Theorem fseq1m1p1 10187

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 9307	. . 3 |- ( N e. NN -> ( N - 1 ) e. NN0 )
2		eqid 2196	. . . 4 |- { <. ( ( N - 1 ) + 1 ) , B >. } = { <. ( ( N - 1 ) + 1 ) , B >. }
3	2	fseq1p1m1 10186	. . 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 9015	. . . . . . . . 9 |- ( N e. NN -> N e. CC )
6		ax-1cn 7989	. . . . . . . . 9 |- 1 e. CC
7		npcan 8252	. . . . . . . . 9 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N - 1 ) + 1 ) = N )
8	5, 6, 7	sylancl 413	. . . . . . . 8 |- ( N e. NN -> ( ( N - 1 ) + 1 ) = N )
9	8	opeq1d 3815	. . . . . . 7 |- ( N e. NN -> <. ( ( N - 1 ) + 1 ) , B >. = <. N , B >. )
10	9	sneqd 3636	. . . . . 6 |- ( N e. NN -> { <. ( ( N - 1 ) + 1 ) , B >. } = { <. N , B >. } )
11		fseq1m1p1.1	. . . . . 6 |- H = { <. N , B >. }
12	10, 11	eqtr4di 2247	. . . . 5 |- ( N e. NN -> { <. ( ( N - 1 ) + 1 ) , B >. } = H )
13	12	uneq2d 3318	. . . 4 |- ( N e. NN -> ( F u. { <. ( ( N - 1 ) + 1 ) , B >. } ) = ( F u. H ) )
14	13	eqeq2d 2208	. . 3 |- ( N e. NN -> ( G = ( F u. { <. ( ( N - 1 ) + 1 ) , B >. } ) <-> G = ( F u. H ) ) )
15	14	3anbi3d 1329	. 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 5941	. . . 4 |- ( N e. NN -> ( 1 ... ( ( N - 1 ) + 1 ) ) = ( 1 ... N ) )
17	16	feq2d 5398	. . 3 |- ( N e. NN -> ( G : ( 1 ... ( ( N - 1 ) + 1 ) ) --> A <-> G : ( 1 ... N ) --> A ) )
18	8	fveq2d 5565	. . . 4 |- ( N e. NN -> ( G ` ( ( N - 1 ) + 1 ) ) = ( G ` N ) )
19	18	eqeq1d 2205	. . 3 |- ( N e. NN -> ( ( G ` ( ( N - 1 ) + 1 ) ) = B <-> ( G ` N ) = B ) )
20	17, 19	3anbi12d 1324	. 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 218	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 105   /\ w3a 980   = wceq 1364   e. wcel 2167   u. cun 3155   {csn 3623   <.cop 3626   |` cres 4666   -->wf 5255   ` cfv 5259  (class class class)co 5925   CCcc 7894   1c1 7897   + caddc 7899   - cmin 8214   NNcn 9007   NN0cn0 9266   ...cfz 10100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-inn 9008  df-n0 9267  df-z 9344  df-uz 9619  df-fz 10101

This theorem is referenced by: (None)