Theorem hashfz 12180

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 10858	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> A e. ZZ )
2		eluzelz 10862	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> B e. ZZ )
3		1z 10668	. . . . . 6 |- 1 e. ZZ
4		zsubcl 10679	. . . . . 6 |- ( ( 1 e. ZZ /\ A e. ZZ ) -> ( 1 - A ) e. ZZ )
5	3, 1, 4	sylancr 663	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> ( 1 - A ) e. ZZ )
6		fzen 11459	. . . . 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 1218	. . . 4 |- ( B e. ( ZZ>= ` A ) -> ( A ... B ) ~~ ( ( A + ( 1 - A ) ) ... ( B + ( 1 - A ) ) ) )
8	1	zcnd 10740	. . . . . 6 |- ( B e. ( ZZ>= ` A ) -> A e. CC )
9		ax-1cn 9332	. . . . . 6 |- 1 e. CC
10		pncan3 9610	. . . . . 6 |- ( ( A e. CC /\ 1 e. CC ) -> ( A + ( 1 - A ) ) = 1 )
11	8, 9, 10	sylancl 662	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> ( A + ( 1 - A ) ) = 1 )
12	2	zcnd 10740	. . . . . . 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 9734	. . . . . 6 |- ( B e. ( ZZ>= ` A ) -> ( B + ( 1 - A ) ) = ( 1 + ( B - A ) ) )
15	12, 8	subcld 9711	. . . . . . 7 |- ( B e. ( ZZ>= ` A ) -> ( B - A ) e. CC )
16		addcom 9547	. . . . . . 7 |- ( ( 1 e. CC /\ ( B - A ) e. CC ) -> ( 1 + ( B - A ) ) = ( ( B - A ) + 1 ) )
17	9, 15, 16	sylancr 663	. . . . . 6 |- ( B e. ( ZZ>= ` A ) -> ( 1 + ( B - A ) ) = ( ( B - A ) + 1 ) )
18	14, 17	eqtrd 2470	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> ( B + ( 1 - A ) ) = ( ( B - A ) + 1 ) )
19	11, 18	oveq12d 6104	. . . 4 |- ( B e. ( ZZ>= ` A ) -> ( ( A + ( 1 - A ) ) ... ( B + ( 1 - A ) ) ) = ( 1 ... ( ( B - A ) + 1 ) ) )
20	7, 19	breqtrd 4311	. . 3 |- ( B e. ( ZZ>= ` A ) -> ( A ... B ) ~~ ( 1 ... ( ( B - A ) + 1 ) ) )
21		hasheni 12111	. . 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 10884	. . 3 |- ( B e. ( ZZ>= ` A ) -> ( B - A ) e. NN0 )
24		peano2nn0 10612	. . 3 |- ( ( B - A ) e. NN0 -> ( ( B - A ) + 1 ) e. NN0 )
25		hashfz1 12109	. . 3 |- ( ( ( B - A ) + 1 ) e. NN0 -> ( # ` ( 1 ... ( ( B - A ) + 1 ) ) ) = ( ( B - A ) + 1 ) )
26	23, 24, 25	3syl 20	. 2 |- ( B e. ( ZZ>= ` A ) -> ( # ` ( 1 ... ( ( B - A ) + 1 ) ) ) = ( ( B - A ) + 1 ) )
27	22, 26	eqtrd 2470	1 |- ( B e. ( ZZ>= ` A ) -> ( # ` ( A ... B ) ) = ( ( B - A ) + 1 ) )

Syntax hints:   -> wi 4   = wceq 1369   e. wcel 1756   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   ~~ cen 7299   CCcc 9272   1c1 9275   + caddc 9277   - cmin 9587   NN0cn0 10571   ZZcz 10638   ZZ>=cuz 10853   ...cfz 11429   #chash 12095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-hash 12096

This theorem is referenced by:  fzsdom2  12181  hashfzo  12182  hashfz0  12185  hashfzdm  12194  fz0hash  12195  0sgmppw  22512  logfaclbnd  22536  ballotlem2  26823  subfacp1lem5  27024  stoweidlem11  29759  stoweidlem26  29774