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Theorem hashfzp1 10823
Description: Value of the numeric cardinality of a (possibly empty) integer range. (Contributed by AV, 19-Jun-2021.)
Assertion
Ref Expression
hashfzp1  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( `  (
( A  +  1 ) ... B ) )  =  ( B  -  A ) )

Proof of Theorem hashfzp1
StepHypRef Expression
1 eluzel2 9552 . . . 4  |-  ( B  e.  ( ZZ>= `  A
)  ->  A  e.  ZZ )
2 eluzelz 9556 . . . 4  |-  ( B  e.  ( ZZ>= `  A
)  ->  B  e.  ZZ )
3 zdceq 9347 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  -> DECID  A  =  B )
41, 2, 3syl2anc 411 . . 3  |-  ( B  e.  ( ZZ>= `  A
)  -> DECID  A  =  B
)
5 exmiddc 837 . . 3  |-  (DECID  A  =  B  ->  ( A  =  B  \/  -.  A  =  B )
)
64, 5syl 14 . 2  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( A  =  B  \/  -.  A  =  B )
)
7 hash0 10795 . . . . 5  |-  ( `  (/) )  =  0
8 eluzelre 9557 . . . . . . . 8  |-  ( B  e.  ( ZZ>= `  A
)  ->  B  e.  RR )
98ltp1d 8906 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  A
)  ->  B  <  ( B  +  1 ) )
10 peano2z 9308 . . . . . . . . 9  |-  ( B  e.  ZZ  ->  ( B  +  1 )  e.  ZZ )
1110ancri 324 . . . . . . . 8  |-  ( B  e.  ZZ  ->  (
( B  +  1 )  e.  ZZ  /\  B  e.  ZZ )
)
12 fzn 10061 . . . . . . . 8  |-  ( ( ( B  +  1 )  e.  ZZ  /\  B  e.  ZZ )  ->  ( B  <  ( B  +  1 )  <-> 
( ( B  + 
1 ) ... B
)  =  (/) ) )
132, 11, 123syl 17 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( B  <  ( B  +  1 )  <->  ( ( B  +  1 ) ... B )  =  (/) ) )
149, 13mpbid 147 . . . . . 6  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( ( B  +  1 ) ... B )  =  (/) )
1514fveq2d 5534 . . . . 5  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( `  (
( B  +  1 ) ... B ) )  =  ( `  (/) ) )
162zcnd 9395 . . . . . 6  |-  ( B  e.  ( ZZ>= `  A
)  ->  B  e.  CC )
1716subidd 8275 . . . . 5  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( B  -  B )  =  0 )
187, 15, 173eqtr4a 2248 . . . 4  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( `  (
( B  +  1 ) ... B ) )  =  ( B  -  B ) )
19 oveq1 5898 . . . . . . 7  |-  ( A  =  B  ->  ( A  +  1 )  =  ( B  + 
1 ) )
2019oveq1d 5906 . . . . . 6  |-  ( A  =  B  ->  (
( A  +  1 ) ... B )  =  ( ( B  +  1 ) ... B ) )
2120fveq2d 5534 . . . . 5  |-  ( A  =  B  ->  ( `  ( ( A  + 
1 ) ... B
) )  =  ( `  ( ( B  + 
1 ) ... B
) ) )
22 oveq2 5899 . . . . 5  |-  ( A  =  B  ->  ( B  -  A )  =  ( B  -  B ) )
2321, 22eqeq12d 2204 . . . 4  |-  ( A  =  B  ->  (
( `  ( ( A  +  1 ) ... B ) )  =  ( B  -  A
)  <->  ( `  ( ( B  +  1 ) ... B ) )  =  ( B  -  B ) ) )
2418, 23imbitrrid 156 . . 3  |-  ( A  =  B  ->  ( B  e.  ( ZZ>= `  A )  ->  ( `  ( ( A  + 
1 ) ... B
) )  =  ( B  -  A ) ) )
25 uzp1 9580 . . . . . . . 8  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( B  =  A  \/  B  e.  ( ZZ>= `  ( A  +  1 ) ) ) )
26 pm2.24 622 . . . . . . . . . 10  |-  ( A  =  B  ->  ( -.  A  =  B  ->  B  e.  ( ZZ>= `  ( A  +  1
