Theorem fzsuc2 10171

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 9652	. 2 |- ( N e. ( ZZ>= ` ( M - 1 ) ) -> ( N = ( M - 1 ) \/ N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) ) )
2		zcn 9348	. . . . . . . 8 |- ( M e. ZZ -> M e. CC )
3		ax-1cn 7989	. . . . . . . 8 |- 1 e. CC
4		npcan 8252	. . . . . . . 8 |- ( ( M e. CC /\ 1 e. CC ) -> ( ( M - 1 ) + 1 ) = M )
5	2, 3, 4	sylancl 413	. . . . . . 7 |- ( M e. ZZ -> ( ( M - 1 ) + 1 ) = M )
6	5	oveq2d 5941	. . . . . 6 |- ( M e. ZZ -> ( M ... ( ( M - 1 ) + 1 ) ) = ( M ... M ) )
7		uncom 3308	. . . . . . . 8 |- ( (/) u. { M } ) = ( { M } u. (/) )
8		un0 3485	. . . . . . . 8 |- ( { M } u. (/) ) = { M }
9	7, 8	eqtri 2217	. . . . . . 7 |- ( (/) u. { M } ) = { M }
10		zre 9347	. . . . . . . . . 10 |- ( M e. ZZ -> M e. RR )
11	10	ltm1d 8976	. . . . . . . . 9 |- ( M e. ZZ -> ( M - 1 ) < M )
12		peano2zm 9381	. . . . . . . . . 10 |- ( M e. ZZ -> ( M - 1 ) e. ZZ )
13		fzn 10134	. . . . . . . . . 10 |- ( ( M e. ZZ /\ ( M - 1 ) e. ZZ ) -> ( ( M - 1 ) < M <-> ( M ... ( M - 1 ) ) = (/) ) )
14	12, 13	mpdan 421	. . . . . . . . 9 |- ( M e. ZZ -> ( ( M - 1 ) < M <-> ( M ... ( M - 1 ) ) = (/) ) )
15	11, 14	mpbid 147	. . . . . . . 8 |- ( M e. ZZ -> ( M ... ( M - 1 ) ) = (/) )
16	5	sneqd 3636	. . . . . . . 8 |- ( M e. ZZ -> { ( ( M - 1 ) + 1 ) } = { M } )
17	15, 16	uneq12d 3319	. . . . . . 7 |- ( M e. ZZ -> ( ( M ... ( M - 1 ) ) u. { ( ( M - 1 ) + 1 ) } ) = ( (/) u. { M } ) )
18		fzsn 10158	. . . . . . 7 |- ( M e. ZZ -> ( M ... M ) = { M } )
19	9, 17, 18	3eqtr4a 2255	. . . . . 6 |- ( M e. ZZ -> ( ( M ... ( M - 1 ) ) u. { ( ( M - 1 ) + 1 ) } ) = ( M ... M ) )
20	6, 19	eqtr4d 2232	. . . . 5 |- ( M e. ZZ -> ( M ... ( ( M - 1 ) + 1 ) ) = ( ( M ... ( M - 1 ) ) u. { ( ( M - 1 ) + 1 ) } ) )
21		oveq1 5932	. . . . . . 7 |- ( N = ( M - 1 ) -> ( N + 1 ) = ( ( M - 1 ) + 1 ) )
22	21	oveq2d 5941	. . . . . 6 |- ( N = ( M - 1 ) -> ( M ... ( N + 1 ) ) = ( M ... ( ( M - 1 ) + 1 ) ) )
23		oveq2 5933	. . . . . . 7 |- ( N = ( M - 1 ) -> ( M ... N ) = ( M ... ( M - 1 ) ) )
24	21	sneqd 3636	. . . . . . 7 |- ( N = ( M - 1 ) -> { ( N + 1 ) } = { ( ( M - 1 ) + 1 ) } )
25	23, 24	uneq12d 3319	. . . . . 6 |- ( N = ( M - 1 ) -> ( ( M ... N ) u. { ( N + 1 ) } ) = ( ( M ... ( M - 1 ) ) u. { ( ( M - 1 ) + 1 ) } ) )
26	22, 25	eqeq12d 2211	. . . . 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 157	. . . 4 |- ( M e. ZZ -> ( N = ( M - 1 ) -> ( M ... ( N + 1 ) ) = ( ( M ... N ) u. { ( N + 1 ) } ) ) )
28	27	imp 124	. . 3 |- ( ( M e. ZZ /\ N = ( M - 1 ) ) -> ( M ... ( N + 1 ) ) = ( ( M ... N ) u. { ( N + 1 ) } ) )
29	5	fveq2d 5565	. . . . . 6 |- ( M e. ZZ -> ( ZZ>= ` ( ( M - 1 ) + 1 ) ) = ( ZZ>= ` M ) )
30	29	eleq2d 2266	. . . . 5 |- ( M e. ZZ -> ( N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) <-> N e. ( ZZ>= ` M ) ) )
31	30	biimpa 296	. . . 4 |- ( ( M e. ZZ /\ N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) ) -> N e. ( ZZ>= ` M ) )
32		fzsuc 10161	. . . 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 798	. 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 286	1 |- ( ( M e. ZZ /\ N e. ( ZZ>= ` ( M - 1 ) ) ) -> ( M ... ( N + 1 ) ) = ( ( M ... N ) u. { ( N + 1 ) } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1364   e. wcel 2167   u. cun 3155   (/)c0 3451   {csn 3623   class class class wbr 4034   ` cfv 5259  (class class class)co 5925   CCcc 7894   1c1 7897   + caddc 7899   < clt 8078   - cmin 8214   ZZcz 9343   ZZ>=cuz 9618   ...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-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:  fseq1p1m1  10186  frecfzennn  10535  zfz1isolemsplit  10947  fsumm1  11598  fprodm1  11780