Theorem hashfz 11056

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 9738	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> A e. ZZ )
2		eluzelz 9743	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> B e. ZZ )
3		1z 9483	. . . . . 6 |- 1 e. ZZ
4		zsubcl 9498	. . . . . 6 |- ( ( 1 e. ZZ /\ A e. ZZ ) -> ( 1 - A ) e. ZZ )
5	3, 1, 4	sylancr 414	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> ( 1 - A ) e. ZZ )
6		fzen 10251	. . . . 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 1271	. . . 4 |- ( B e. ( ZZ>= ` A ) -> ( A ... B ) ~~ ( ( A + ( 1 - A ) ) ... ( B + ( 1 - A ) ) ) )
8	1	zcnd 9581	. . . . . 6 |- ( B e. ( ZZ>= ` A ) -> A e. CC )
9		ax-1cn 8103	. . . . . 6 |- 1 e. CC
10		pncan3 8365	. . . . . 6 |- ( ( A e. CC /\ 1 e. CC ) -> ( A + ( 1 - A ) ) = 1 )
11	8, 9, 10	sylancl 413	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> ( A + ( 1 - A ) ) = 1 )
12		1cnd 8173	. . . . . 6 |- ( B e. ( ZZ>= ` A ) -> 1 e. CC )
13	2	zcnd 9581	. . . . . . 7 |- ( B e. ( ZZ>= ` A ) -> B e. CC )
14	13, 8	subcld 8468	. . . . . 6 |- ( B e. ( ZZ>= ` A ) -> ( B - A ) e. CC )
15	13, 12, 8	addsub12d 8491	. . . . . 6 |- ( B e. ( ZZ>= ` A ) -> ( B + ( 1 - A ) ) = ( 1 + ( B - A ) ) )
16	12, 14, 15	comraddd 8314	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> ( B + ( 1 - A ) ) = ( ( B - A ) + 1 ) )
17	11, 16	oveq12d 6025	. . . 4 |- ( B e. ( ZZ>= ` A ) -> ( ( A + ( 1 - A ) ) ... ( B + ( 1 - A ) ) ) = ( 1 ... ( ( B - A ) + 1 ) ) )
18	7, 17	breqtrd 4109	. . 3 |- ( B e. ( ZZ>= ` A ) -> ( A ... B ) ~~ ( 1 ... ( ( B - A ) + 1 ) ) )
19	1, 2	fzfigd 10665	. . . 4 |- ( B e. ( ZZ>= ` A ) -> ( A ... B ) e. Fin )
20		1zzd 9484	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> 1 e. ZZ )
21	2, 1	zsubcld 9585	. . . . . 6 |- ( B e. ( ZZ>= ` A ) -> ( B - A ) e. ZZ )
22	21	peano2zd 9583	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> ( ( B - A ) + 1 ) e. ZZ )
23	20, 22	fzfigd 10665	. . . 4 |- ( B e. ( ZZ>= ` A ) -> ( 1 ... ( ( B - A ) + 1 ) ) e. Fin )
24		hashen 11018	. . . 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 411	. . 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 9766	. . 3 |- ( B e. ( ZZ>= ` A ) -> ( B - A ) e. NN0 )
28		peano2nn0 9420	. . 3 |- ( ( B - A ) e. NN0 -> ( ( B - A ) + 1 ) e. NN0 )
29		hashfz1 11017	. . 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 2262	1 |- ( B e. ( ZZ>= ` A ) -> (♯ ` ( A ... B ) ) = ( ( B - A ) + 1 ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   e. wcel 2200   class class class wbr 4083   ` cfv 5318  (class class class)co 6007   ~~ cen 6893   Fincfn 6895   CCcc 8008   1c1 8011   + caddc 8013   - cmin 8328   NN0cn0 9380   ZZcz 9457   ZZ>=cuz 9733   ...cfz 10216  ♯chash 11009

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-addass 8112  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-0id 8118  ax-rnegex 8119  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-1o 6568  df-er 6688  df-en 6896  df-dom 6897  df-fin 6898  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-inn 9122  df-n0 9381  df-z 9458  df-uz 9734  df-fz 10217  df-ihash 11010

This theorem is referenced by:  hashfzo  11057  hashfzp1  11059  hashfz0  11060  0sgmppw  15683  gausslemma2dlem5  15761