) ) ) )
2726eqcoms 2192 . . . . . . . . 9  |-  ( B  =  A  ->  ( -.  A  =  B  ->  B  e.  ( ZZ>= `  ( A  +  1
) ) ) )
28 ax-1 6 . . . . . . . . 9  |-  ( B  e.  ( ZZ>= `  ( A  +  1 ) )  ->  ( -.  A  =  B  ->  B  e.  ( ZZ>= `  ( A  +  1 ) ) ) )
2927, 28jaoi 717 . . . . . . . 8  |-  ( ( B  =  A  \/  B  e.  ( ZZ>= `  ( A  +  1
) ) )  -> 
( -.  A  =  B  ->  B  e.  ( ZZ>= `  ( A  +  1 ) ) ) )
3025, 29syl 14 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( -.  A  =  B  ->  B  e.  ( ZZ>= `  ( A  +  1 ) ) ) )
3130impcom 125 . . . . . 6  |-  ( ( -.  A  =  B  /\  B  e.  (
ZZ>= `  A ) )  ->  B  e.  (
ZZ>= `  ( A  + 
1 ) ) )
32 hashfz 10820 . . . . . 6  |-  ( B  e.  ( ZZ>= `  ( A  +  1 ) )  ->  ( `  (
( A  +  1 ) ... B ) )  =  ( ( B  -  ( A  +  1 ) )  +  1 ) )
3331, 32syl 14 . . . . 5  |-  ( ( -.  A  =  B  /\  B  e.  (
ZZ>= `  A ) )  ->  ( `  ( ( A  +  1 ) ... B ) )  =  ( ( B  -  ( A  + 
1 ) )  +  1 ) )
341zcnd 9395 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  A
)  ->  A  e.  CC )
35 1cnd 7992 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  A
)  ->  1  e.  CC )
3616, 34, 35nppcan2d 8313 . . . . . 6  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( ( B  -  ( A  +  1 ) )  +  1 )  =  ( B  -  A
) )
3736adantl 277 . . . . 5  |-  ( ( -.  A  =  B  /\  B  e.  (
ZZ>= `  A ) )  ->  ( ( B  -  ( A  + 
1 ) )  +  1 )  =  ( B  -  A ) )
3833, 37eqtrd 2222 . . . 4  |-  ( ( -.  A  =  B  /\  B  e.  (
ZZ>= `  A ) )  ->  ( `  ( ( A  +  1 ) ... B ) )  =  ( B  -  A ) )
3938ex 115 . . 3  |-  ( -.  A  =  B  -> 
( B  e.  (
ZZ>= `  A )  -> 
( `  ( ( A  +  1 ) ... B ) )  =  ( B  -  A
) ) )
4024, 39jaoi 717 . 2  |-  ( ( A  =  B  \/  -.  A  =  B
)  ->  ( B  e.  ( ZZ>= `  A )  ->  ( `  ( ( A  +  1 ) ... B ) )  =  ( B  -  A ) ) )
416, 40mpcom 36 1  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( `  (
( A  +  1 ) ... B ) )  =  ( B  -  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709  DECID wdc 835    = wceq 1364    e. wcel 2160   (/)c0 3437   class class class wbr 4018   ` cfv 5231  (class class class)co 5891   0cc0 7830   1c1 7831    + caddc 7833    < clt 8011    - cmin 8147   ZZcz 9272   ZZ>=cuz 9547   ...cfz 10027  ♯chash 10774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7921  ax-resscn 7922  ax-1cn 7923  ax-1re 7924  ax-icn 7925  ax-addcl 7926  ax-addrcl 7927  ax-mulcl 7928  ax-addcom 7930  ax-addass 7932  ax-distr 7934  ax-i2m1 7935  ax-0lt1 7936  ax-0id 7938  ax-rnegex 7939  ax-cnre 7941  ax-pre-ltirr 7942  ax-pre-ltwlin 7943  ax-pre-lttrn 7944  ax-pre-apti 7945  ax-pre-ltadd 7946
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-frec 6410  df-1o 6435  df-er 6553  df-en 6759  df-dom 6760  df-fin 6761  df-pnf 8013  df-mnf 8014  df-xr 8015  df-ltxr 8016  df-le 8017  df-sub 8149  df-neg 8150  df-inn 8939  df-n0 9196  df-z 9273  df-uz 9548  df-fz 10028  df-ihash 10775
This theorem is referenced by: (None)
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