Theorem fzdifsuc2 39525

Description: Remove a successor from the end of a finite set of sequential integers. Similar to fzdifsuc 12400, but with a weaker condition. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Assertion

Ref	Expression
fzdifsuc2	|- ( N e. ( ZZ>= ` ( M - 1 ) ) -> ( M ... N ) = ( ( M ... ( N + 1 ) ) \ { ( N + 1 ) } ) )

Proof of Theorem fzdifsuc2

Step	Hyp	Ref	Expression
1		simpr 477	. . . . . 6 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ N = ( M - 1 ) ) -> N = ( M - 1 ) )
2		zre 11381	. . . . . . . 8 |- ( M e. ZZ -> M e. RR )
3	2	ad2antlr 763	. . . . . . 7 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ N = ( M - 1 ) ) -> M e. RR )
4	3	ltm1d 10956	. . . . . 6 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ N = ( M - 1 ) ) -> ( M - 1 ) < M )
5	1, 4	eqbrtrd 4675	. . . . 5 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ N = ( M - 1 ) ) -> N < M )
6		simplr 792	. . . . . 6 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ N = ( M - 1 ) ) -> M e. ZZ )
7		eluzelz 11697	. . . . . . 7 |- ( N e. ( ZZ>= ` ( M - 1 ) ) -> N e. ZZ )
8	7	ad2antrr 762	. . . . . 6 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ N = ( M - 1 ) ) -> N e. ZZ )
9		fzn 12357	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( N < M <-> ( M ... N ) = (/) ) )
10	6, 8, 9	syl2anc 693	. . . . 5 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ N = ( M - 1 ) ) -> ( N < M <-> ( M ... N ) = (/) ) )
11	5, 10	mpbid 222	. . . 4 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ N = ( M - 1 ) ) -> ( M ... N ) = (/) )
12		difid 3948	. . . . . 6 |- ( { M } \ { M } ) = (/)
13	12	a1i 11	. . . . 5 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ N = ( M - 1 ) ) -> ( { M } \ { M } ) = (/) )
14	13	eqcomd 2628	. . . 4 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ N = ( M - 1 ) ) -> (/) = ( { M } \ { M } ) )
15		oveq1 6657	. . . . . . . . 9 |- ( N = ( M - 1 ) -> ( N + 1 ) = ( ( M - 1 ) + 1 ) )
16	15	adantl 482	. . . . . . . 8 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ N = ( M - 1 ) ) -> ( N + 1 ) = ( ( M - 1 ) + 1 ) )
17	2	recnd 10068	. . . . . . . . . 10 |- ( M e. ZZ -> M e. CC )
18	17	ad2antlr 763	. . . . . . . . 9 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ N = ( M - 1 ) ) -> M e. CC )
19		1cnd 10056	. . . . . . . . 9 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ N = ( M - 1 ) ) -> 1 e. CC )
20	18, 19	npcand 10396	. . . . . . . 8 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ N = ( M - 1 ) ) -> ( ( M - 1 ) + 1 ) = M )
21	16, 20	eqtrd 2656	. . . . . . 7 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ N = ( M - 1 ) ) -> ( N + 1 ) = M )
22	21	oveq2d 6666	. . . . . 6 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ N = ( M - 1 ) ) -> ( M ... ( N + 1 ) ) = ( M ... M ) )
23		fzsn 12383	. . . . . . 7 |- ( M e. ZZ -> ( M ... M ) = { M } )
24	23	ad2antlr 763	. . . . . 6 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ N = ( M - 1 ) ) -> ( M ... M ) = { M } )
25	22, 24	eqtr2d 2657	. . . . 5 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ N = ( M - 1 ) ) -> { M } = ( M ... ( N + 1 ) ) )
26	21	eqcomd 2628	. . . . . 6 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ N = ( M - 1 ) ) -> M = ( N + 1 ) )
27	26	sneqd 4189	. . . . 5 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ N = ( M - 1 ) ) -> { M } = { ( N + 1 ) } )
28	25, 27	difeq12d 3729	. . . 4 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ N = ( M - 1 ) ) -> ( { M } \ { M } ) = ( ( M ... ( N + 1 ) ) \ { ( N + 1 ) } ) )
29	11, 14, 28	3eqtrd 2660	. . 3 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ N = ( M - 1 ) ) -> ( M ... N ) = ( ( M ... ( N + 1 ) ) \ { ( N + 1 ) } ) )
30		simplr 792	. . . . . 6 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ -. N = ( M - 1 ) ) -> M e. ZZ )
31	7	ad2antrr 762	. . . . . 6 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ -. N = ( M - 1 ) ) -> N e. ZZ )
32	2	ad2antlr 763	. . . . . . . . 9 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ -. N = ( M - 1 ) ) -> M e. RR )
