, 40seqcl 12096 . . . . . . . . 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 2878 . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  S  ( x  .+  Z )  =  Z )
4443adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  A. x  e.  S  ( x  .+  Z )  =  Z )
45 oveq1 6292 . . . . . . . . . . 11  |-  ( x  =  (  seq M
(  .+  ,  F
) `  ( K  -  1 ) )  ->  ( x  .+  Z )  =  ( (  seq M ( 
.+  ,  F ) `
 ( K  - 
1 ) )  .+  Z ) )
4645eqeq1d 2469 . . . . . . . . . 10  |-  ( x  =  (  seq M
(  .+  ,  F
) `  ( K  -  1 ) )  ->  ( ( x 
.+  Z )  =  Z  <->  ( (  seq M (  .+  ,  F ) `  ( K  -  1 ) )  .+  Z )  =  Z ) )
4746rspcv 3210 . . . . . . . . 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 60 . . . . . . . 8  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( (  seq M (  .+  ,  F ) `  ( K  -  1 ) )  .+  Z )  =  Z )
4919, 48eqtrd 2508 . . . . . . 7  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( (  seq M (  .+  ,  F ) `  ( K  -  1 ) )  .+  ( F `
 K ) )  =  Z )
5017, 49eqtrd 2508 . . . . . 6  |-  ( (
ph  /\  K  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  (  seq M (  .+  ,  F ) `  K
)  =  Z )
5150ex 434 . . . . 5  |-  ( ph  ->  ( K  e.  (
ZZ>= `  ( M  + 
1 ) )  -> 
(  seq M (  .+  ,  F ) `  K
)  =  Z ) )
52 uzp1 11116 . . . . . 6  |-  ( K  e.  ( ZZ>= `  M
)  ->  ( K  =  M  \/  K  e.  ( ZZ>= `  ( M  +  1 ) ) ) )
533, 52syl 16 . . . . 5  |-  ( ph  ->  ( K  =  M  \/  K  e.  (
ZZ>= `  ( M  + 
1 ) ) ) )
5413, 51, 53mpjaod 381 . . . 4  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 K )  =  Z )
5554, 8eqtr4d 2511 . . 3  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 K )  =  ( F `  K
) )
56 eqidd 2468 . . 3  |-  ( (
ph  /\  x  e.  ( ( K  + 
1 ) ... N
) )  ->  ( F `  x )  =  ( F `  x ) )
573, 55, 30, 56seqfveq2 12098 . 2  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 N )  =  (  seq K ( 
.+  ,  F ) `
 N ) )
58 fvex 5876 . . . . . 6  |-  ( F `
 K )  e. 
_V
5958elsnc 4051 . . . . 5  |-  ( ( F `  K )  e.  { Z }  <->  ( F `  K )  =  Z )
608, 59sylibr 212 . . . 4  |-  ( ph  ->  ( F `  K
)  e.  { Z } )
61 simprl 755 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  { Z }  /\  y  e.  S )
)  ->  x  e.  { Z } )
62 elsn 4041 . . . . . . . 8  |-  ( x  e.  { Z }  <->  x  =  Z )
6361, 62sylib 196 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  { Z }  /\  y  e.  S )
)  ->  x  =  Z )
6463oveq1d 6300 . . . . . 6  |-  ( (
ph  /\  ( x  e.  { Z }  /\  y  e.  S )
)  ->  ( x  .+  y )  =  ( Z  .+  y ) )
65 simprr 756 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  { Z }  /\  y  e.  S )
)  ->  y  e.  S )
66 seqz.3 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  S )  ->  ( Z  .+  x )  =  Z )
6766ralrimiva 2878 . . . . . . . 8  |-  ( ph  ->  A. x  e.  S  ( Z  .+  x )  =  Z )
6867adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  { Z }  /\  y  e.  S )
)  ->  A. x  e.  S  ( Z  .+  x )  =  Z )
69 oveq2 6293 . . . . . . . . 9  |-  ( x  =  y  ->  ( Z  .+  x )  =  ( Z  .+  y
) )
7069eqeq1d 2469 . . . . . . . 8  |-  ( x  =  y  ->  (
( Z  .+  x
)  =  Z  <->  ( Z  .+  y )  =  Z ) )
7170rspcv 3210 . . . . . . 7  |-  ( y  e.  S  ->  ( A. x  e.  S  ( Z  .+  x )  =  Z  ->  ( Z  .+  y )  =  Z ) )
7265, 68, 71sylc 60 . . . . . 6  |-  ( (
ph  /\  ( x  e.  { Z }  /\  y  e.  S )
)  ->  ( Z  .+  y )  =  Z )
7364, 72eqtrd 2508 . . . . 5  |-  ( (
ph  /\  ( x  e.  { Z }  /\  y  e.  S )
)  ->  ( x  .+  y )  =  Z )
74 ovex 6310 . . . . . 6  |-  ( x 
.+  y )  e. 
_V
7574elsnc 4051 . . . . 5  |-  ( ( x  .+  y )  e.  { Z }  <->  ( x  .+  y )  =  Z )
7673, 75sylibr 212 . . . 4  |-  ( (
ph  /\  ( x  e.  { Z }  /\  y  e.  S )
)  ->  ( x  .+  y )  e.  { Z } )
77 peano2uz 11135 . . . . . . . 8  |-  ( K  e.  ( ZZ>= `  M
)  ->  ( K  +  1 )  e.  ( ZZ>= `  M )
)
783, 77syl 16 . . . . . . 7  |-  ( ph  ->  ( K  +  1 )  e.  ( ZZ>= `  M ) )
79 fzss1 11723 . . . . . . 7  |-  ( ( K  +  1 )  e.  ( ZZ>= `  M
)  ->  ( ( K  +  1 ) ... N )  C_  ( M ... N ) )
8078, 79syl 16 . . . . . 6  |-  ( ph  ->  ( ( K  + 
1 ) ... N
)  C_  ( M ... N ) )
8180sselda 3504 . . . . 5  |-  ( (
ph  /\  x  e.  ( ( K  + 
1 ) ... N
) )  ->  x  e.  ( M ... N
) )
8281, 36syldan 470 . . . 4  |-  ( (
ph  /\  x  e.  ( ( K  + 
1 ) ... N
) )  ->  ( F `  x )  e.  S )
8360, 76, 30, 82seqcl2 12094 . . 3  |-  ( ph  ->  (  seq K ( 
.+  ,  F ) `
 N )  e. 
{ Z } )
84 elsni 4052 . . 3  |-  ( (  seq K (  .+  ,  F ) `  N
)  e.  { Z }  ->  (  seq K
(  .+  ,  F
) `  N )  =  Z )
8583, 84syl 16 . 2  |-  ( ph  ->  (  seq K ( 
.+  ,  F ) `
 N )  =  Z )
8657, 85eqtrd 2508 1  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 N )  =  Z )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814    C_ wss 3476   {csn 4027   ` cfv 5588  (class class class)co 6285   CCcc 9491   1c1 9494    + caddc 9496    - cmin 9806   ZZcz 10865   ZZ>=cuz 11083   ...cfz 11673    seqcseq 12076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-n0 10797  df-z 10866  df-uz 11084  df-fz 11674  df-seq 12077
This theorem is referenced by:  bcval5  12365  elqaalem2  22542  lgsne0  23433
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