Theorem hashfz 11262

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 9926	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> A e. ZZ )
2		eluzelz 9931	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> B e. ZZ )
3		1z 9670	. . . . . 6 |- 1 e. ZZ
4		zsubcl 9685	. . . . . 6 |- ( ( 1 e. ZZ /\ A e. ZZ ) -> ( 1 - A ) e. ZZ )
5	3, 1, 4	sylancr 418	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> ( 1 - A ) e. ZZ )
6		fzen 10447	. . . . 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 1278	. . . 4 |- ( B e. ( ZZ>= ` A ) -> ( A ... B ) ~~ ( ( A + ( 1 - A ) ) ... ( B + ( 1 - A ) ) ) )
8	1	zcnd 9769	. . . . . 6 |- ( B e. ( ZZ>= ` A ) -> A e. CC )
9		ax-1cn 8272	. . . . . 6 |- 1 e. CC
10		pncan3 8534	. . . . . 6 |- ( ( A e. CC /\ 1 e. CC ) -> ( A + ( 1 - A ) ) = 1 )
11	8, 9, 10	sylancl 417	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> ( A + ( 1 - A ) ) = 1 )
12		1cnd 8342	. . . . . 6 |- ( B e. ( ZZ>= ` A ) -> 1 e. CC )
13	2	zcnd 9769	. . . . . . 7 |- ( B e. ( ZZ>= ` A ) -> B e. CC )
14	13, 8	subcld 8637	. . . . . 6 |- ( B e. ( ZZ>= ` A ) -> ( B - A ) e. CC )
15	13, 12, 8	addsub12d 8660	. . . . . 6 |- ( B e. ( ZZ>= ` A ) -> ( B + ( 1 - A ) ) = ( 1 + ( B - A ) ) )
16	12, 14, 15	comraddd 8483	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> ( B + ( 1 - A ) ) = ( ( B - A ) + 1 ) )
17	11, 16	oveq12d 6103	. . . 4 |- ( B e. ( ZZ>= ` A ) -> ( ( A + ( 1 - A ) ) ... ( B + ( 1 - A ) ) ) = ( 1 ... ( ( B - A ) + 1 ) ) )
18	7, 17	breqtrd 4156	. . 3 |- ( B e. ( ZZ>= ` A ) -> ( A ... B ) ~~ ( 1 ... ( ( B - A ) + 1 ) ) )
19	1, 2	fzfigd 10868	. . . 4 |- ( B e. ( ZZ>= ` A ) -> ( A ... B ) e. Fin )
20		1zzd 9671	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> 1 e. ZZ )
21	2, 1	zsubcld 9773	. . . . . 6 |- ( B e. ( ZZ>= ` A ) -> ( B - A ) e. ZZ )
22	21	peano2zd 9771	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> ( ( B - A ) + 1 ) e. ZZ )
23	20, 22	fzfigd 10868	. . . 4 |- ( B e. ( ZZ>= ` A ) -> ( 1 ... ( ( B - A ) + 1 ) ) e. Fin )
24		hashen 11223	. . . 4 |- ( ( ( A ... B ) e. Fin /\ ( 1 ... ( ( B - A ) + 1 ) ) e. Fin ) -> ( (♯ ` ( A ... B ) ) = (♯ ` ( 1 ... ( ( B - A ) + 1 ) ) ) <-> ( A ... B ) ~~ ( 1 ... ( ( B - A ) + 1 ) ) ) )
25	19, 23, 24	syl2anc 415	. . 3 |- ( B e. ( ZZ>= ` A ) -> ( (♯ ` ( A ... B ) ) = (♯ ` ( 1 ... ( ( B - A ) + 1 ) ) ) <-> ( A ... B ) ~~ ( 1 ... ( ( B - A ) + 1 ) ) ) )
26	18, 25	mpbird 167	. 2 |- ( B e. ( ZZ>= ` A ) -> (♯ ` ( A ... B ) ) = (♯ ` ( 1 ... ( ( B - A ) + 1 ) ) ) )
27		uznn0sub 9954	. . 3 |- ( B e. ( ZZ>= ` A ) -> ( B - A ) e. NN0 )
28		peano2nn0 9603	. . 3 |- ( ( B - A ) e. NN0 -> ( ( B - A ) + 1 ) e. NN0 )
29		hashfz1 11222	. . 3 |- ( ( ( B - A ) + 1 ) e. NN0 -> (♯ ` ( 1 ... ( ( B - A ) + 1 ) ) ) = ( ( B - A ) + 1 ) )
30	27, 28, 29	3syl 17	. 2 |- ( B e. ( ZZ>= ` A ) -> (♯ ` ( 1 ... ( ( B - A ) + 1 ) ) ) = ( ( B - A ) + 1 ) )
31	26, 30	eqtrd 2271	1 |- ( B e. ( ZZ>= ` A ) -> (♯ ` ( A ... B ) ) = ( ( B - A ) + 1 ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   = wceq 1402   e. wcel 2209   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   ~~ cen 7020   Fincfn 7022   CCcc 8177   1c1 8180   + caddc 8182   - cmin 8497   NN0cn0 9563   ZZcz 9644   ZZ>=cuz 9921   ...cfz 10411  ♯chash 11214

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-fz 10412  df-ihash 11215

This theorem is used by:  hashfzo  11263  hashfzp1  11265  hashfz0  11266  ballotfilem2  13228  gzsumgsum  14155  0sgmppw  16107  gausslemma2dlem5  16185  lgseisenlem4  16192