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Theorem hashfz 12437
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 11076 . . . . 5  |-  ( B  e.  ( ZZ>= `  A
)  ->  A  e.  ZZ )
2 eluzelz 11080 . . . . 5  |-  ( B  e.  ( ZZ>= `  A
)  ->  B  e.  ZZ )
3 1z 10883 . . . . . 6  |-  1  e.  ZZ
4 zsubcl 10894 . . . . . 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 11692 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  (
1  -  A )  e.  ZZ )  -> 
( A ... B
)  ~~  ( ( A  +  ( 1  -  A ) ) ... ( B  +  ( 1  -  A
) ) ) )
71, 2, 5, 6syl3anc 1223 . . . 4  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( A ... B )  ~~  (
( A  +  ( 1  -  A ) ) ... ( B  +  ( 1  -  A ) ) ) )
81zcnd 10956 . . . . . 6  |-  ( B  e.  ( ZZ>= `  A
)  ->  A  e.  CC )
9 ax-1cn 9539 . . . . . 6  |-  1  e.  CC
10 pncan3 9817 . . . . . 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 10956 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  A
)  ->  B  e.  CC )
139a1i 11 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  A
)  ->  1  e.  CC )
1412, 13, 8addsub12d 9942 . . . . . 6  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( B  +  ( 1  -  A ) )  =  ( 1  +  ( B  -  A ) ) )
1512, 8subcld 9919 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( B  -  A )  e.  CC )
16 addcom 9754 . . . . . . 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 2501 . . . . 5  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( B  +  ( 1  -  A ) )  =  ( ( B  -  A )  +  1 ) )
1911, 18oveq12d 6293 . . . 4  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( ( A  +  ( 1  -  A ) ) ... ( B  +  ( 1  -  A
) ) )  =  ( 1 ... (
( B  -  A
)  +  1 ) ) )
207, 19breqtrd 4464 . . 3  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( A ... B )  ~~  (
1 ... ( ( B  -  A )  +  1 ) ) )
21 hasheni 12376 . . 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 11102 . . 3  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( B  -  A )  e.  NN0 )
24 peano2nn0 10825 . . 3  |-  ( ( B  -  A )  e.  NN0  ->  ( ( B  -  A )  +  1 )  e. 
NN0 )
25 hashfz1 12374 . . 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 2501 1  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( # `  ( A ... B ) )  =  ( ( B  -  A )  +  1 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   class class class wbr 4440   ` cfv 5579  (class class class)co 6275    ~~ cen 7503   CCcc 9479   1c1 9482    + caddc 9484    - cmin 9794   NN0cn0 10784   ZZcz 10853   ZZ>=cuz 11071   ...cfz 11661   #chash 12360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-hash 12361
This theorem is referenced by:  fzsdom2  12438  hashfzo  12439  hashfz0  12442  hashfzdm  12451  fz0hash  12452  0sgmppw  23194  logfaclbnd  23218  ballotlem2  27917  subfacp1lem5  28118  fzisoeu  30896  stoweidlem11  31130  stoweidlem26  31145
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