Theorem fzsuc2 9859

Description: Join a successor to the end of a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Mar-2014.)

Assertion

Ref	Expression
fzsuc2	|- ( ( M e. ZZ /\ N e. ( ZZ>= ` ( M - 1 ) ) ) -> ( M ... ( N + 1 ) ) = ( ( M ... N ) u. { ( N + 1 ) } ) )

Proof of Theorem fzsuc2

Step	Hyp	Ref	Expression
1		uzp1 9359	. 2 |- ( N e. ( ZZ>= ` ( M - 1 ) ) -> ( N = ( M - 1 ) \/ N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) ) )
2		zcn 9059	. . . . . . . 8 |- ( M e. ZZ -> M e. CC )
3		ax-1cn 7713	. . . . . . . 8 |- 1 e. CC
4		npcan 7971	. . . . . . . 8 |- ( ( M e. CC /\ 1 e. CC ) -> ( ( M - 1 ) + 1 ) = M )
5	2, 3, 4	sylancl 409	. . . . . . 7 |- ( M e. ZZ -> ( ( M - 1 ) + 1 ) = M )
6	5	oveq2d 5790	. . . . . 6 |- ( M e. ZZ -> ( M ... ( ( M - 1 ) + 1 ) ) = ( M ... M ) )
7		uncom 3220	. . . . . . . 8 |- ( (/) u. { M } ) = ( { M } u. (/) )
8		un0 3396	. . . . . . . 8 |- ( { M } u. (/) ) = { M }
9	7, 8	eqtri 2160	. . . . . . 7 |- ( (/) u. { M } ) = { M }
10		zre 9058	. . . . . . . . . 10 |- ( M e. ZZ -> M e. RR )
11	10	ltm1d 8690	. . . . . . . . 9 |- ( M e. ZZ -> ( M - 1 ) < M )
12		peano2zm 9092	. . . . . . . . . 10 |- ( M e. ZZ -> ( M - 1 ) e. ZZ )
13		fzn 9822	. . . . . . . . . 10 |- ( ( M e. ZZ /\ ( M - 1 ) e. ZZ ) -> ( ( M - 1 ) < M <-> ( M ... ( M - 1 ) ) = (/) ) )
14	12, 13	mpdan 417	. . . . . . . . 9 |- ( M e. ZZ -> ( ( M - 1 ) < M <-> ( M ... ( M - 1 ) ) = (/) ) )
15	11, 14	mpbid 146	. . . . . . . 8 |- ( M e. ZZ -> ( M ... ( M - 1 ) ) = (/) )
16	5	sneqd 3540	. . . . . . . 8 |- ( M e. ZZ -> { ( ( M - 1 ) + 1 ) } = { M } )
17	15, 16	uneq12d 3231	. . . . . . 7 |- ( M e. ZZ -> ( ( M ... ( M - 1 ) ) u. { ( ( M - 1 ) + 1 ) } ) = ( (/) u. { M } ) )
18		fzsn 9846	. . . . . . 7 |- ( M e. ZZ -> ( M ... M ) = { M } )
19	9, 17, 18	3eqtr4a 2198	. . . . . 6 |- ( M e. ZZ -> ( ( M ... ( M - 1 ) ) u. { ( ( M - 1 ) + 1 ) } ) = ( M ... M ) )
20	6, 19	eqtr4d 2175	. . . . 5 |- ( M e. ZZ -> ( M ... ( ( M - 1 ) + 1 ) ) = ( ( M ... ( M - 1 ) ) u. { ( ( M - 1 ) + 1 ) } ) )
21		oveq1 5781	. . . . . . 7 |- ( N = ( M - 1 ) -> ( N + 1 ) = ( ( M - 1 ) + 1 ) )
22	21	oveq2d 5790	. . . . . 6 |- ( N = ( M - 1 ) -> ( M ... ( N + 1 ) ) = ( M ... ( ( M - 1 ) + 1 ) ) )
23		oveq2 5782	. . . . . . 7 |- ( N = ( M - 1 ) -> ( M ... N ) = ( M ... ( M - 1 ) ) )
24	21	sneqd 3540	. . . . . . 7 |- ( N = ( M - 1 ) -> { ( N + 1 ) } = { ( ( M - 1 ) + 1 ) } )
25	23, 24	uneq12d 3231	. . . . . 6 |- ( N = ( M - 1 ) -> ( ( M ... N ) u. { ( N + 1 ) } ) = ( ( M ... ( M - 1 ) ) u. { ( ( M - 1 ) + 1 ) } ) )
26	22, 25	eqeq12d 2154	. . . . 5 |- ( N = ( M - 1 ) -> ( ( M ... ( N + 1 ) ) = ( ( M ... N ) u. { ( N + 1 ) } ) <-> ( M ... ( ( M - 1 ) + 1 ) ) = ( ( M ... ( M - 1 ) ) u. { ( ( M - 1 ) + 1 ) } ) ) )
27	20, 26	syl5ibrcom 156	. . . 4 |- ( M e. ZZ -> ( N = ( M - 1 ) -> ( M ... ( N + 1 ) ) = ( ( M ... N ) u. { ( N + 1 ) } ) ) )
28	27	imp 123	. . 3 |- ( ( M e. ZZ /\ N = ( M - 1 ) ) -> ( M ... ( N + 1 ) ) = ( ( M ... N ) u. { ( N + 1 ) } ) )
29	5	fveq2d 5425	. . . . . 6 |- ( M e. ZZ -> ( ZZ>= ` ( ( M - 1 ) + 1 ) ) = ( ZZ>= ` M ) )
30	29	eleq2d 2209	. . . . 5 |- ( M e. ZZ -> ( N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) <-> N e. ( ZZ>= ` M ) ) )
31	30	biimpa 294	. . . 4 |- ( ( M e. ZZ /\ N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) ) -> N e. ( ZZ>= ` M ) )
32		fzsuc 9849	. . . 4 |- ( N e. ( ZZ>= ` M ) -> ( M ... ( N + 1 ) ) = ( ( M ... N ) u. { ( N + 1 ) } ) )
33	31, 32	syl 14	. . 3 |- ( ( M e. ZZ /\ N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) ) -> ( M ... ( N + 1 ) ) = ( ( M ... N ) u. { ( N + 1 ) } ) )
34	28, 33	jaodan 786	. 2 |- ( ( M e. ZZ /\ ( N = ( M - 1 ) \/ N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) ) ) -> ( M ... ( N + 1 ) ) = ( ( M ... N ) u. { ( N + 1 ) } ) )
35	1, 34	sylan2 284	1 |- ( ( M e. ZZ /\ N e. ( ZZ>= ` ( M - 1 ) ) ) -> ( M ... ( N + 1 ) ) = ( ( M ... N ) u. { ( N + 1 ) } ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697   = wceq 1331   e. wcel 1480   u. cun 3069   (/)c0 3363   {csn 3527   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   CCcc 7618   1c1 7621   + caddc 7623   < clt 7800   - cmin 7933   ZZcz 9054   ZZ>=cuz 9326   ...cfz 9790

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-addcom 7720  ax-addass 7722  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-inn 8721  df-n0 8978  df-z 9055  df-uz 9327  df-fz 9791

This theorem is referenced by:  fseq1p1m1  9874  frecfzennn  10199  zfz1isolemsplit  10581  fsumm1  11185