Theorem frecfzen2 10041

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 9181	. . . 4 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
2		eluzelz 9185	. . . 4 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
3		1z 8932	. . . . 5 |- 1 e. ZZ
4		zsubcl 8947	. . . . 5 |- ( ( 1 e. ZZ /\ M e. ZZ ) -> ( 1 - M ) e. ZZ )
5	3, 1, 4	sylancr 408	. . . 4 |- ( N e. ( ZZ>= ` M ) -> ( 1 - M ) e. ZZ )
6		fzen 9664	. . . 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 1184	. . 3 |- ( N e. ( ZZ>= ` M ) -> ( M ... N ) ~~ ( ( M + ( 1 - M ) ) ... ( N + ( 1 - M ) ) ) )
8	1	zcnd 9026	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. CC )
9		ax-1cn 7588	. . . . 5 |- 1 e. CC
10		pncan3 7841	. . . . 5 |- ( ( M e. CC /\ 1 e. CC ) -> ( M + ( 1 - M ) ) = 1 )
11	8, 9, 10	sylancl 407	. . . 4 |- ( N e. ( ZZ>= ` M ) -> ( M + ( 1 - M ) ) = 1 )
12		zcn 8911	. . . . . . 7 |- ( N e. ZZ -> N e. CC )
13		zcn 8911	. . . . . . 7 |- ( M e. ZZ -> M e. CC )
14		addsubass 7843	. . . . . . . 8 |- ( ( N e. CC /\ 1 e. CC /\ M e. CC ) -> ( ( N + 1 ) - M ) = ( N + ( 1 - M ) ) )
15	9, 14	mp3an2 1271	. . . . . . 7 |- ( ( N e. CC /\ M e. CC ) -> ( ( N + 1 ) - M ) = ( N + ( 1 - M ) ) )
16	12, 13, 15	syl2an 285	. . . . . 6 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( ( N + 1 ) - M ) = ( N + ( 1 - M ) ) )
17	2, 1, 16	syl2anc 406	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> ( ( N + 1 ) - M ) = ( N + ( 1 - M ) ) )
18	17	eqcomd 2105	. . . 4 |- ( N e. ( ZZ>= ` M ) -> ( N + ( 1 - M ) ) = ( ( N + 1 ) - M ) )
19	11, 18	oveq12d 5724	. . 3 |- ( N e. ( ZZ>= ` M ) -> ( ( M + ( 1 - M ) ) ... ( N + ( 1 - M ) ) ) = ( 1 ... ( ( N + 1 ) - M ) ) )
20	7, 19	breqtrd 3899	. 2 |- ( N e. ( ZZ>= ` M ) -> ( M ... N ) ~~ ( 1 ... ( ( N + 1 ) - M ) ) )
21		peano2uz 9228	. . 3 |- ( N e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( ZZ>= ` M ) )
22		uznn0sub 9207	. . 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 10040	. . 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 6608	. 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 406	1 |- ( N e. ( ZZ>= ` M ) -> ( M ... N ) ~~ ( `' G ` ( ( N + 1 ) - M ) ) )

Syntax hints:   -> wi 4   = wceq 1299   e. wcel 1448   class class class wbr 3875   |-> cmpt 3929   `'ccnv 4476   ` cfv 5059  (class class class)co 5706  freccfrec 6217   ~~ cen 6562   CCcc 7498   0cc0 7500   1c1 7501   + caddc 7503   - cmin 7804   NN0cn0 8829   ZZcz 8906   ZZ>=cuz 9176   ...cfz 9631

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-addcom 7595  ax-addass 7597  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-0id 7603  ax-rnegex 7604  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611

This theorem depends on definitions:  df-bi 116  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-1o 6243  df-er 6359  df-en 6565  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-inn 8579  df-n0 8830  df-z 8907  df-uz 9177  df-fz 9632

This theorem is referenced by:  fzfig  10044