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Theorem fseq1m1p1 9716
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 8870 . . 3  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
2 eqid 2100 . . . 4  |-  { <. ( ( N  -  1 )  +  1 ) ,  B >. }  =  { <. ( ( N  -  1 )  +  1 ) ,  B >. }
32fseq1p1m1 9715 . . 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 8586 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  CC )
6 ax-1cn 7588 . . . . . . . . 9  |-  1  e.  CC
7 npcan 7842 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
85, 6, 7sylancl 407 . . . . . . . 8  |-  ( N  e.  NN  ->  (
( N  -  1 )  +  1 )  =  N )
98opeq1d 3658 . . . . . . 7  |-  ( N  e.  NN  ->  <. (
( N  -  1 )  +  1 ) ,  B >.  =  <. N ,  B >. )
109sneqd 3487 . . . . . 6  |-  ( N  e.  NN  ->  { <. ( ( N  -  1 )  +  1 ) ,  B >. }  =  { <. N ,  B >. } )
11 fseq1m1p1.1 . . . . . 6  |-  H  =  { <. N ,  B >. }
1210, 11syl6eqr 2150 . . . . 5  |-  ( N  e.  NN  ->  { <. ( ( N  -  1 )  +  1 ) ,  B >. }  =  H )
1312uneq2d 3177 . . . 4  |-  ( N  e.  NN  ->  ( F  u.  { <. (
( N  -  1 )  +  1 ) ,  B >. } )  =  ( F  u.  H ) )
1413eqeq2d 2111 . . 3  |-  ( N  e.  NN  ->  ( G  =  ( F  u.  { <. ( ( N  -  1 )  +  1 ) ,  B >. } )  <->  G  =  ( F  u.  H
) ) )
15143anbi3d 1264 . 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 5722 . . . 4  |-  ( N  e.  NN  ->  (
1 ... ( ( N  -  1 )  +  1 ) )  =  ( 1 ... N
) )
1716feq2d 5196 . . 3  |-  ( N  e.  NN  ->  ( G : ( 1 ... ( ( N  - 
1 )  +  1 ) ) --> A  <->  G :
( 1 ... N
) --> A ) )
188fveq2d 5357 . . . 4  |-  ( N  e.  NN  ->  ( G `  ( ( N  -  1 )  +  1 ) )  =  ( G `  N ) )
1918eqeq1d 2108 . . 3  |-  ( N  e.  NN  ->  (
( G `  (
( N  -  1 )  +  1 ) )  =  B  <->  ( G `  N )  =  B ) )
2017, 193anbi12d 1259 . 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 217 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 104    /\ w3a 930    = wceq 1299    e. wcel 1448    u. cun 3019   {csn 3474   <.cop 3477    |` cres 4479   -->wf 5055   ` cfv 5059  (class class class)co 5706   CCcc 7498   1c1 7501    + caddc 7503    - cmin 7804   NNcn 8578   NN0cn0 8829   ...cfz 9631
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-addcom 7595  ax-addass 7597  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-0id 7603  ax-rnegex 7604  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611
This theorem depends on definitions:  df-bi 116  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-inn 8579  df-n0 8830  df-z 8907  df-uz 9177  df-fz 9632
This theorem is referenced by: (None)
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