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Theorem hashfz 11381
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 10235 . . . . 5  |-  ( B  e.  ( ZZ>= `  A
)  ->  A  e.  ZZ )
2 eluzelz 10238 . . . . 5  |-  ( B  e.  ( ZZ>= `  A
)  ->  B  e.  ZZ )
3 1z 10053 . . . . . 6  |-  1  e.  ZZ
4 zsubcl 10061 . . . . . 6  |-  ( ( 1  e.  ZZ  /\  A  e.  ZZ )  ->  ( 1  -  A
)  e.  ZZ )
53, 1, 4sylancr 644 . . . . 5  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( 1  -  A )  e.  ZZ )
6 fzen 10811 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  (
1  -  A )  e.  ZZ )  -> 
( A ... B
)  ~~  ( ( A  +  ( 1  -  A ) ) ... ( B  +  ( 1  -  A
) ) ) )
71, 2, 5, 6syl3anc 1182 . . . 4  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( A ... B )  ~~  (
( A  +  ( 1  -  A ) ) ... ( B  +  ( 1  -  A ) ) ) )
81zcnd 10118 . . . . . 6  |-  ( B  e.  ( ZZ>= `  A
)  ->  A  e.  CC )
9 ax-1cn 8795 . . . . . 6  |-  1  e.  CC
10 pncan3 9059 . . . . . 6  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A  +  ( 1  -  A ) )  =  1 )
118, 9, 10sylancl 643 . . . . 5  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( A  +  ( 1  -  A ) )  =  1 )
122zcnd 10118 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  A
)  ->  B  e.  CC )
139a1i 10 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  A
)  ->  1  e.  CC )
1412, 13, 8addsub12d 9180 . . . . . 6  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( B  +  ( 1  -  A ) )  =  ( 1  +  ( B  -  A ) ) )
1512, 8subcld 9157 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( B  -  A )  e.  CC )
16 addcom 8998 . . . . . . 7  |-  ( ( 1  e.  CC  /\  ( B  -  A
)  e.  CC )  ->  ( 1  +  ( B  -  A
) )  =  ( ( B  -  A
)  +  1 ) )
179, 15, 16sylancr 644 . . . . . 6  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( 1  +  ( B  -  A ) )  =  ( ( B  -  A )  +  1 ) )
1814, 17eqtrd 2315 . . . . 5  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( B  +  ( 1  -  A ) )  =  ( ( B  -  A )  +  1 ) )
1911, 18oveq12d 5876 . . . 4  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( ( A  +  ( 1  -  A ) ) ... ( B  +  ( 1  -  A
) ) )  =  ( 1 ... (
( B  -  A
)  +  1 ) ) )
207, 19breqtrd 4047 . . 3  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( A ... B )  ~~  (
1 ... ( ( B  -  A )  +  1 ) ) )
21 hasheni 11347 . . 3  |-  ( ( A ... B ) 
~~  ( 1 ... ( ( B  -  A )  +  1 ) )  ->  ( # `
 ( A ... B ) )  =  ( # `  (
1 ... ( ( B  -  A )  +  1 ) ) ) )
2220, 21syl 15 . 2  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( # `  ( A ... B ) )  =  ( # `  (
1 ... ( ( B  -  A )  +  1 ) ) ) )
23 uznn0sub 10259 . . 3  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( B  -  A )  e.  NN0 )
24 peano2nn0 10004 . . 3  |-  ( ( B  -  A )  e.  NN0  ->  ( ( B  -  A )  +  1 )  e. 
NN0 )
25 hashfz1 11345 . . 3  |-  ( ( ( B  -  A
)  +  1 )  e.  NN0  ->  ( # `  ( 1 ... (
( B  -  A
)  +  1 ) ) )  =  ( ( B  -  A
)  +  1 ) )
2623, 24, 253syl 18 . 2  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( # `  (
1 ... ( ( B  -  A )  +  1 ) ) )  =  ( ( B  -  A )  +  1 ) )
2722, 26eqtrd 2315 1  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( # `  ( A ... B ) )  =  ( ( B  -  A )  +  1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858    ~~ cen 6860   CCcc 8735   1c1 8738    + caddc 8740    - cmin 9037   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782   #chash 11337
This theorem is referenced by:  fzsdom2  11382  hashfzo  11383  0sgmppw  20437  logfaclbnd  20461  ballotlem2  23047  subfacp1lem5  23715  stoweidlem11  27760  stoweidlem26  27775
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-hash 11338
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