Theorem hashfz 11647

Description: Value of the numeric cardinality of a nonempty integer range. (Contributed by Stefan O'Rear, 12-Sep-2014.) (Proof shortened by Mario Carneiro, 15-Apr-2015.)

Assertion

Ref	Expression
hashfz	|- ( B e. ( ZZ>= ` A ) -> ( # ` ( A ... B ) ) = ( ( B - A ) + 1 ) )

Proof of Theorem hashfz

Step	Hyp	Ref	Expression
1		eluzel2 10449	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> A e. ZZ )
2		eluzelz 10452	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> B e. ZZ )
3		1z 10267	. . . . . 6 |- 1 e. ZZ
4		zsubcl 10275	. . . . . 6 |- ( ( 1 e. ZZ /\ A e. ZZ ) -> ( 1 - A ) e. ZZ )
5	3, 1, 4	sylancr 645	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> ( 1 - A ) e. ZZ )
6		fzen 11028	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ ( 1 - A ) e. ZZ ) -> ( A ... B ) ~~ ( ( A + ( 1 - A ) ) ... ( B + ( 1 - A ) ) ) )
7	1, 2, 5, 6	syl3anc 1184	. . . 4 |- ( B e. ( ZZ>= ` A ) -> ( A ... B ) ~~ ( ( A + ( 1 - A ) ) ... ( B + ( 1 - A ) ) ) )
8	1	zcnd 10332	. . . . . 6 |- ( B e. ( ZZ>= ` A ) -> A e. CC )
9		ax-1cn 9004	. . . . . 6 |- 1 e. CC
10		pncan3 9269	. . . . . 6 |- ( ( A e. CC /\ 1 e. CC ) -> ( A + ( 1 - A ) ) = 1 )
11	8, 9, 10	sylancl 644	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> ( A + ( 1 - A ) ) = 1 )
12	2	zcnd 10332	. . . . . . 7 |- ( B e. ( ZZ>= ` A ) -> B e. CC )
13	9	a1i 11	. . . . . . 7 |- ( B e. ( ZZ>= ` A ) -> 1 e. CC )
14	12, 13, 8	addsub12d 9390	. . . . . 6 |- ( B e. ( ZZ>= ` A ) -> ( B + ( 1 - A ) ) = ( 1 + ( B - A ) ) )
15	12, 8	subcld 9367	. . . . . . 7 |- ( B e. ( ZZ>= ` A ) -> ( B - A ) e. CC )
16		addcom 9208	. . . . . . 7 |- ( ( 1 e. CC /\ ( B - A ) e. CC ) -> ( 1 + ( B - A ) ) = ( ( B - A ) + 1 ) )
17	9, 15, 16	sylancr 645	. . . . . 6 |- ( B e. ( ZZ>= ` A ) -> ( 1 + ( B - A ) ) = ( ( B - A ) + 1 ) )
18	14, 17	eqtrd 2436	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> ( B + ( 1 - A ) ) = ( ( B - A ) + 1 ) )
19	11, 18	oveq12d 6058	. . . 4 |- ( B e. ( ZZ>= ` A ) -> ( ( A + ( 1 - A ) ) ... ( B + ( 1 - A ) ) ) = ( 1 ... ( ( B - A ) + 1 ) ) )
20	7, 19	breqtrd 4196	. . 3 |- ( B e. ( ZZ>= ` A ) -> ( A ... B ) ~~ ( 1 ... ( ( B - A ) + 1 ) ) )
21		hasheni 11587	. . 3 |- ( ( A ... B ) ~~ ( 1 ... ( ( B - A ) + 1 ) ) -> ( # ` ( A ... B ) ) = ( # ` ( 1 ... ( ( B - A ) + 1 ) ) ) )
22	20, 21	syl 16	. 2 |- ( B e. ( ZZ>= ` A ) -> ( # ` ( A ... B ) ) = ( # ` ( 1 ... ( ( B - A ) + 1 ) ) ) )
23		uznn0sub 10473	. . 3 |- ( B e. ( ZZ>= ` A ) -> ( B - A ) e. NN0 )
24		peano2nn0 10216	. . 3 |- ( ( B - A ) e. NN0 -> ( ( B - A ) + 1 ) e. NN0 )
25		hashfz1 11585	. . 3 |- ( ( ( B - A ) + 1 ) e. NN0 -> ( # ` ( 1 ... ( ( B - A ) + 1 ) ) ) = ( ( B - A ) + 1 ) )
26	23, 24, 25	3syl 19	. 2 |- ( B e. ( ZZ>= ` A ) -> ( # ` ( 1 ... ( ( B - A ) + 1 ) ) ) = ( ( B - A ) + 1 ) )
27	22, 26	eqtrd 2436	1 |- ( B e. ( ZZ>= ` A ) -> ( # ` ( A ... B ) ) = ( ( B - A ) + 1 ) )

Syntax hints:   -> wi 4   = wceq 1649   e. wcel 1721   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   ~~ cen 7065   CCcc 8944   1c1 8947   + caddc 8949   - cmin 9247   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999   #chash 11573

This theorem is referenced by:  fzsdom2  11648  hashfzo  11649  0sgmppw  20935  logfaclbnd  20959  ballotlem2  24699  subfacp1lem5  24823  stoweidlem11  27627  stoweidlem26  27642  hashfzdm  27997

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-hash 11574