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Theorem hashfz 11647
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
StepHypRef Expression
1 eluzel2 10449 . . . . 5  |-  ( B  e.  ( ZZ>= `  A
)  ->  A  e.  ZZ )
2 eluzelz 10452 . . . . 5  |-  ( B  e.  ( ZZ>= `  A
)  ->  B  e.  ZZ )
3 1z 10267 . . . . . 6  |-  1  e.  ZZ
4 zsubcl 10275 . . . . . 6  |-  ( ( 1  e.  ZZ  /\  A  e.  ZZ )  ->  ( 1  -  A
)  e.  ZZ )
53, 1, 4sylancr 645 . . . . 5  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( 1  -  A )  e.  ZZ )
6 fzen 11028 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  (
1  -  A )  e.  ZZ )  -> 
( A ... B
)  ~~  ( ( A  +  ( 1  -  A ) ) ... ( B  +  ( 1  -  A
) ) ) )
71, 2, 5, 6syl3anc 1184 . . . 4  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( A ... B )  ~~  (
( A  +  ( 1  -  A ) ) ... ( B  +  ( 1  -  A ) ) ) )
81zcnd 10332 . . . . . 6  |-  ( B  e.  ( ZZ>= `  A
)  ->  A  e.  CC )
9 ax-1cn 9004 . . . . . 6  |-  1  e.  CC
10 pncan3 9269 . . . . . 6  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A  +  ( 1  -  A ) )  =  1 )
118, 9, 10sylancl 644 . . . . 5  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( A  +  ( 1  -  A ) )  =  1 )
122zcnd 10332 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  A
)  ->  B  e.  CC )
139a1i 11 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  A
)  ->  1  e.  CC )
1412, 13, 8addsub12d 9390 . . . . . 6  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( B  +  ( 1  -  A ) )  =  ( 1  +  ( B  -  A ) ) )
1512, 8subcld 9367 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( B  -  A )  e.  CC )
16 addcom 9208 . . . . . . 7  |-  ( ( 1  e.  CC  /\  ( B  -  A
)  e.  CC )  ->  ( 1  +  ( B  -  A
) )  =  ( ( B  -  A
)  +  1 ) )
179, 15, 16sylancr 645 . . . . . 6  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( 1  +  ( B  -  A ) )  =  ( ( B  -  A )  +  1 ) )
1814, 17eqtrd 2436 . . . . 5  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( B  +  ( 1  -  A ) )  =  ( ( B  -  A )  +  1 ) )
1911, 18oveq12d 6058 . . . 4  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( ( A  +  ( 1  -  A ) ) ... ( B  +  ( 1  -  A
) ) )  =  ( 1 ... (
( B  -  A
)  +  1 ) ) )
207, 19breqtrd 4196 . . 3  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( A ... B )  ~~  (
1 ... ( ( B  -  A )  +  1 ) ) )
21 hasheni 11587 . . 3  |-  ( ( A ... B ) 
~~  ( 1 ... ( ( B  -  A )  +  1 ) )  ->  ( # `
 ( A ... B ) )  =  ( # `  (
1 ... ( ( B  -  A )  +  1 ) ) ) )
2220, 21syl 16 . 2  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( # `  ( A ... B ) )  =  ( # `  (
1 ... ( ( B  -  A )  +  1 ) ) ) )
23 uznn0sub 10473 . . 3  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( B  -  A )  e.  NN0 )
24 peano2nn0 10216 . . 3  |-  ( ( B  -  A )  e.  NN0  ->  ( ( B  -  A )  +  1 )  e. 
NN0 )
25 hashfz1 11585 . . 3  |-  ( ( ( B  -  A
)  +  1 )  e.  NN0  ->  ( # `  ( 1 ... (
( B  -  A
)  +  1 ) ) )  =  ( ( B  -  A
)  +  1 ) )
2623, 24, 253syl 19 . 2  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( # `  (
1 ... ( ( B  -  A )  +  1 ) ) )  =  ( ( B  -  A )  +  1 ) )
2722, 26eqtrd 2436 1  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( # `  ( A ... B ) )  =  ( ( B  -  A )  +  1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   class class class wbr 4172   ` cfv 5413  (class class class)co 6040    ~~ cen 7065   CCcc 8944   1c1 8947    + caddc 8949    - cmin 9247   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999   #chash 11573
This theorem is referenced by:  fzsdom2  11648  hashfzo  11649  0sgmppw  20935  logfaclbnd  20959  ballotlem2  24699  subfacp1lem5  24823  stoweidlem11  27627  stoweidlem26  27642  hashfzdm  27997
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-hash 11574
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