Theorem frecfzen2 10572

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 9653	. . . 4 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
2		eluzelz 9657	. . . 4 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
3		1z 9398	. . . . 5 |- 1 e. ZZ
4		zsubcl 9413	. . . . 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 10165	. . . 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 1250	. . 3 |- ( N e. ( ZZ>= ` M ) -> ( M ... N ) ~~ ( ( M + ( 1 - M ) ) ... ( N + ( 1 - M ) ) ) )
8	1	zcnd 9496	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. CC )
9		ax-1cn 8018	. . . . 5 |- 1 e. CC
10		pncan3 8280	. . . . 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 9377	. . . . . . 7 |- ( N e. ZZ -> N e. CC )
13		zcn 9377	. . . . . . 7 |- ( M e. ZZ -> M e. CC )
14		addsubass 8282	. . . . . . . 8 |- ( ( N e. CC /\ 1 e. CC /\ M e. CC ) -> ( ( N + 1 ) - M ) = ( N + ( 1 - M ) ) )
15	9, 14	mp3an2 1338	. . . . . . 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 2211	. . . 4 |- ( N e. ( ZZ>= ` M ) -> ( N + ( 1 - M ) ) = ( ( N + 1 ) - M ) )
19	11, 18	oveq12d 5962	. . 3 |- ( N e. ( ZZ>= ` M ) -> ( ( M + ( 1 - M ) ) ... ( N + ( 1 - M ) ) ) = ( 1 ... ( ( N + 1 ) - M ) ) )
20	7, 19	breqtrd 4070	. 2 |- ( N e. ( ZZ>= ` M ) -> ( M ... N ) ~~ ( 1 ... ( ( N + 1 ) - M ) ) )
21		peano2uz 9704	. . 3 |- ( N e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( ZZ>= ` M ) )
22		uznn0sub 9680	. . 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 10571	. . 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 6876	. 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 1373   e. wcel 2176   class class class wbr 4044   |-> cmpt 4105   `'ccnv 4674   ` cfv 5271  (class class class)co 5944  freccfrec 6476   ~~ cen 6825   CCcc 7923   0cc0 7925   1c1 7926   + caddc 7928   - cmin 8243   NN0cn0 9295   ZZcz 9372   ZZ>=cuz 9648   ...cfz 10130

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-frec 6477  df-1o 6502  df-er 6620  df-en 6828  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-inn 9037  df-n0 9296  df-z 9373  df-uz 9649  df-fz 10131

This theorem is referenced by:  fzfig  10575