Theorem hashfz 11086

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 9760	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> A e. ZZ )
2		eluzelz 9765	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> B e. ZZ )
3		1z 9505	. . . . . 6 |- 1 e. ZZ
4		zsubcl 9520	. . . . . 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 10278	. . . . 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 1273	. . . 4 |- ( B e. ( ZZ>= ` A ) -> ( A ... B ) ~~ ( ( A + ( 1 - A ) ) ... ( B + ( 1 - A ) ) ) )
8	1	zcnd 9603	. . . . . 6 |- ( B e. ( ZZ>= ` A ) -> A e. CC )
9		ax-1cn 8125	. . . . . 6 |- 1 e. CC
10		pncan3 8387	. . . . . 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 8195	. . . . . 6 |- ( B e. ( ZZ>= ` A ) -> 1 e. CC )
13	2	zcnd 9603	. . . . . . 7 |- ( B e. ( ZZ>= ` A ) -> B e. CC )
14	13, 8	subcld 8490	. . . . . 6 |- ( B e. ( ZZ>= ` A ) -> ( B - A ) e. CC )
15	13, 12, 8	addsub12d 8513	. . . . . 6 |- ( B e. ( ZZ>= ` A ) -> ( B + ( 1 - A ) ) = ( 1 + ( B - A ) ) )
16	12, 14, 15	comraddd 8336	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> ( B + ( 1 - A ) ) = ( ( B - A ) + 1 ) )
17	11, 16	oveq12d 6036	. . . 4 |- ( B e. ( ZZ>= ` A ) -> ( ( A + ( 1 - A ) ) ... ( B + ( 1 - A ) ) ) = ( 1 ... ( ( B - A ) + 1 ) ) )
18	7, 17	breqtrd 4114	. . 3 |- ( B e. ( ZZ>= ` A ) -> ( A ... B ) ~~ ( 1 ... ( ( B - A ) + 1 ) ) )
19	1, 2	fzfigd 10694	. . . 4 |- ( B e. ( ZZ>= ` A ) -> ( A ... B ) e. Fin )
20		1zzd 9506	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> 1 e. ZZ )
21	2, 1	zsubcld 9607	. . . . . 6 |- ( B e. ( ZZ>= ` A ) -> ( B - A ) e. ZZ )
22	21	peano2zd 9605	. . . . 5 |- ( B e. ( ZZ>= ` A ) -> ( ( B - A ) + 1 ) e. ZZ )
23	20, 22	fzfigd 10694	. . . 4 |- ( B e. ( ZZ>= ` A ) -> ( 1 ... ( ( B - A ) + 1 ) ) e. Fin )
24		hashen 11047	. . . 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 9788	. . 3 |- ( B e. ( ZZ>= ` A ) -> ( B - A ) e. NN0 )
28		peano2nn0 9442	. . 3 |- ( ( B - A ) e. NN0 -> ( ( B - A ) + 1 ) e. NN0 )
29		hashfz1 11046	. . 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 2264	1 |- ( B e. ( ZZ>= ` A ) -> (♯ ` ( A ... B ) ) = ( ( B - A ) + 1 ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1397   e. wcel 2202   class class class wbr 4088   ` cfv 5326  (class class class)co 6018   ~~ cen 6907   Fincfn 6909   CCcc 8030   1c1 8033   + caddc 8035   - cmin 8350   NN0cn0 9402   ZZcz 9479   ZZ>=cuz 9755   ...cfz 10243  ♯chash 11038

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-1o 6582  df-er 6702  df-en 6910  df-dom 6911  df-fin 6912  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-n0 9403  df-z 9480  df-uz 9756  df-fz 10244  df-ihash 11039

This theorem is referenced by:  hashfzo  11087  hashfzp1  11089  hashfz0  11090  0sgmppw  15736  gausslemma2dlem5  15814  gsumgfsum  16736