Theorem hashfz 12437

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 11076	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> A e. ZZ )
2		eluzelz 11080	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> B e. ZZ )
3		1z 10883	. . . . . 6 |- 1 e. ZZ
4		zsubcl 10894	. . . . . 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 11692	. . . . 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 1223	. . . 4 |- ( B e. ( ZZ>= ` A ) -> ( A ... B ) ~~ ( ( A + ( 1 - A ) ) ... ( B + ( 1 - A ) ) ) )
8	1	zcnd 10956	. . . . . 6 |- ( B e. ( ZZ>= ` A ) -> A e. CC )
9		ax-1cn 9539	. . . . . 6 |- 1 e. CC
10		pncan3 9817	. . . . . 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 10956	. . . . . . 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 9942	. . . . . 6 |- ( B e. ( ZZ>= ` A ) -> ( B + ( 1 - A ) ) = ( 1 + ( B - A ) ) )
15	12, 8	subcld 9919	. . . . . . 7 |- ( B e. ( ZZ>= ` A ) -> ( B - A ) e. CC )
16		addcom 9754	. . . . . . 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 2501	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> ( B + ( 1 - A ) ) = ( ( B - A ) + 1 ) )
19	11, 18	oveq12d 6293	. . . 4 |- ( B e. ( ZZ>= ` A ) -> ( ( A + ( 1 - A ) ) ... ( B + ( 1 - A ) ) ) = ( 1 ... ( ( B - A ) + 1 ) ) )
20	7, 19	breqtrd 4464	. . 3 |- ( B e. ( ZZ>= ` A ) -> ( A ... B ) ~~ ( 1 ... ( ( B - A ) + 1 ) ) )
21		hasheni 12376	. . 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 11102	. . 3 |- ( B e. ( ZZ>= ` A ) -> ( B - A ) e. NN0 )
24		peano2nn0 10825	. . 3 |- ( ( B - A ) e. NN0 -> ( ( B - A ) + 1 ) e. NN0 )
25		hashfz1 12374	. . 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 2501	1 |- ( B e. ( ZZ>= ` A ) -> ( # ` ( A ... B ) ) = ( ( B - A ) + 1 ) )

Syntax hints:   -> wi 4   = wceq 1374   e. wcel 1762   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   ~~ cen 7503   CCcc 9479   1c1 9482   + caddc 9484   - cmin 9794   NN0cn0 10784   ZZcz 10853   ZZ>=cuz 11071   ...cfz 11661   #chash 12360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-hash 12361

This theorem is referenced by:  fzsdom2  12438  hashfzo  12439  hashfz0  12442  hashfzdm  12451  fz0hash  12452  0sgmppw  23194  logfaclbnd  23218  ballotlem2  27917  subfacp1lem5  28118  fzisoeu  30896  stoweidlem11  31130  stoweidlem26  31145