Theorem fseq1m1p1 10217

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 9336	. . 3 |- ( N e. NN -> ( N - 1 ) e. NN0 )
2		eqid 2205	. . . 4 |- { <. ( ( N - 1 ) + 1 ) , B >. } = { <. ( ( N - 1 ) + 1 ) , B >. }
3	2	fseq1p1m1 10216	. . 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 9044	. . . . . . . . 9 |- ( N e. NN -> N e. CC )
6		ax-1cn 8018	. . . . . . . . 9 |- 1 e. CC
7		npcan 8281	. . . . . . . . 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 3825	. . . . . . 7 |- ( N e. NN -> <. ( ( N - 1 ) + 1 ) , B >. = <. N , B >. )
10	9	sneqd 3646	. . . . . 6 |- ( N e. NN -> { <. ( ( N - 1 ) + 1 ) , B >. } = { <. N , B >. } )
11		fseq1m1p1.1	. . . . . 6 |- H = { <. N , B >. }
12	10, 11	eqtr4di 2256	. . . . 5 |- ( N e. NN -> { <. ( ( N - 1 ) + 1 ) , B >. } = H )
13	12	uneq2d 3327	. . . 4 |- ( N e. NN -> ( F u. { <. ( ( N - 1 ) + 1 ) , B >. } ) = ( F u. H ) )
14	13	eqeq2d 2217	. . 3 |- ( N e. NN -> ( G = ( F u. { <. ( ( N - 1 ) + 1 ) , B >. } ) <-> G = ( F u. H ) ) )
15	14	3anbi3d 1331	. 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 5960	. . . 4 |- ( N e. NN -> ( 1 ... ( ( N - 1 ) + 1 ) ) = ( 1 ... N ) )
17	16	feq2d 5413	. . 3 |- ( N e. NN -> ( G : ( 1 ... ( ( N - 1 ) + 1 ) ) --> A <-> G : ( 1 ... N ) --> A ) )
18	8	fveq2d 5580	. . . 4 |- ( N e. NN -> ( G ` ( ( N - 1 ) + 1 ) ) = ( G ` N ) )
19	18	eqeq1d 2214	. . 3 |- ( N e. NN -> ( ( G ` ( ( N - 1 ) + 1 ) ) = B <-> ( G ` N ) = B ) )
20	17, 19	3anbi12d 1326	. 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 981   = wceq 1373   e. wcel 2176   u. cun 3164   {csn 3633   <.cop 3636   |` cres 4677   -->wf 5267   ` cfv 5271  (class class class)co 5944   CCcc 7923   1c1 7926   + caddc 7928   - cmin 8243   NNcn 9036   NN0cn0 9295   ...cfz 10130

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-inn 9037  df-n0 9296  df-z 9373  df-uz 9649  df-fz 10131

This theorem is referenced by: (None)