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Theorem hashfzp1 10895
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 9597 . . . 4 |- ( B e. ( ZZ>= ` A ) -> A e. ZZ )
2 eluzelz 9601 . . . 4 |- ( B e. ( ZZ>= ` A ) -> B e. ZZ )
3 zdceq 9392 . . . 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 10867 . . . . 5 |- (♯ ` (/) ) = 0
8 eluzelre 9602 . . . . . . . 8 |- ( B e. ( ZZ>= ` A ) -> B e. RR )
98ltp1d 8949 . . . . . . 7 |- ( B e. ( ZZ>= ` A ) -> B < ( B + 1 ) )
10 peano2z 9353 . . . . . . . . 9 |- ( B e. ZZ -> ( B + 1 ) e. ZZ )
1110ancri 324 . . . . . . . 8 |- ( B e. ZZ -> ( ( B + 1 ) e. ZZ /\ B e. ZZ ) )
12 fzn 10108 . . . . . . . 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 5558 . . . . 5 |- ( B e. ( ZZ>= ` A ) -> (♯ ` ( ( B + 1 ) ... B ) ) = (♯ ` (/) ) )
162zcnd 9440 . . . . . 6 |- ( B e. ( ZZ>= ` A ) -> B e. CC )
1716subidd 8318 . . . . 5 |- ( B e. ( ZZ>= ` A ) -> ( B - B ) = 0 )
187, 15, 173eqtr4a 2252 . . . 4 |- ( B e. ( ZZ>= ` A ) -> (♯ ` ( ( B + 1 ) ... B ) ) = ( B - B ) )
19 oveq1 5925 . . . . . . 7 |- ( A = B -> ( A + 1 ) = ( B + 1 ) )
2019oveq1d 5933 . . . . . 6 |- ( A = B -> ( ( A + 1 ) ... B ) = ( ( B + 1 ) ... B ) )
2120fveq2d 5558 . . . . 5 |- ( A = B -> (♯ ` ( ( A + 1 ) ... B ) ) = (♯ ` ( ( B + 1 ) ... B ) ) )
22 oveq2 5926 . . . . 5 |- ( A = B -> ( B - A ) = ( B - B ) )
2321, 22eqeq12d 2208 . . . 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 9626 . . . . . . . 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 2196 . . . . . . . . 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 10892 . . . . . 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 9440 . . . . . . 7 |- ( B e. ( ZZ>= ` A ) -> A e. CC )
35 1cnd 8035 . . . . . . 7 |- ( B e. ( ZZ>= ` A ) -> 1 e. CC )
3616, 34, 35nppcan2d 8356 . . . . . 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 2226 . . . 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 2164   (/)c0 3446   class class class wbr 4029   ` cfv 5254  (class class class)co 5918   0cc0 7872   1c1 7873   + caddc 7875   < clt 8054   - cmin 8190   ZZcz 9317   ZZ>=cuz 9592   ...cfz 10074  ♯chash 10846
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 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988
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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-1o 6469  df-er 6587  df-en 6795  df-dom 6796  df-fin 6797  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-inn 8983  df-n0 9241  df-z 9318  df-uz 9593  df-fz 10075  df-ihash 10847
This theorem is referenced by: (None)
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