Theorem frecfzen2 10501

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 9600	. . . 4 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
2		eluzelz 9604	. . . 4 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
3		1z 9346	. . . . 5 |- 1 e. ZZ
4		zsubcl 9361	. . . . 5 |- ( ( 1 e. ZZ /\ M e. ZZ ) -> ( 1 - M ) e. ZZ )
5	3, 1, 4	sylancr 414	. . . 4 |- ( N e. ( ZZ>= ` M ) -> ( 1 - M ) e. ZZ )
6		fzen 10112	. . . 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 1249	. . 3 |- ( N e. ( ZZ>= ` M ) -> ( M ... N ) ~~ ( ( M + ( 1 - M ) ) ... ( N + ( 1 - M ) ) ) )
8	1	zcnd 9443	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. CC )
9		ax-1cn 7967	. . . . 5 |- 1 e. CC
10		pncan3 8229	. . . . 5 |- ( ( M e. CC /\ 1 e. CC ) -> ( M + ( 1 - M ) ) = 1 )
11	8, 9, 10	sylancl 413	. . . 4 |- ( N e. ( ZZ>= ` M ) -> ( M + ( 1 - M ) ) = 1 )
12		zcn 9325	. . . . . . 7 |- ( N e. ZZ -> N e. CC )
13		zcn 9325	. . . . . . 7 |- ( M e. ZZ -> M e. CC )
14		addsubass 8231	. . . . . . . 8 |- ( ( N e. CC /\ 1 e. CC /\ M e. CC ) -> ( ( N + 1 ) - M ) = ( N + ( 1 - M ) ) )
15	9, 14	mp3an2 1336	. . . . . . 7 |- ( ( N e. CC /\ M e. CC ) -> ( ( N + 1 ) - M ) = ( N + ( 1 - M ) ) )
16	12, 13, 15	syl2an 289	. . . . . 6 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( ( N + 1 ) - M ) = ( N + ( 1 - M ) ) )
17	2, 1, 16	syl2anc 411	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> ( ( N + 1 ) - M ) = ( N + ( 1 - M ) ) )
18	17	eqcomd 2199	. . . 4 |- ( N e. ( ZZ>= ` M ) -> ( N + ( 1 - M ) ) = ( ( N + 1 ) - M ) )
19	11, 18	oveq12d 5937	. . 3 |- ( N e. ( ZZ>= ` M ) -> ( ( M + ( 1 - M ) ) ... ( N + ( 1 - M ) ) ) = ( 1 ... ( ( N + 1 ) - M ) ) )
20	7, 19	breqtrd 4056	. 2 |- ( N e. ( ZZ>= ` M ) -> ( M ... N ) ~~ ( 1 ... ( ( N + 1 ) - M ) ) )
21		peano2uz 9651	. . 3 |- ( N e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( ZZ>= ` M ) )
22		uznn0sub 9627	. . 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 10500	. . 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 6840	. 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 411	1 |- ( N e. ( ZZ>= ` M ) -> ( M ... N ) ~~ ( `' G ` ( ( N + 1 ) - M ) ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   class class class wbr 4030   |-> cmpt 4091   `'ccnv 4659   ` cfv 5255  (class class class)co 5919  freccfrec 6445   ~~ cen 6794   CCcc 7872   0cc0 7874   1c1 7875   + caddc 7877   - cmin 8192   NN0cn0 9243   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-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  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-csb 3082  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-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  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-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-1o 6471  df-er 6589  df-en 6797  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:  fzfig  10504