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Theorem hashfz 10574
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 9338 . . . . 5  |-  ( B  e.  ( ZZ>= `  A
)  ->  A  e.  ZZ )
2 eluzelz 9342 . . . . 5  |-  ( B  e.  ( ZZ>= `  A
)  ->  B  e.  ZZ )
3 1z 9087 . . . . . 6  |-  1  e.  ZZ
4 zsubcl 9102 . . . . . 6  |-  ( ( 1  e.  ZZ  /\  A  e.  ZZ )  ->  ( 1  -  A
)  e.  ZZ )
53, 1, 4sylancr 410 . . . . 5  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( 1  -  A )  e.  ZZ )
6 fzen 9830 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  (
1  -  A )  e.  ZZ )  -> 
( A ... B
)  ~~  ( ( A  +  ( 1  -  A ) ) ... ( B  +  ( 1  -  A
) ) ) )
71, 2, 5, 6syl3anc 1216 . . . 4  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( A ... B )  ~~  (
( A  +  ( 1  -  A ) ) ... ( B  +  ( 1  -  A ) ) ) )
81zcnd 9181 . . . . . 6  |-  ( B  e.  ( ZZ>= `  A
)  ->  A  e.  CC )
9 ax-1cn 7720 . . . . . 6  |-  1  e.  CC
10 pncan3 7977 . . . . . 6  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A  +  ( 1  -  A ) )  =  1 )
118, 9, 10sylancl 409 . . . . 5  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( A  +  ( 1  -  A ) )  =  1 )
12 1cnd 7789 . . . . . 6  |-  ( B  e.  ( ZZ>= `  A
)  ->  1  e.  CC )
132zcnd 9181 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  A
)  ->  B  e.  CC )
1413, 8subcld 8080 . . . . . 6  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( B  -  A )  e.  CC )
1513, 12, 8addsub12d 8103 . . . . . 6  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( B  +  ( 1  -  A ) )  =  ( 1  +  ( B  -  A ) ) )
1612, 14, 15comraddd 7926 . . . . 5  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( B  +  ( 1  -  A ) )  =  ( ( B  -  A )  +  1 ) )
1711, 16oveq12d 5792 . . . 4  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( ( A  +  ( 1  -  A ) ) ... ( B  +  ( 1  -  A
) ) )  =  ( 1 ... (
( B  -  A
)  +  1 ) ) )
187, 17breqtrd 3954 . . 3  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( A ... B )  ~~  (
1 ... ( ( B  -  A )  +  1 ) ) )
191, 2fzfigd 10211 . . . 4  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( A ... B )  e.  Fin )
20 1zzd 9088 . . . . 5  |-  ( B  e.  ( ZZ>= `  A
)  ->  1  e.  ZZ )
212, 1zsubcld 9185 . . . . . 6  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( B  -  A )  e.  ZZ )
2221peano2zd 9183 . . . . 5  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( ( B  -  A )  +  1 )  e.  ZZ )
2320, 22fzfigd 10211 . . . 4  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( 1 ... ( ( B  -  A )  +  1 ) )  e. 
Fin )
24 hashen 10537 . . . 4  |-  ( ( ( A ... B
)  e.  Fin  /\  ( 1 ... (
( B  -  A
)  +  1 ) )  e.  Fin )  ->  ( ( `  ( A ... B ) )  =  ( `  (
1 ... ( ( B  -  A )  +  1 ) ) )  <-> 
( A ... B
)  ~~  ( 1 ... ( ( B  -  A )  +  1 ) ) ) )
2519, 23, 24syl2anc 408 . . 3  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( ( `  ( A ... B
) )  =  ( `  ( 1 ... (
( B  -  A
)  +  1 ) ) )  <->  ( A ... B )  ~~  (
1 ... ( ( B  -  A )  +  1 ) ) ) )
2618, 25mpbird 166 . 2  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( `  ( A ... B ) )  =  ( `  (
1 ... ( ( B  -  A )  +  1 ) ) ) )
27 uznn0sub 9364 . . 3  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( B  -  A )  e.  NN0 )
28 peano2nn0 9024 . . 3  |-  ( ( B  -  A )  e.  NN0  ->  ( ( B  -  A )  +  1 )  e. 
NN0 )
29 hashfz1 10536 . . 3  |-  ( ( ( B  -  A
)  +  1 )  e.  NN0  ->  ( `  (
1 ... ( ( B  -  A )  +  1 ) ) )  =  ( ( B  -  A )  +  1 ) )
3027, 28, 293syl 17 . 2  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( `  (
1 ... ( ( B  -  A )  +  1 ) ) )  =  ( ( B  -  A )  +  1 ) )
3126, 30eqtrd 2172 1  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( `  ( A ... B ) )  =  ( ( B  -  A )  +  1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1331    e. wcel 1480   class class class wbr 3929   ` cfv 5123  (class class class)co 5774    ~~ cen 6632   Fincfn 6634   CCcc 7625   1c1 7628    + caddc 7630    - cmin 7940   NN0cn0 8984   ZZcz 9061   ZZ>=cuz 9333   ...cfz 9797  ♯chash 10528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-addcom 7727  ax-addass 7729  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-0id 7735  ax-rnegex 7736  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-1o 6313  df-er 6429  df-en 6635  df-dom 6636  df-fin 6637  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-inn 8728  df-n0 8985  df-z 9062  df-uz 9334  df-fz 9798  df-ihash 10529
This theorem is referenced by:  hashfzo  10575  hashfzp1  10577  hashfz0  10578
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