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Theorem fseq1m1p1 11867
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 10911 . . 3  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
2 eqid 2429 . . . 4  |-  { <. ( ( N  -  1 )  +  1 ) ,  B >. }  =  { <. ( ( N  -  1 )  +  1 ) ,  B >. }
32fseq1p1m1 11866 . . 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 17 . 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 10617 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  CC )
6 ax-1cn 9596 . . . . . . . . 9  |-  1  e.  CC
7 npcan 9883 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
85, 6, 7sylancl 666 . . . . . . . 8  |-  ( N  e.  NN  ->  (
( N  -  1 )  +  1 )  =  N )
98opeq1d 4196 . . . . . . 7  |-  ( N  e.  NN  ->  <. (
( N  -  1 )  +  1 ) ,  B >.  =  <. N ,  B >. )
109sneqd 4014 . . . . . 6  |-  ( N  e.  NN  ->  { <. ( ( N  -  1 )  +  1 ) ,  B >. }  =  { <. N ,  B >. } )
11 fseq1m1p1.1 . . . . . 6  |-  H  =  { <. N ,  B >. }
1210, 11syl6eqr 2488 . . . . 5  |-  ( N  e.  NN  ->  { <. ( ( N  -  1 )  +  1 ) ,  B >. }  =  H )
1312uneq2d 3626 . . . 4  |-  ( N  e.  NN  ->  ( F  u.  { <. (
( N  -  1 )  +  1 ) ,  B >. } )  =  ( F  u.  H ) )
1413eqeq2d 2443 . . 3  |-  ( N  e.  NN  ->  ( G  =  ( F  u.  { <. ( ( N  -  1 )  +  1 ) ,  B >. } )  <->  G  =  ( F  u.  H
) ) )
15143anbi3d 1341 . 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 6321 . . . 4  |-  ( N  e.  NN  ->  (
1 ... ( ( N  -  1 )  +  1 ) )  =  ( 1 ... N
) )
1716feq2d 5733 . . 3  |-  ( N  e.  NN  ->  ( G : ( 1 ... ( ( N  - 
1 )  +  1 ) ) --> A  <->  G :
( 1 ... N
) --> A ) )
188fveq2d 5885 . . . 4  |-  ( N  e.  NN  ->  ( G `  ( ( N  -  1 )  +  1 ) )  =  ( G `  N ) )
1918eqeq1d 2431 . . 3  |-  ( N  e.  NN  ->  (
( G `  (
( N  -  1 )  +  1 ) )  =  B  <->  ( G `  N )  =  B ) )
2017, 193anbi12d 1336 . 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 286 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 187    /\ w3a 982    = wceq 1437    e. wcel 1870    u. cun 3440   {csn 4002   <.cop 4008    |` cres 4856   -->wf 5597   ` cfv 5601  (class class class)co 6305   CCcc 9536   1c1 9539    + caddc 9541    - cmin 9859   NNcn 10609   NN0cn0 10869   ...cfz 11782
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
This theorem is referenced by: (None)
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