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Theorem hashfzp1 10570
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 9331 . . . 4 |- ( B e. ( ZZ>= ` A ) -> A e. ZZ )
2 eluzelz 9335 . . . 4 |- ( B e. ( ZZ>= ` A ) -> B e. ZZ )
3 zdceq 9126 . . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> DECID A = B )
41, 2, 3syl2anc 408 . . 3 |- ( B e. ( ZZ>= ` A ) -> DECID A = B )
5 exmiddc 821 . . 3 |- (DECID A = B -> ( A = B \/ -. A = B ) )
64, 5syl 14 . 2 |- ( B e. ( ZZ>= ` A ) -> ( A = B \/ -. A = B ) )
7 hash0 10543 . . . . 5 |- (♯ ` (/) ) = 0
8 eluzelre 9336 . . . . . . . 8 |- ( B e. ( ZZ>= ` A ) -> B e. RR )
98ltp1d 8688 . . . . . . 7 |- ( B e. ( ZZ>= ` A ) -> B < ( B + 1 ) )
10 peano2z 9090 . . . . . . . . 9 |- ( B e. ZZ -> ( B + 1 ) e. ZZ )
1110ancri 322 . . . . . . . 8 |- ( B e. ZZ -> ( ( B + 1 ) e. ZZ /\ B e. ZZ ) )
12 fzn 9822 . . . . . . . 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 5425 . . . . 5 |- ( B e. ( ZZ>= ` A ) -> (♯ ` ( ( B + 1 ) ... B ) ) = (♯ ` (/) ) )
162zcnd 9174 . . . . . 6 |- ( B e. ( ZZ>= ` A ) -> B e. CC )
1716subidd 8061 . . . . 5 |- ( B e. ( ZZ>= ` A ) -> ( B - B ) = 0 )
187, 15, 173eqtr4a 2198 . . . 4 |- ( B e. ( ZZ>= ` A ) -> (♯ ` ( ( B + 1 ) ... B ) ) = ( B - B ) )
19 oveq1 5781 . . . . . . 7 |- ( A = B -> ( A + 1 ) = ( B + 1 ) )
2019oveq1d 5789 . . . . . 6 |- ( A = B -> ( ( A + 1 ) ... B ) = ( ( B + 1 ) ... B ) )
2120fveq2d 5425 . . . . 5 |- ( A = B -> (♯ ` ( ( A + 1 ) ... B ) ) = (♯ ` ( ( B + 1 ) ... B ) ) )
22 oveq2 5782 . . . . 5 |- ( A = B -> ( B - A ) = ( B - B ) )
2321, 22eqeq12d 2154 . . . 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 9359 . . . . . . . 8 |- ( B e. ( ZZ>= ` A ) -> ( B = A \/ B e. ( ZZ>= ` ( A + 1 ) ) ) )
26 pm2.24 610 . . . . . . . . . 10 |- ( A = B -> ( -. A = B -> B e. ( ZZ>= ` ( A + 1 ) ) ) )
2726eqcoms 2142 . . . . . . . . 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 705 . . . . . . . 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 10567 . . . . . 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 9174 . . . . . . 7 |- ( B e. ( ZZ>= ` A ) -> A e. CC )
35 1cnd 7782 . . . . . . 7 |- ( B e. ( ZZ>= ` A ) -> 1 e. CC )
3616, 34, 35nppcan2d 8099 . . . . . 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 2172 . . . 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 705 . 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 697  DECID wdc 819   = wceq 1331   e. wcel 1480   (/)c0 3363   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   0cc0 7620   1c1 7621   + caddc 7623   < clt 7800   - cmin 7933   ZZcz 9054   ZZ>=cuz 9326   ...cfz 9790  ♯chash 10521
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-addcom 7720  ax-addass 7722  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-1o 6313  df-er 6429  df-en 6635  df-dom 6636  df-fin 6637  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-inn 8721  df-n0 8978  df-z 9055  df-uz 9327  df-fz 9791  df-ihash 10522
This theorem is referenced by: (None)
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