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Theorem fseq1m1p1 11753
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 10837 . . 3  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
2 eqid 2467 . . . 4  |-  { <. ( ( N  -  1 )  +  1 ) ,  B >. }  =  { <. ( ( N  -  1 )  +  1 ) ,  B >. }
32fseq1p1m1 11752 . . 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 16 . 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 10544 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  CC )
6 ax-1cn 9550 . . . . . . . . 9  |-  1  e.  CC
7 npcan 9829 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
85, 6, 7sylancl 662 . . . . . . . 8  |-  ( N  e.  NN  ->  (
( N  -  1 )  +  1 )  =  N )
98opeq1d 4219 . . . . . . 7  |-  ( N  e.  NN  ->  <. (
( N  -  1 )  +  1 ) ,  B >.  =  <. N ,  B >. )
109sneqd 4039 . . . . . 6  |-  ( N  e.  NN  ->  { <. ( ( N  -  1 )  +  1 ) ,  B >. }  =  { <. N ,  B >. } )
11 fseq1m1p1.1 . . . . . 6  |-  H  =  { <. N ,  B >. }
1210, 11syl6eqr 2526 . . . . 5  |-  ( N  e.  NN  ->  { <. ( ( N  -  1 )  +  1 ) ,  B >. }  =  H )
1312uneq2d 3658 . . . 4  |-  ( N  e.  NN  ->  ( F  u.  { <. (
( N  -  1 )  +  1 ) ,  B >. } )  =  ( F  u.  H ) )
1413eqeq2d 2481 . . 3  |-  ( N  e.  NN  ->  ( G  =  ( F  u.  { <. ( ( N  -  1 )  +  1 ) ,  B >. } )  <->  G  =  ( F  u.  H
) ) )
15143anbi3d 1305 . 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 6300 . . . 4  |-  ( N  e.  NN  ->  (
1 ... ( ( N  -  1 )  +  1 ) )  =  ( 1 ... N
) )
1716feq2d 5718 . . 3  |-  ( N  e.  NN  ->  ( G : ( 1 ... ( ( N  - 
1 )  +  1 ) ) --> A  <->  G :
( 1 ... N
) --> A ) )
188fveq2d 5870 . . . 4  |-  ( N  e.  NN  ->  ( G `  ( ( N  -  1 )  +  1 ) )  =  ( G `  N ) )
1918eqeq1d 2469 . . 3  |-  ( N  e.  NN  ->  (
( G `  (
( N  -  1 )  +  1 ) )  =  B  <->  ( G `  N )  =  B ) )
2017, 193anbi12d 1300 . 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 283 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 973    = wceq 1379    e. wcel 1767    u. cun 3474   {csn 4027   <.cop 4033    |` cres 5001   -->wf 5584   ` cfv 5588  (class class class)co 6284   CCcc 9490   1c1 9493    + caddc 9495    - cmin 9805   NNcn 10536   NN0cn0 10795   ...cfz 11672
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 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
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 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673
This theorem is referenced by: (None)
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