Theorem hashfz 12292

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 10969	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> A e. ZZ )
2		eluzelz 10973	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> B e. ZZ )
3		1z 10779	. . . . . 6 |- 1 e. ZZ
4		zsubcl 10790	. . . . . 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 11570	. . . . 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 1219	. . . 4 |- ( B e. ( ZZ>= ` A ) -> ( A ... B ) ~~ ( ( A + ( 1 - A ) ) ... ( B + ( 1 - A ) ) ) )
8	1	zcnd 10851	. . . . . 6 |- ( B e. ( ZZ>= ` A ) -> A e. CC )
9		ax-1cn 9443	. . . . . 6 |- 1 e. CC
10		pncan3 9721	. . . . . 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 10851	. . . . . . 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 9845	. . . . . 6 |- ( B e. ( ZZ>= ` A ) -> ( B + ( 1 - A ) ) = ( 1 + ( B - A ) ) )
15	12, 8	subcld 9822	. . . . . . 7 |- ( B e. ( ZZ>= ` A ) -> ( B - A ) e. CC )
16		addcom 9658	. . . . . . 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 2492	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> ( B + ( 1 - A ) ) = ( ( B - A ) + 1 ) )
19	11, 18	oveq12d 6210	. . . 4 |- ( B e. ( ZZ>= ` A ) -> ( ( A + ( 1 - A ) ) ... ( B + ( 1 - A ) ) ) = ( 1 ... ( ( B - A ) + 1 ) ) )
20	7, 19	breqtrd 4416	. . 3 |- ( B e. ( ZZ>= ` A ) -> ( A ... B ) ~~ ( 1 ... ( ( B - A ) + 1 ) ) )
21		hasheni 12222	. . 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 10995	. . 3 |- ( B e. ( ZZ>= ` A ) -> ( B - A ) e. NN0 )
24		peano2nn0 10723	. . 3 |- ( ( B - A ) e. NN0 -> ( ( B - A ) + 1 ) e. NN0 )
25		hashfz1 12220	. . 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 2492	1 |- ( B e. ( ZZ>= ` A ) -> ( # ` ( A ... B ) ) = ( ( B - A ) + 1 ) )

Syntax hints:   -> wi 4   = wceq 1370   e. wcel 1758   class class class wbr 4392   ` cfv 5518  (class class class)co 6192   ~~ cen 7409   CCcc 9383   1c1 9386   + caddc 9388   - cmin 9698   NN0cn0 10682   ZZcz 10749   ZZ>=cuz 10964   ...cfz 11540   #chash 12206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-1o 7022  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-card 8212  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-n0 10683  df-z 10750  df-uz 10965  df-fz 11541  df-hash 12207

This theorem is referenced by:  fzsdom2  12293  hashfzo  12294  hashfz0  12297  hashfzdm  12306  fz0hash  12307  0sgmppw  22655  logfaclbnd  22679  ballotlem2  27007  subfacp1lem5  27208  stoweidlem11  29946  stoweidlem26  29961