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Theorem hashfzp1 10702
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 9445 . . . 4  |-  ( B  e.  ( ZZ>= `  A
)  ->  A  e.  ZZ )
2 eluzelz 9449 . . . 4  |-  ( B  e.  ( ZZ>= `  A
)  ->  B  e.  ZZ )
3 zdceq 9240 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  -> DECID  A  =  B )
41, 2, 3syl2anc 409 . . 3  |-  ( B  e.  ( ZZ>= `  A
)  -> DECID  A  =  B
)
5 exmiddc 822 . . 3  |-  (DECID  A  =  B  ->  ( A  =  B  \/  -.  A  =  B )
)
64, 5syl 14 . 2  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( A  =  B  \/  -.  A  =  B )
)
7 hash0 10675 . . . . 5  |-  ( `  (/) )  =  0
8 eluzelre 9450 . . . . . . . 8  |-  ( B  e.  ( ZZ>= `  A
)  ->  B  e.  RR )
98ltp1d 8802 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  A
)  ->  B  <  ( B  +  1 ) )
10 peano2z 9204 . . . . . . . . 9  |-  ( B  e.  ZZ  ->  ( B  +  1 )  e.  ZZ )
1110ancri 322 . . . . . . . 8  |-  ( B  e.  ZZ  ->  (
( B  +  1 )  e.  ZZ  /\  B  e.  ZZ )
)
12 fzn 9945 . . . . . . . 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 146 . . . . . 6  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( ( B  +  1 ) ... B )  =  (/) )
1514fveq2d 5473 . . . . 5  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( `  (
( B  +  1 ) ... B ) )  =  ( `  (/) ) )
162zcnd 9288 . . . . . 6  |-  ( B  e.  ( ZZ>= `  A
)  ->  B  e.  CC )
1716subidd 8175 . . . . 5  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( B  -  B )  =  0 )
187, 15, 173eqtr4a 2216 . . . 4  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( `  (
( B  +  1 ) ... B ) )  =  ( B  -  B ) )
19 oveq1 5832 . . . . . . 7  |-  ( A  =  B  ->  ( A  +  1 )  =  ( B  + 
1 ) )
2019oveq1d 5840 . . . . . 6  |-  ( A  =  B  ->  (
( A  +  1 ) ... B )  =  ( ( B  +  1 ) ... B ) )
2120fveq2d 5473 . . . . 5  |-  ( A  =  B  ->  ( `  ( ( A  + 
1 ) ... B
) )  =  ( `  ( ( B  + 
1 ) ... B
) ) )
22 oveq2 5833 . . . . 5  |-  ( A  =  B  ->  ( B  -  A )  =  ( B  -  B ) )
2321, 22eqeq12d 2172 . . . 4  |-  ( A  =  B  ->  (
( `  ( ( A  +  1 ) ... B ) )  =  ( B  -  A
)  <->  ( `  ( ( B  +  1 ) ... B ) )  =  ( B  -  B ) ) )
2418, 23syl5ibr 155 . . 3  |-  ( A  =  B  ->  ( B  e.  ( ZZ>= `  A )  ->  ( `  ( ( A  + 
1 ) ... B
) )  =  ( B  -  A ) ) )
25 uzp1 9473 . . . . . . . 8  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( B  =  A  \/  B  e.  ( ZZ>= `  ( A  +  1 ) ) ) )
26 pm2.24 611 . . . . . . . . . 10  |-  ( A  =  B  ->  ( -.  A  =  B  ->  B  e.  ( ZZ>= `  ( A  +  1
) ) ) )
2726eqcoms 2160 . . . . . . . . 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 706 . . . . . . . 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 124 . . . . . 6  |-  ( ( -.  A  =  B  /\  B  e.  (
ZZ>= `  A ) )  ->  B  e.  (
ZZ>= `  ( A  + 
1 ) ) )
32 hashfz 10699 . . . . . 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 9288 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  A
)  ->  A  e.  CC )
35 1cnd 7895 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  A
)  ->  1  e.  CC )
3616, 34, 35nppcan2d 8213 . . . . . 6  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( ( B  -  ( A  +  1 ) )  +  1 )  =  ( B  -  A
) )
3736adantl 275 . . . . 5  |-  ( ( -.  A  =  B  /\  B  e.  (
ZZ>= `  A ) )  ->  ( ( B  -  ( A  + 
1 ) )  +  1 )  =  ( B  -  A ) )
3833, 37eqtrd 2190 . . . 4  |-  ( ( -.  A  =  B  /\  B  e.  (
ZZ>= `  A ) )  ->  ( `  ( ( A  +  1 ) ... B ) )  =  ( B  -  A ) )
3938ex 114 . . 3  |-  ( -.  A  =  B  -> 
( B  e.  (
ZZ>= `  A )  -> 
( `  ( ( A  +  1 ) ... B ) )  =  ( B  -  A
) ) )
4024, 39jaoi 706 . 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 103    <-> wb 104    \/ wo 698  DECID wdc 820    = wceq 1335    e. wcel 2128   (/)c0 3394   class class class wbr 3966   ` cfv 5171  (class class class)co 5825   0cc0 7733   1c1 7734    + caddc 7736    < clt 7913    - cmin 8047   ZZcz 9168   ZZ>=cuz 9440   ...cfz 9913  ♯chash 10653
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4080  ax-sep 4083  ax-nul 4091  ax-pow 4136  ax-pr 4170  ax-un 4394  ax-setind 4497  ax-iinf 4548  ax-cnex 7824  ax-resscn 7825  ax-1cn 7826  ax-1re 7827  ax-icn 7828  ax-addcl 7829  ax-addrcl 7830  ax-mulcl 7831  ax-addcom 7833  ax-addass 7835  ax-distr 7837  ax-i2m1 7838  ax-0lt1 7839  ax-0id 7841  ax-rnegex 7842  ax-cnre 7844  ax-pre-ltirr 7845  ax-pre-ltwlin 7846  ax-pre-lttrn 7847  ax-pre-apti 7848  ax-pre-ltadd 7849
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-int 3809  df-iun 3852  df-br 3967  df-opab 4027  df-mpt 4028  df-tr 4064  df-id 4254  df-iord 4327  df-on 4329  df-ilim 4330  df-suc 4332  df-iom 4551  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-iota 5136  df-fun 5173  df-fn 5174  df-f 5175  df-f1 5176  df-fo 5177  df-f1o 5178  df-fv 5179  df-riota 5781  df-ov 5828  df-oprab 5829  df-mpo 5830  df-1st 6089  df-2nd 6090  df-recs 6253  df-frec 6339  df-1o 6364  df-er 6481  df-en 6687  df-dom 6688  df-fin 6689  df-pnf 7915  df-mnf 7916  df-xr 7917  df-ltxr 7918  df-le 7919  df-sub 8049  df-neg 8050  df-inn 8835  df-n0 9092  df-z 9169  df-uz 9441  df-fz 9914  df-ihash 10654
This theorem is referenced by: (None)
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