Theorem hashfz 12459

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 11090	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> A e. ZZ )
2		eluzelz 11094	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> B e. ZZ )
3		1z 10895	. . . . . 6 |- 1 e. ZZ
4		zsubcl 10907	. . . . . 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 11707	. . . . 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 1227	. . . 4 |- ( B e. ( ZZ>= ` A ) -> ( A ... B ) ~~ ( ( A + ( 1 - A ) ) ... ( B + ( 1 - A ) ) ) )
8	1	zcnd 10970	. . . . . 6 |- ( B e. ( ZZ>= ` A ) -> A e. CC )
9		ax-1cn 9548	. . . . . 6 |- 1 e. CC
10		pncan3 9828	. . . . . 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 10970	. . . . . . 7 |- ( B e. ( ZZ>= ` A ) -> B e. CC )
13		1cnd 9610	. . . . . . 7 |- ( B e. ( ZZ>= ` A ) -> 1 e. CC )
14	12, 13, 8	addsub12d 9954	. . . . . 6 |- ( B e. ( ZZ>= ` A ) -> ( B + ( 1 - A ) ) = ( 1 + ( B - A ) ) )
15	12, 8	subcld 9931	. . . . . . 7 |- ( B e. ( ZZ>= ` A ) -> ( B - A ) e. CC )
16		addcom 9764	. . . . . . 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 2482	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> ( B + ( 1 - A ) ) = ( ( B - A ) + 1 ) )
19	11, 18	oveq12d 6295	. . . 4 |- ( B e. ( ZZ>= ` A ) -> ( ( A + ( 1 - A ) ) ... ( B + ( 1 - A ) ) ) = ( 1 ... ( ( B - A ) + 1 ) ) )
20	7, 19	breqtrd 4457	. . 3 |- ( B e. ( ZZ>= ` A ) -> ( A ... B ) ~~ ( 1 ... ( ( B - A ) + 1 ) ) )
21		hasheni 12395	. . 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 11116	. . 3 |- ( B e. ( ZZ>= ` A ) -> ( B - A ) e. NN0 )
24		peano2nn0 10837	. . 3 |- ( ( B - A ) e. NN0 -> ( ( B - A ) + 1 ) e. NN0 )
25		hashfz1 12393	. . 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 2482	1 |- ( B e. ( ZZ>= ` A ) -> ( # ` ( A ... B ) ) = ( ( B - A ) + 1 ) )

Syntax hints:   -> wi 4   = wceq 1381   e. wcel 1802   class class class wbr 4433   ` cfv 5574  (class class class)co 6277   ~~ cen 7511   CCcc 9488   1c1 9491   + caddc 9493   - cmin 9805   NN0cn0 10796   ZZcz 10865   ZZ>=cuz 11085   ...cfz 11676   #chash 12379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-card 8318  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-n0 10797  df-z 10866  df-uz 11086  df-fz 11677  df-hash 12380

This theorem is referenced by:  fzsdom2  12460  hashfzo  12461  hashfz0  12464  0sgmppw  23338  logfaclbnd  23362  ballotlem2  28293  subfacp1lem5  28494  fzisoeu  31445  stoweidlem11  31678  stoweidlem26  31693