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Theorem hashfzp1 10821
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 9550 . . . 4 |- ( B e. ( ZZ>= ` A ) -> A e. ZZ )
2 eluzelz 9554 . . . 4 |- ( B e. ( ZZ>= ` A ) -> B e. ZZ )
3 zdceq 9345 . . . 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 10793 . . . . 5 |- (♯ ` (/) ) = 0
8 eluzelre 9555 . . . . . . . 8 |- ( B e. ( ZZ>= ` A ) -> B e. RR )
98ltp1d 8904 . . . . . . 7 |- ( B e. ( ZZ>= ` A ) -> B < ( B + 1 ) )
10 peano2z 9306 . . . . . . . . 9 |- ( B e. ZZ -> ( B + 1 ) e. ZZ )
1110ancri 324 . . . . . . . 8 |- ( B e. ZZ -> ( ( B + 1 ) e. ZZ /\ B e. ZZ ) )
12 fzn 10059 . . . . . . . 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 5533 . . . . 5 |- ( B e. ( ZZ>= ` A ) -> (♯ ` ( ( B + 1 ) ... B ) ) = (♯ ` (/) ) )
162zcnd 9393 . . . . . 6 |- ( B e. ( ZZ>= ` A ) -> B e. CC )
1716subidd 8273 . . . . 5 |- ( B e. ( ZZ>= ` A ) -> ( B - B ) = 0 )
187, 15, 173eqtr4a 2247 . . . 4 |- ( B e. ( ZZ>= ` A ) -> (♯ ` ( ( B + 1 ) ... B ) ) = ( B - B ) )
19 oveq1 5897 . . . . . . 7 |- ( A = B -> ( A + 1 ) = ( B + 1 ) )
2019oveq1d 5905 . . . . . 6 |- ( A = B -> ( ( A + 1 ) ... B ) = ( ( B + 1 ) ... B ) )
2120fveq2d 5533 . . . . 5 |- ( A = B -> (♯ ` ( ( A + 1 ) ... B ) ) = (♯ ` ( ( B + 1 ) ... B ) ) )
22 oveq2 5898 . . . . 5 |- ( A = B -> ( B - A ) = ( B - B ) )
2321, 22eqeq12d 2203 . . . 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 9578 . . . . . . . 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 2191 . . . . . . . . 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 10818 . . . . . 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 9393 . . . . . . 7 |- ( B e. ( ZZ>= ` A ) -> A e. CC )
35 1cnd 7990 . . . . . . 7 |- ( B e. ( ZZ>= ` A ) -> 1 e. CC )
3616, 34, 35nppcan2d 8311 . . . . . 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 2221 . . . 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 2159   (/)c0 3436   class class class wbr 4017   ` cfv 5230  (class class class)co 5890   0cc0 7828   1c1 7829   + caddc 7831   < clt 8009   - cmin 8145   ZZcz 9270   ZZ>=cuz 9545   ...cfz 10025  ♯chash 10772
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 2161  ax-14 2162  ax-ext 2170  ax-coll 4132  ax-sep 4135  ax-nul 4143  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550  ax-iinf 4601  ax-cnex 7919  ax-resscn 7920  ax-1cn 7921  ax-1re 7922  ax-icn 7923  ax-addcl 7924  ax-addrcl 7925  ax-mulcl 7926  ax-addcom 7928  ax-addass 7930  ax-distr 7932  ax-i2m1 7933  ax-0lt1 7934  ax-0id 7936  ax-rnegex 7937  ax-cnre 7939  ax-pre-ltirr 7940  ax-pre-ltwlin 7941  ax-pre-lttrn 7942  ax-pre-apti 7943  ax-pre-ltadd 7944
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 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-nel 2455  df-ral 2472  df-rex 2473  df-reu 2474  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-nul 3437  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-iun 3902  df-br 4018  df-opab 4079  df-mpt 4080  df-tr 4116  df-id 4307  df-iord 4380  df-on 4382  df-ilim 4383  df-suc 4385  df-iom 4604  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-riota 5846  df-ov 5893  df-oprab 5894  df-mpo 5895  df-1st 6158  df-2nd 6159  df-recs 6323  df-frec 6409  df-1o 6434  df-er 6552  df-en 6758  df-dom 6759  df-fin 6760  df-pnf 8011  df-mnf 8012  df-xr 8013  df-ltxr 8014  df-le 8015  df-sub 8147  df-neg 8148  df-inn 8937  df-n0 9194  df-z 9271  df-uz 9546  df-fz 10026  df-ihash 10773
This theorem is referenced by: (None)
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