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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
StepHypRef 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 )
53, 1, 4sylancr 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
) ) ) )
71, 2, 5, 6syl3anc 1219 . . . 4  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( A ... B )  ~~  (
( A  +  ( 1  -  A ) ) ... ( B  +  ( 1  -  A ) ) ) )
81zcnd 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 )
118, 9, 10sylancl 662 . . . . 5  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( A  +  ( 1  -  A ) )  =  1 )
122zcnd 10851 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  A
)  ->  B  e.  CC )
139a1i 11 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  A
)  ->  1  e.  CC )
1412, 13, 8addsub12d 9845 . . . . . 6  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( B  +  ( 1  -  A ) )  =  ( 1  +  ( B  -  A ) ) )
1512, 8subcld 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 ) )
179, 15, 16sylancr 663 . . . . . 6  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( 1  +  ( B  -  A ) )  =  ( ( B  -  A )  +  1 ) )
1814, 17eqtrd 2492 . . . . 5  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( B  +  ( 1  -  A ) )  =  ( ( B  -  A )  +  1 ) )
1911, 18oveq12d 6210 . . . 4  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( ( A  +  ( 1  -  A ) ) ... ( B  +  ( 1  -  A
) ) )  =  ( 1 ... (
( B  -  A
)  +  1 ) ) )
207, 19breqtrd 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 ) ) ) )
2220, 21syl 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 ) )
2623, 24, 253syl 20 . 2  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( # `  (
1 ... ( ( B  -  A )  +  1 ) ) )  =  ( ( B  -  A )  +  1 ) )
2722, 26eqtrd 2492 1  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( # `  ( A ... B ) )  =  ( ( B  -  A )  +  1 ) )
Colors of variables: wff setvar class
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
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