Theorem fzsuc2 10413

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 9888	. 2 |- ( N e. ( ZZ>= ` ( M - 1 ) ) -> ( N = ( M - 1 ) \/ N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) ) )
2		zcn 9582	. . . . . . . 8 |- ( M e. ZZ -> M e. CC )
3		ax-1cn 8220	. . . . . . . 8 |- 1 e. CC
4		npcan 8482	. . . . . . . 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 6066	. . . . . 6 |- ( M e. ZZ -> ( M ... ( ( M - 1 ) + 1 ) ) = ( M ... M ) )
7		uncom 3363	. . . . . . . 8 |- ( (/) u. { M } ) = ( { M } u. (/) )
8		un0 3542	. . . . . . . 8 |- ( { M } u. (/) ) = { M }
9	7, 8	eqtri 2253	. . . . . . 7 |- ( (/) u. { M } ) = { M }
10		zre 9581	. . . . . . . . . 10 |- ( M e. ZZ -> M e. RR )
11	10	ltm1d 9206	. . . . . . . . 9 |- ( M e. ZZ -> ( M - 1 ) < M )
12		peano2zm 9615	. . . . . . . . . 10 |- ( M e. ZZ -> ( M - 1 ) e. ZZ )
13		fzn 10376	. . . . . . . . . 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 3702	. . . . . . . 8 |- ( M e. ZZ -> { ( ( M - 1 ) + 1 ) } = { M } )
17	15, 16	uneq12d 3374	. . . . . . 7 |- ( M e. ZZ -> ( ( M ... ( M - 1 ) ) u. { ( ( M - 1 ) + 1 ) } ) = ( (/) u. { M } ) )
18		fzsn 10400	. . . . . . 7 |- ( M e. ZZ -> ( M ... M ) = { M } )
19	9, 17, 18	3eqtr4a 2291	. . . . . 6 |- ( M e. ZZ -> ( ( M ... ( M - 1 ) ) u. { ( ( M - 1 ) + 1 ) } ) = ( M ... M ) )
20	6, 19	eqtr4d 2268	. . . . 5 |- ( M e. ZZ -> ( M ... ( ( M - 1 ) + 1 ) ) = ( ( M ... ( M - 1 ) ) u. { ( ( M - 1 ) + 1 ) } ) )
21		oveq1 6057	. . . . . . 7 |- ( N = ( M - 1 ) -> ( N + 1 ) = ( ( M - 1 ) + 1 ) )
22	21	oveq2d 6066	. . . . . 6 |- ( N = ( M - 1 ) -> ( M ... ( N + 1 ) ) = ( M ... ( ( M - 1 ) + 1 ) ) )
23		oveq2 6058	. . . . . . 7 |- ( N = ( M - 1 ) -> ( M ... N ) = ( M ... ( M - 1 ) ) )
24	21	sneqd 3702	. . . . . . 7 |- ( N = ( M - 1 ) -> { ( N + 1 ) } = { ( ( M - 1 ) + 1 ) } )
25	23, 24	uneq12d 3374	. . . . . 6 |- ( N = ( M - 1 ) -> ( ( M ... N ) u. { ( N + 1 ) } ) = ( ( M ... ( M - 1 ) ) u. { ( ( M - 1 ) + 1 ) } ) )
26	22, 25	eqeq12d 2247	. . . . 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 5674	. . . . . 6 |- ( M e. ZZ -> ( ZZ>= ` ( ( M - 1 ) + 1 ) ) = ( ZZ>= ` M ) )
30	29	eleq2d 2302	. . . . 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 10403	. . . 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 805	. 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 716   = wceq 1398   e. wcel 2203   u. cun 3209   (/)c0 3508   {csn 3689   class class class wbr 4109   ` cfv 5352  (class class class)co 6050   CCcc 8125   1c1 8128   + caddc 8130   < clt 8308   - cmin 8444   ZZcz 9577   ZZ>=cuz 9853   ...cfz 10342

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854  df-fz 10343

This theorem is referenced by:  fseq1p1m1  10428  frecfzennn  10788  zfz1isolemsplit  11210  fsumm1  12102  fprodm1  12284  gfsump1  16868