Theorem fzsuc2 10148

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 9629	. 2 |- ( N e. ( ZZ>= ` ( M - 1 ) ) -> ( N = ( M - 1 ) \/ N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) ) )
2		zcn 9325	. . . . . . . 8 |- ( M e. ZZ -> M e. CC )
3		ax-1cn 7967	. . . . . . . 8 |- 1 e. CC
4		npcan 8230	. . . . . . . 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 5935	. . . . . 6 |- ( M e. ZZ -> ( M ... ( ( M - 1 ) + 1 ) ) = ( M ... M ) )
7		uncom 3304	. . . . . . . 8 |- ( (/) u. { M } ) = ( { M } u. (/) )
8		un0 3481	. . . . . . . 8 |- ( { M } u. (/) ) = { M }
9	7, 8	eqtri 2214	. . . . . . 7 |- ( (/) u. { M } ) = { M }
10		zre 9324	. . . . . . . . . 10 |- ( M e. ZZ -> M e. RR )
11	10	ltm1d 8953	. . . . . . . . 9 |- ( M e. ZZ -> ( M - 1 ) < M )
12		peano2zm 9358	. . . . . . . . . 10 |- ( M e. ZZ -> ( M - 1 ) e. ZZ )
13		fzn 10111	. . . . . . . . . 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 3632	. . . . . . . 8 |- ( M e. ZZ -> { ( ( M - 1 ) + 1 ) } = { M } )
17	15, 16	uneq12d 3315	. . . . . . 7 |- ( M e. ZZ -> ( ( M ... ( M - 1 ) ) u. { ( ( M - 1 ) + 1 ) } ) = ( (/) u. { M } ) )
18		fzsn 10135	. . . . . . 7 |- ( M e. ZZ -> ( M ... M ) = { M } )
19	9, 17, 18	3eqtr4a 2252	. . . . . 6 |- ( M e. ZZ -> ( ( M ... ( M - 1 ) ) u. { ( ( M - 1 ) + 1 ) } ) = ( M ... M ) )
20	6, 19	eqtr4d 2229	. . . . 5 |- ( M e. ZZ -> ( M ... ( ( M - 1 ) + 1 ) ) = ( ( M ... ( M - 1 ) ) u. { ( ( M - 1 ) + 1 ) } ) )
21		oveq1 5926	. . . . . . 7 |- ( N = ( M - 1 ) -> ( N + 1 ) = ( ( M - 1 ) + 1 ) )
22	21	oveq2d 5935	. . . . . 6 |- ( N = ( M - 1 ) -> ( M ... ( N + 1 ) ) = ( M ... ( ( M - 1 ) + 1 ) ) )
23		oveq2 5927	. . . . . . 7 |- ( N = ( M - 1 ) -> ( M ... N ) = ( M ... ( M - 1 ) ) )
24	21	sneqd 3632	. . . . . . 7 |- ( N = ( M - 1 ) -> { ( N + 1 ) } = { ( ( M - 1 ) + 1 ) } )
25	23, 24	uneq12d 3315	. . . . . 6 |- ( N = ( M - 1 ) -> ( ( M ... N ) u. { ( N + 1 ) } ) = ( ( M ... ( M - 1 ) ) u. { ( ( M - 1 ) + 1 ) } ) )
26	22, 25	eqeq12d 2208	. . . . 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 5559	. . . . . 6 |- ( M e. ZZ -> ( ZZ>= ` ( ( M - 1 ) + 1 ) ) = ( ZZ>= ` M ) )
30	29	eleq2d 2263	. . . . 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 10138	. . . 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 2164   u. cun 3152   (/)c0 3447   {csn 3619   class class class wbr 4030   ` cfv 5255  (class class class)co 5919   CCcc 7872   1c1 7875   + caddc 7877   < clt 8056   - cmin 8192   ZZcz 9320   ZZ>=cuz 9595   ...cfz 10077

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-inn 8985  df-n0 9244  df-z 9321  df-uz 9596  df-fz 10078

This theorem is referenced by:  fseq1p1m1  10163  frecfzennn  10500  zfz1isolemsplit  10912  fsumm1  11562  fprodm1  11744