33		1red 10055	. . . . . . . . 9 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ -. N = ( M - 1 ) ) -> 1 e. RR )
34	32, 33	resubcld 10458	. . . . . . . 8 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ -. N = ( M - 1 ) ) -> ( M - 1 ) e. RR )
35	31	zred 11482	. . . . . . . 8 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ -. N = ( M - 1 ) ) -> N e. RR )
36		eluzle 11700	. . . . . . . . 9 |- ( N e. ( ZZ>= ` ( M - 1 ) ) -> ( M - 1 ) <_ N )
37	36	ad2antrr 762	. . . . . . . 8 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ -. N = ( M - 1 ) ) -> ( M - 1 ) <_ N )
38		neqne 2802	. . . . . . . . 9 |- ( -. N = ( M - 1 ) -> N =/= ( M - 1 ) )
39	38	adantl 482	. . . . . . . 8 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ -. N = ( M - 1 ) ) -> N =/= ( M - 1 ) )
40	34, 35, 37, 39	leneltd 10191	. . . . . . 7 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ -. N = ( M - 1 ) ) -> ( M - 1 ) < N )
41		zlem1lt 11429	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M <_ N <-> ( M - 1 ) < N ) )
42	30, 31, 41	syl2anc 693	. . . . . . 7 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ -. N = ( M - 1 ) ) -> ( M <_ N <-> ( M - 1 ) < N ) )
43	40, 42	mpbird 247	. . . . . 6 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ -. N = ( M - 1 ) ) -> M <_ N )
44	30, 31, 43	3jca 1242	. . . . 5 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ -. N = ( M - 1 ) ) -> ( M e. ZZ /\ N e. ZZ /\ M <_ N ) )
45		eluz2 11693	. . . . 5 |- ( N e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ N e. ZZ /\ M <_ N ) )
46	44, 45	sylibr 224	. . . 4 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ -. N = ( M - 1 ) ) -> N e. ( ZZ>= ` M ) )
47		fzdifsuc 12400	. . . 4 |- ( N e. ( ZZ>= ` M ) -> ( M ... N ) = ( ( M ... ( N + 1 ) ) \ { ( N + 1 ) } ) )
48	46, 47	syl 17	. . 3 |- ( ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) /\ -. N = ( M - 1 ) ) -> ( M ... N ) = ( ( M ... ( N + 1 ) ) \ { ( N + 1 ) } ) )
49	29, 48	pm2.61dan 832	. 2 |- ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ M e. ZZ ) -> ( M ... N ) = ( ( M ... ( N + 1 ) ) \ { ( N + 1 ) } ) )
50		eluzel2 11692	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
51	50	con3i 150	. . . . . 6 |- ( -. M e. ZZ -> -. N e. ( ZZ>= ` M ) )
52		fzn0 12355	. . . . . 6 |- ( ( M ... N ) =/= (/) <-> N e. ( ZZ>= ` M ) )
53	51, 52	sylnibr 319	. . . . 5 |- ( -. M e. ZZ -> -. ( M ... N ) =/= (/) )
54		nne 2798	. . . . 5 |- ( -. ( M ... N ) =/= (/) <-> ( M ... N ) = (/) )
55	53, 54	sylib 208	. . . 4 |- ( -. M e. ZZ -> ( M ... N ) = (/) )
56		eluzel2 11692	. . . . . . . . 9 |- ( ( N + 1 ) e. ( ZZ>= ` M ) -> M e. ZZ )
57	56	con3i 150	. . . . . . . 8 |- ( -. M e. ZZ -> -. ( N + 1 ) e. ( ZZ>= ` M ) )
58		fzn0 12355	. . . . . . . 8 |- ( ( M ... ( N + 1 ) ) =/= (/) <-> ( N + 1 ) e. ( ZZ>= ` M ) )
59	57, 58	sylnibr 319	. . . . . . 7 |- ( -. M e. ZZ -> -. ( M ... ( N + 1 ) ) =/= (/) )
60		nne 2798	. . . . . . 7 |- ( -. ( M ... ( N + 1 ) ) =/= (/) <-> ( M ... ( N + 1 ) ) = (/) )
61	59, 60	sylib 208	. . . . . 6 |- ( -. M e. ZZ -> ( M ... ( N + 1 ) ) = (/) )
62	61	difeq1d 3727	. . . . 5 |- ( -. M e. ZZ -> ( ( M ... ( N + 1 ) ) \ { ( N + 1 ) } ) = ( (/) \ { ( N + 1 ) } ) )
63		0dif 3977	. . . . . 6 |- ( (/) \ { ( N + 1 ) } ) = (/)
64	63	a1i 11	. . . . 5 |- ( -. M e. ZZ -> ( (/) \ { ( N + 1 ) } ) = (/) )
65	62, 64	eqtr2d 2657	. . . 4 |- ( -. M e. ZZ -> (/) = ( ( M ... ( N + 1 ) ) \ { ( N + 1 ) } ) )
66	55, 65	eqtrd 2656	. . 3 |- ( -. M e. ZZ -> ( M ... N ) = ( ( M ... ( N + 1 ) ) \ { ( N + 1 ) } ) )
67	66	adantl 482	. 2 |- ( ( N e. ( ZZ>= ` ( M - 1 ) ) /\ -. M e. ZZ ) -> ( M ... N ) = ( ( M ... ( N + 1 ) ) \ { ( N + 1 ) } ) )
68	49, 67	pm2.61dan 832	1 |- ( N e. ( ZZ>= ` ( M - 1 ) ) -> ( M ... N ) = ( ( M ... ( N + 1 ) ) \ { ( N + 1 ) } ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   (/)c0 3915   {csn 4177   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327

This theorem is referenced by:  dvnmul  40158