Theorem fzsuc2 10035

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 9520	. 2 |- ( N e. ( ZZ>= ` ( M - 1 ) ) -> ( N = ( M - 1 ) \/ N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) ) )
2		zcn 9217	. . . . . . . 8 |- ( M e. ZZ -> M e. CC )
3		ax-1cn 7867	. . . . . . . 8 |- 1 e. CC
4		npcan 8128	. . . . . . . 8 |- ( ( M e. CC /\ 1 e. CC ) -> ( ( M - 1 ) + 1 ) = M )
5	2, 3, 4	sylancl 411	. . . . . . 7 |- ( M e. ZZ -> ( ( M - 1 ) + 1 ) = M )
6	5	oveq2d 5869	. . . . . 6 |- ( M e. ZZ -> ( M ... ( ( M - 1 ) + 1 ) ) = ( M ... M ) )
7		uncom 3271	. . . . . . . 8 |- ( (/) u. { M } ) = ( { M } u. (/) )
8		un0 3448	. . . . . . . 8 |- ( { M } u. (/) ) = { M }
9	7, 8	eqtri 2191	. . . . . . 7 |- ( (/) u. { M } ) = { M }
10		zre 9216	. . . . . . . . . 10 |- ( M e. ZZ -> M e. RR )
11	10	ltm1d 8848	. . . . . . . . 9 |- ( M e. ZZ -> ( M - 1 ) < M )
12		peano2zm 9250	. . . . . . . . . 10 |- ( M e. ZZ -> ( M - 1 ) e. ZZ )
13		fzn 9998	. . . . . . . . . 10 |- ( ( M e. ZZ /\ ( M - 1 ) e. ZZ ) -> ( ( M - 1 ) < M <-> ( M ... ( M - 1 ) ) = (/) ) )
14	12, 13	mpdan 419	. . . . . . . . 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 3596	. . . . . . . 8 |- ( M e. ZZ -> { ( ( M - 1 ) + 1 ) } = { M } )
17	15, 16	uneq12d 3282	. . . . . . 7 |- ( M e. ZZ -> ( ( M ... ( M - 1 ) ) u. { ( ( M - 1 ) + 1 ) } ) = ( (/) u. { M } ) )
18		fzsn 10022	. . . . . . 7 |- ( M e. ZZ -> ( M ... M ) = { M } )
19	9, 17, 18	3eqtr4a 2229	. . . . . 6 |- ( M e. ZZ -> ( ( M ... ( M - 1 ) ) u. { ( ( M - 1 ) + 1 ) } ) = ( M ... M ) )
20	6, 19	eqtr4d 2206	. . . . 5 |- ( M e. ZZ -> ( M ... ( ( M - 1 ) + 1 ) ) = ( ( M ... ( M - 1 ) ) u. { ( ( M - 1 ) + 1 ) } ) )
21		oveq1 5860	. . . . . . 7 |- ( N = ( M - 1 ) -> ( N + 1 ) = ( ( M - 1 ) + 1 ) )
22	21	oveq2d 5869	. . . . . 6 |- ( N = ( M - 1 ) -> ( M ... ( N + 1 ) ) = ( M ... ( ( M - 1 ) + 1 ) ) )
23		oveq2 5861	. . . . . . 7 |- ( N = ( M - 1 ) -> ( M ... N ) = ( M ... ( M - 1 ) ) )
24	21	sneqd 3596	. . . . . . 7 |- ( N = ( M - 1 ) -> { ( N + 1 ) } = { ( ( M - 1 ) + 1 ) } )
25	23, 24	uneq12d 3282	. . . . . 6 |- ( N = ( M - 1 ) -> ( ( M ... N ) u. { ( N + 1 ) } ) = ( ( M ... ( M - 1 ) ) u. { ( ( M - 1 ) + 1 ) } ) )
26	22, 25	eqeq12d 2185	. . . . 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 5500	. . . . . 6 |- ( M e. ZZ -> ( ZZ>= ` ( ( M - 1 ) + 1 ) ) = ( ZZ>= ` M ) )
30	29	eleq2d 2240	. . . . 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 10025	. . . 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 792	. 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 703   = wceq 1348   e. wcel 2141   u. cun 3119   (/)c0 3414   {csn 3583   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   CCcc 7772   1c1 7775   + caddc 7777   < clt 7954   - cmin 8090   ZZcz 9212   ZZ>=cuz 9487   ...cfz 9965

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-fz 9966

This theorem is referenced by:  fseq1p1m1  10050  frecfzennn  10382  zfz1isolemsplit  10773  fsumm1  11379  fprodm1  11561