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Theorem hashfzp1 11094
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 9765 . . . 4 |- ( B e. ( ZZ>= ` A ) -> A e. ZZ )
2 eluzelz 9770 . . . 4 |- ( B e. ( ZZ>= ` A ) -> B e. ZZ )
3 zdceq 9560 . . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> DECID A = B )
41, 2, 3syl2anc 411 . . 3 |- ( B e. ( ZZ>= ` A ) -> DECID A = B )
5 exmiddc 843 . . 3 |- (DECID A = B -> ( A = B \/ -. A = B ) )
64, 5syl 14 . 2 |- ( B e. ( ZZ>= ` A ) -> ( A = B \/ -. A = B ) )
7 hash0 11064 . . . . 5 |- (♯ ` (/) ) = 0
8 eluzelre 9771 . . . . . . . 8 |- ( B e. ( ZZ>= ` A ) -> B e. RR )
98ltp1d 9115 . . . . . . 7 |- ( B e. ( ZZ>= ` A ) -> B < ( B + 1 ) )
10 peano2z 9520 . . . . . . . . 9 |- ( B e. ZZ -> ( B + 1 ) e. ZZ )
1110ancri 324 . . . . . . . 8 |- ( B e. ZZ -> ( ( B + 1 ) e. ZZ /\ B e. ZZ ) )
12 fzn 10282 . . . . . . . 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 5646 . . . . 5 |- ( B e. ( ZZ>= ` A ) -> (♯ ` ( ( B + 1 ) ... B ) ) = (♯ ` (/) ) )
162zcnd 9608 . . . . . 6 |- ( B e. ( ZZ>= ` A ) -> B e. CC )
1716subidd 8483 . . . . 5 |- ( B e. ( ZZ>= ` A ) -> ( B - B ) = 0 )
187, 15, 173eqtr4a 2289 . . . 4 |- ( B e. ( ZZ>= ` A ) -> (♯ ` ( ( B + 1 ) ... B ) ) = ( B - B ) )
19 oveq1 6030 . . . . . . 7 |- ( A = B -> ( A + 1 ) = ( B + 1 ) )
2019oveq1d 6038 . . . . . 6 |- ( A = B -> ( ( A + 1 ) ... B ) = ( ( B + 1 ) ... B ) )
2120fveq2d 5646 . . . . 5 |- ( A = B -> (♯ ` ( ( A + 1 ) ... B ) ) = (♯ ` ( ( B + 1 ) ... B ) ) )
22 oveq2 6031 . . . . 5 |- ( A = B -> ( B - A ) = ( B - B ) )
2321, 22eqeq12d 2245 . . . 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 9795 . . . . . . . 8 |- ( B e. ( ZZ>= ` A ) -> ( B = A \/ B e. ( ZZ>= ` ( A + 1 ) ) ) )
26 pm2.24 626 . . . . . . . . . 10 |- ( A = B -> ( -. A = B -> B e. ( ZZ>= ` ( A + 1 ) ) ) )
2726eqcoms 2233 . . . . . . . . 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 723 . . . . . . . 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 11091 . . . . . 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 9608 . . . . . . 7 |- ( B e. ( ZZ>= ` A ) -> A e. CC )
35 1cnd 8200 . . . . . . 7 |- ( B e. ( ZZ>= ` A ) -> 1 e. CC )
3616, 34, 35nppcan2d 8521 . . . . . 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 2263 . . . 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 723 . 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 715  DECID wdc 841   = wceq 1397   e. wcel 2201   (/)c0 3493   class class class wbr 4089   ` cfv 5328  (class class class)co 6023   0cc0 8037   1c1 8038   + caddc 8040   < clt 8219   - cmin 8355   ZZcz 9484   ZZ>=cuz 9760   ...cfz 10248  ♯chash 11043
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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-iinf 4688  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-addcom 8137  ax-addass 8139  ax-distr 8141  ax-i2m1 8142  ax-0lt1 8143  ax-0id 8145  ax-rnegex 8146  ax-cnre 8148  ax-pre-ltirr 8149  ax-pre-ltwlin 8150  ax-pre-lttrn 8151  ax-pre-apti 8152  ax-pre-ltadd 8153
This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-tr 4189  df-id 4392  df-iord 4465  df-on 4467  df-ilim 4468  df-suc 4470  df-iom 4691  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-recs 6476  df-frec 6562  df-1o 6587  df-er 6707  df-en 6915  df-dom 6916  df-fin 6917  df-pnf 8221  df-mnf 8222  df-xr 8223  df-ltxr 8224  df-le 8225  df-sub 8357  df-neg 8358  df-inn 9149  df-n0 9408  df-z 9485  df-uz 9761  df-fz 10249  df-ihash 11044
This theorem is referenced by:  2lgslem1  15849
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