Theorem frecfzen2 10200

Description: The cardinality of a finite set of sequential integers with arbitrary endpoints. (Contributed by Jim Kingdon, 18-May-2020.)

Hypothesis

Ref	Expression
frecfzennn.1	|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )

Assertion

Ref	Expression
frecfzen2	|- ( N e. ( ZZ>= ` M ) -> ( M ... N ) ~~ ( `' G ` ( ( N + 1 ) - M ) ) )

Proof of Theorem frecfzen2

Step	Hyp	Ref	Expression
1		eluzel2 9331	. . . 4 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
2		eluzelz 9335	. . . 4 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
3		1z 9080	. . . . 5 |- 1 e. ZZ
4		zsubcl 9095	. . . . 5 |- ( ( 1 e. ZZ /\ M e. ZZ ) -> ( 1 - M ) e. ZZ )
5	3, 1, 4	sylancr 410	. . . 4 |- ( N e. ( ZZ>= ` M ) -> ( 1 - M ) e. ZZ )
6		fzen 9823	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ /\ ( 1 - M ) e. ZZ ) -> ( M ... N ) ~~ ( ( M + ( 1 - M ) ) ... ( N + ( 1 - M ) ) ) )
7	1, 2, 5, 6	syl3anc 1216	. . 3 |- ( N e. ( ZZ>= ` M ) -> ( M ... N ) ~~ ( ( M + ( 1 - M ) ) ... ( N + ( 1 - M ) ) ) )
8	1	zcnd 9174	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. CC )
9		ax-1cn 7713	. . . . 5 |- 1 e. CC
10		pncan3 7970	. . . . 5 |- ( ( M e. CC /\ 1 e. CC ) -> ( M + ( 1 - M ) ) = 1 )
11	8, 9, 10	sylancl 409	. . . 4 |- ( N e. ( ZZ>= ` M ) -> ( M + ( 1 - M ) ) = 1 )
12		zcn 9059	. . . . . . 7 |- ( N e. ZZ -> N e. CC )
13		zcn 9059	. . . . . . 7 |- ( M e. ZZ -> M e. CC )
14		addsubass 7972	. . . . . . . 8 |- ( ( N e. CC /\ 1 e. CC /\ M e. CC ) -> ( ( N + 1 ) - M ) = ( N + ( 1 - M ) ) )
15	9, 14	mp3an2 1303	. . . . . . 7 |- ( ( N e. CC /\ M e. CC ) -> ( ( N + 1 ) - M ) = ( N + ( 1 - M ) ) )
16	12, 13, 15	syl2an 287	. . . . . 6 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( ( N + 1 ) - M ) = ( N + ( 1 - M ) ) )
17	2, 1, 16	syl2anc 408	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> ( ( N + 1 ) - M ) = ( N + ( 1 - M ) ) )
18	17	eqcomd 2145	. . . 4 |- ( N e. ( ZZ>= ` M ) -> ( N + ( 1 - M ) ) = ( ( N + 1 ) - M ) )
19	11, 18	oveq12d 5792	. . 3 |- ( N e. ( ZZ>= ` M ) -> ( ( M + ( 1 - M ) ) ... ( N + ( 1 - M ) ) ) = ( 1 ... ( ( N + 1 ) - M ) ) )
20	7, 19	breqtrd 3954	. 2 |- ( N e. ( ZZ>= ` M ) -> ( M ... N ) ~~ ( 1 ... ( ( N + 1 ) - M ) ) )
21		peano2uz 9378	. . 3 |- ( N e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( ZZ>= ` M ) )
22		uznn0sub 9357	. . 3 |- ( ( N + 1 ) e. ( ZZ>= ` M ) -> ( ( N + 1 ) - M ) e. NN0 )
23		frecfzennn.1	. . . 4 |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )
24	23	frecfzennn 10199	. . 3 |- ( ( ( N + 1 ) - M ) e. NN0 -> ( 1 ... ( ( N + 1 ) - M ) ) ~~ ( `' G ` ( ( N + 1 ) - M ) ) )
25	21, 22, 24	3syl 17	. 2 |- ( N e. ( ZZ>= ` M ) -> ( 1 ... ( ( N + 1 ) - M ) ) ~~ ( `' G ` ( ( N + 1 ) - M ) ) )
26		entr 6678	. 2 |- ( ( ( M ... N ) ~~ ( 1 ... ( ( N + 1 ) - M ) ) /\ ( 1 ... ( ( N + 1 ) - M ) ) ~~ ( `' G ` ( ( N + 1 ) - M ) ) ) -> ( M ... N ) ~~ ( `' G ` ( ( N + 1 ) - M ) ) )
27	20, 25, 26	syl2anc 408	1 |- ( N e. ( ZZ>= ` M ) -> ( M ... N ) ~~ ( `' G ` ( ( N + 1 ) - M ) ) )

Syntax hints:   -> wi 4   = wceq 1331   e. wcel 1480   class class class wbr 3929   |-> cmpt 3989   `'ccnv 4538   ` cfv 5123  (class class class)co 5774  freccfrec 6287   ~~ cen 6632   CCcc 7618   0cc0 7620   1c1 7621   + caddc 7623   - cmin 7933   NN0cn0 8977   ZZcz 9054   ZZ>=cuz 9326   ...cfz 9790

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-addcom 7720  ax-addass 7722  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-1o 6313  df-er 6429  df-en 6635  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-inn 8721  df-n0 8978  df-z 9055  df-uz 9327  df-fz 9791

This theorem is referenced by:  fzfig  10203