Theorem fseq1m1p1 10030

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

Step	Hyp	Ref	Expression
1		nnm1nn0 9155	. . 3 |- ( N e. NN -> ( N - 1 ) e. NN0 )
2		eqid 2165	. . . 4 |- { <. ( ( N - 1 ) + 1 ) , B >. } = { <. ( ( N - 1 ) + 1 ) , B >. }
3	2	fseq1p1m1 10029	. . 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 ) ) ) ) ) )
4	1, 3	syl 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 8865	. . . . . . . . 9 |- ( N e. NN -> N e. CC )
6		ax-1cn 7846	. . . . . . . . 9 |- 1 e. CC
7		npcan 8107	. . . . . . . . 9 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N - 1 ) + 1 ) = N )
8	5, 6, 7	sylancl 410	. . . . . . . 8 |- ( N e. NN -> ( ( N - 1 ) + 1 ) = N )
9	8	opeq1d 3764	. . . . . . 7 |- ( N e. NN -> <. ( ( N - 1 ) + 1 ) , B >. = <. N , B >. )
10	9	sneqd 3589	. . . . . 6 |- ( N e. NN -> { <. ( ( N - 1 ) + 1 ) , B >. } = { <. N , B >. } )
11		fseq1m1p1.1	. . . . . 6 |- H = { <. N , B >. }
12	10, 11	eqtr4di 2217	. . . . 5 |- ( N e. NN -> { <. ( ( N - 1 ) + 1 ) , B >. } = H )
13	12	uneq2d 3276	. . . 4 |- ( N e. NN -> ( F u. { <. ( ( N - 1 ) + 1 ) , B >. } ) = ( F u. H ) )
14	13	eqeq2d 2177	. . 3 |- ( N e. NN -> ( G = ( F u. { <. ( ( N - 1 ) + 1 ) , B >. } ) <-> G = ( F u. H ) ) )
15	14	3anbi3d 1308	. 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 ) ) ) )
16	8	oveq2d 5858	. . . 4 |- ( N e. NN -> ( 1 ... ( ( N - 1 ) + 1 ) ) = ( 1 ... N ) )
17	16	feq2d 5325	. . 3 |- ( N e. NN -> ( G : ( 1 ... ( ( N - 1 ) + 1 ) ) --> A <-> G : ( 1 ... N ) --> A ) )
18	8	fveq2d 5490	. . . 4 |- ( N e. NN -> ( G ` ( ( N - 1 ) + 1 ) ) = ( G ` N ) )
19	18	eqeq1d 2174	. . 3 |- ( N e. NN -> ( ( G ` ( ( N - 1 ) + 1 ) ) = B <-> ( G ` N ) = B ) )
20	17, 19	3anbi12d 1303	. 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 ) ) ) ) ) )
21	4, 15, 20	3bitr3d 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 ) ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   u. cun 3114   {csn 3576   <.cop 3579   |` cres 4606   -->wf 5184   ` cfv 5188  (class class class)co 5842   CCcc 7751   1c1 7754   + caddc 7756   - cmin 8069   NNcn 8857   NN0cn0 9114   ...cfz 9944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-inn 8858  df-n0 9115  df-z 9192  df-uz 9467  df-fz 9945

This theorem is referenced by: (None)