Theorem fseq1m1p1 10097

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 9219	. . 3 |- ( N e. NN -> ( N - 1 ) e. NN0 )
2		eqid 2177	. . . 4 |- { <. ( ( N - 1 ) + 1 ) , B >. } = { <. ( ( N - 1 ) + 1 ) , B >. }
3	2	fseq1p1m1 10096	. . 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 8929	. . . . . . . . 9 |- ( N e. NN -> N e. CC )
6		ax-1cn 7906	. . . . . . . . 9 |- 1 e. CC
7		npcan 8168	. . . . . . . . 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 3786	. . . . . . 7 |- ( N e. NN -> <. ( ( N - 1 ) + 1 ) , B >. = <. N , B >. )
10	9	sneqd 3607	. . . . . 6 |- ( N e. NN -> { <. ( ( N - 1 ) + 1 ) , B >. } = { <. N , B >. } )
11		fseq1m1p1.1	. . . . . 6 |- H = { <. N , B >. }
12	10, 11	eqtr4di 2228	. . . . 5 |- ( N e. NN -> { <. ( ( N - 1 ) + 1 ) , B >. } = H )
13	12	uneq2d 3291	. . . 4 |- ( N e. NN -> ( F u. { <. ( ( N - 1 ) + 1 ) , B >. } ) = ( F u. H ) )
14	13	eqeq2d 2189	. . 3 |- ( N e. NN -> ( G = ( F u. { <. ( ( N - 1 ) + 1 ) , B >. } ) <-> G = ( F u. H ) ) )
15	14	3anbi3d 1318	. 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 5893	. . . 4 |- ( N e. NN -> ( 1 ... ( ( N - 1 ) + 1 ) ) = ( 1 ... N ) )
17	16	feq2d 5355	. . 3 |- ( N e. NN -> ( G : ( 1 ... ( ( N - 1 ) + 1 ) ) --> A <-> G : ( 1 ... N ) --> A ) )
18	8	fveq2d 5521	. . . 4 |- ( N e. NN -> ( G ` ( ( N - 1 ) + 1 ) ) = ( G ` N ) )
19	18	eqeq1d 2186	. . 3 |- ( N e. NN -> ( ( G ` ( ( N - 1 ) + 1 ) ) = B <-> ( G ` N ) = B ) )
20	17, 19	3anbi12d 1313	. 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 978   = wceq 1353   e. wcel 2148   u. cun 3129   {csn 3594   <.cop 3597   |` cres 4630   -->wf 5214   ` cfv 5218  (class class class)co 5877   CCcc 7811   1c1 7814   + caddc 7816   - cmin 8130   NNcn 8921   NN0cn0 9178   ...cfz 10010

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-inn 8922  df-n0 9179  df-z 9256  df-uz 9531  df-fz 10011

This theorem is referenced by: (None)