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Theorem fseq1m1p1 10858
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 10005 . . 3  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
2 eqid 2283 . . . 4  |-  { <. ( ( N  -  1 )  +  1 ) ,  B >. }  =  { <. ( ( N  -  1 )  +  1 ) ,  B >. }
32fseq1p1m1 10857 . . 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 15 . 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 9754 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  CC )
6 ax-1cn 8795 . . . . . . . . 9  |-  1  e.  CC
7 npcan 9060 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
85, 6, 7sylancl 643 . . . . . . . 8  |-  ( N  e.  NN  ->  (
( N  -  1 )  +  1 )  =  N )
98opeq1d 3802 . . . . . . 7  |-  ( N  e.  NN  ->  <. (
( N  -  1 )  +  1 ) ,  B >.  =  <. N ,  B >. )
109sneqd 3653 . . . . . 6  |-  ( N  e.  NN  ->  { <. ( ( N  -  1 )  +  1 ) ,  B >. }  =  { <. N ,  B >. } )
11 fseq1m1p1.1 . . . . . 6  |-  H  =  { <. N ,  B >. }
1210, 11syl6eqr 2333 . . . . 5  |-  ( N  e.  NN  ->  { <. ( ( N  -  1 )  +  1 ) ,  B >. }  =  H )
1312uneq2d 3329 . . . 4  |-  ( N  e.  NN  ->  ( F  u.  { <. (
( N  -  1 )  +  1 ) ,  B >. } )  =  ( F  u.  H ) )
1413eqeq2d 2294 . . 3  |-  ( N  e.  NN  ->  ( G  =  ( F  u.  { <. ( ( N  -  1 )  +  1 ) ,  B >. } )  <->  G  =  ( F  u.  H
) ) )
15143anbi3d 1258 . 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 5874 . . . 4  |-  ( N  e.  NN  ->  (
1 ... ( ( N  -  1 )  +  1 ) )  =  ( 1 ... N
) )
1716feq2d 5380 . . 3  |-  ( N  e.  NN  ->  ( G : ( 1 ... ( ( N  - 
1 )  +  1 ) ) --> A  <->  G :
( 1 ... N
) --> A ) )
188fveq2d 5529 . . . 4  |-  ( N  e.  NN  ->  ( G `  ( ( N  -  1 )  +  1 ) )  =  ( G `  N ) )
1918eqeq1d 2291 . . 3  |-  ( N  e.  NN  ->  (
( G `  (
( N  -  1 )  +  1 ) )  =  B  <->  ( G `  N )  =  B ) )
2017, 193anbi12d 1253 . 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 274 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 176    /\ w3a 934    = wceq 1623    e. wcel 1684    u. cun 3150   {csn 3640   <.cop 3643    |` cres 4691   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   1c1 8738    + caddc 8740    - cmin 9037   NNcn 9746   NN0cn0 9965   ...cfz 10782
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783
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