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Theorem hashfzp1 10817
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 9546 . . . 4  |-  ( B  e.  ( ZZ>= `  A
)  ->  A  e.  ZZ )
2 eluzelz 9550 . . . 4  |-  ( B  e.  ( ZZ>= `  A
)  ->  B  e.  ZZ )
3 zdceq 9341 . . . 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 10789 . . . . 5  |-  ( `  (/) )  =  0
8 eluzelre 9551 . . . . . . . 8  |-  ( B  e.  ( ZZ>= `  A
)  ->  B  e.  RR )
98ltp1d 8900 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  A
)  ->  B  <  ( B  +  1 ) )
10 peano2z 9302 . . . . . . . . 9  |-  ( B  e.  ZZ  ->  ( B  +  1 )  e.  ZZ )
1110ancri 324 . . . . . . . 8  |-  ( B  e.  ZZ  ->  (
( B  +  1 )  e.  ZZ  /\  B  e.  ZZ )
)
12 fzn 10055 . . . . . . . 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 5531 . . . . 5  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( `  (
( B  +  1 ) ... B ) )  =  ( `  (/) ) )
162zcnd 9389 . . . . . 6  |-  ( B  e.  ( ZZ>= `  A
)  ->  B  e.  CC )
1716subidd 8269 . . . . 5  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( B  -  B )  =  0 )
187, 15, 173eqtr4a 2246 . . . 4  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( `  (
( B  +  1 ) ... B ) )  =  ( B  -  B ) )
19 oveq1 5895 . . . . . . 7  |-  ( A  =  B  ->  ( A  +  1 )  =  ( B  + 
1 ) )
2019oveq1d 5903 . . . . . 6  |-  ( A  =  B  ->  (
( A  +  1 ) ... B )  =  ( ( B  +  1 ) ... B ) )
2120fveq2d 5531 . . . . 5  |-  ( A  =  B  ->  ( `  ( ( A  + 
1 ) ... B
) )  =  ( `  ( ( B  + 
1 ) ... B
) ) )
22 oveq2 5896 . . . . 5  |-  ( A  =  B  ->  ( B  -  A )  =  ( B  -  B ) )
2321, 22eqeq12d 2202 . . . 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 9574 . . . . . . . 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 2190 . . . . . . . . 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 10814 . . . . . 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 9389 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  A
)  ->  A  e.  CC )
35 1cnd 7986 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  A
)  ->  1  e.  CC )
3616, 34, 35nppcan2d 8307 . . . . . 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 2220 . . . 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 1363    e. wcel 2158   (/)c0 3434   class class class wbr 4015   ` cfv 5228  (class class class)co 5888   0cc0 7824   1c1 7825    + caddc 7827    < clt 8005    - cmin 8141   ZZcz 9266   ZZ>=cuz 9541   ...cfz 10021  ♯chash 10768
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-addcom 7924  ax-addass 7926  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-0id 7932  ax-rnegex 7933  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-frec 6405  df-1o 6430  df-er 6548  df-en 6754  df-dom 6755  df-fin 6756  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-inn 8933  df-n0 9190  df-z 9267  df-uz 9542  df-fz 10022  df-ihash 10769
This theorem is referenced by: (None)
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