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Theorem fseq1m1p1 9188
Description: Add/remove an item to/from the end of a finite sequence. (Contributed by Paul Chapman, 17-Nov-2012.)
Hypothesis
Ref Expression
fseq1m1p1.1  |-  H  =  { <. N ,  B >. }
Assertion
Ref Expression
fseq1m1p1  |-  ( N  e.  NN  ->  (
( F : ( 1 ... ( N  -  1 ) ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H
) )  <->  ( G : ( 1 ... N ) --> A  /\  ( G `  N )  =  B  /\  F  =  ( G  |`  ( 1 ... ( N  -  1 ) ) ) ) ) )

Proof of Theorem fseq1m1p1
StepHypRef Expression
1 nnm1nn0 8396 . . 3  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
2 eqid 2082 . . . 4  |-  { <. ( ( N  -  1 )  +  1 ) ,  B >. }  =  { <. ( ( N  -  1 )  +  1 ) ,  B >. }
32fseq1p1m1 9187 . . 3  |-  ( ( N  -  1 )  e.  NN0  ->  ( ( F : ( 1 ... ( N  - 
1 ) ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  { <. (
( N  -  1 )  +  1 ) ,  B >. } ) )  <->  ( G :
( 1 ... (
( N  -  1 )  +  1 ) ) --> A  /\  ( G `  ( ( N  -  1 )  +  1 ) )  =  B  /\  F  =  ( G  |`  ( 1 ... ( N  -  1 ) ) ) ) ) )
41, 3syl 14 . 2  |-  ( N  e.  NN  ->  (
( F : ( 1 ... ( N  -  1 ) ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  { <. ( ( N  - 
1 )  +  1 ) ,  B >. } ) )  <->  ( G : ( 1 ... ( ( N  - 
1 )  +  1 ) ) --> A  /\  ( G `  ( ( N  -  1 )  +  1 ) )  =  B  /\  F  =  ( G  |`  ( 1 ... ( N  -  1 ) ) ) ) ) )
5 nncn 8114 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  CC )
6 ax-1cn 7131 . . . . . . . . 9  |-  1  e.  CC
7 npcan 7384 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
85, 6, 7sylancl 404 . . . . . . . 8  |-  ( N  e.  NN  ->  (
( N  -  1 )  +  1 )  =  N )
98opeq1d 3584 . . . . . . 7  |-  ( N  e.  NN  ->  <. (
( N  -  1 )  +  1 ) ,  B >.  =  <. N ,  B >. )
109sneqd 3419 . . . . . 6  |-  ( N  e.  NN  ->  { <. ( ( N  -  1 )  +  1 ) ,  B >. }  =  { <. N ,  B >. } )
11 fseq1m1p1.1 . . . . . 6  |-  H  =  { <. N ,  B >. }
1210, 11syl6eqr 2132 . . . . 5  |-  ( N  e.  NN  ->  { <. ( ( N  -  1 )  +  1 ) ,  B >. }  =  H )
1312uneq2d 3127 . . . 4  |-  ( N  e.  NN  ->  ( F  u.  { <. (
( N  -  1 )  +  1 ) ,  B >. } )  =  ( F  u.  H ) )
1413eqeq2d 2093 . . 3  |-  ( N  e.  NN  ->  ( G  =  ( F  u.  { <. ( ( N  -  1 )  +  1 ) ,  B >. } )  <->  G  =  ( F  u.  H
) ) )
15143anbi3d 1250 . 2  |-  ( N  e.  NN  ->  (
( F : ( 1 ... ( N  -  1 ) ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  { <. ( ( N  - 
1 )  +  1 ) ,  B >. } ) )  <->  ( F : ( 1 ... ( N  -  1 ) ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H ) ) ) )
168oveq2d 5559 . . . 4  |-  ( N  e.  NN  ->  (
1 ... ( ( N  -  1 )  +  1 ) )  =  ( 1 ... N
) )
1716feq2d 5066 . . 3  |-  ( N  e.  NN  ->  ( G : ( 1 ... ( ( N  - 
1 )  +  1 ) ) --> A  <->  G :
( 1 ... N
) --> A ) )
188fveq2d 5213 . . . 4  |-  ( N  e.  NN  ->  ( G `  ( ( N  -  1 )  +  1 ) )  =  ( G `  N ) )
1918eqeq1d 2090 . . 3  |-  ( N  e.  NN  ->  (
( G `  (
( N  -  1 )  +  1 ) )  =  B  <->  ( G `  N )  =  B ) )
2017, 193anbi12d 1245 . 2  |-  ( N  e.  NN  ->  (
( G : ( 1 ... ( ( N  -  1 )  +  1 ) ) --> A  /\  ( G `
 ( ( N  -  1 )  +  1 ) )  =  B  /\  F  =  ( G  |`  (
1 ... ( N  - 
1 ) ) ) )  <->  ( G :
( 1 ... N
) --> A  /\  ( G `  N )  =  B  /\  F  =  ( G  |`  (
1 ... ( N  - 
1 ) ) ) ) ) )
214, 15, 203bitr3d 216 1  |-  ( N  e.  NN  ->  (
( F : ( 1 ... ( N  -  1 ) ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H
) )  <->  ( G : ( 1 ... N ) --> A  /\  ( G `  N )  =  B  /\  F  =  ( G  |`  ( 1 ... ( N  -  1 ) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    /\ w3a 920    = wceq 1285    e. wcel 1434    u. cun 2972   {csn 3406   <.cop 3409    |` cres 4373   -->wf 4928   ` cfv 4932  (class class class)co 5543   CCcc 7041   1c1 7044    + caddc 7046    - cmin 7346   NNcn 8106   NN0cn0 8355   ...cfz 9105
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-addass 7140  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-apti 7153  ax-pre-ltadd 7154
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-inn 8107  df-n0 8356  df-z 8433  df-uz 8701  df-fz 9106
This theorem is referenced by: (None)
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