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Theorem hashfzp1 10759
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 9492 . . . 4 |- ( B e. ( ZZ>= ` A ) -> A e. ZZ )
2 eluzelz 9496 . . . 4 |- ( B e. ( ZZ>= ` A ) -> B e. ZZ )
3 zdceq 9287 . . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> DECID A = B )
41, 2, 3syl2anc 409 . . 3 |- ( B e. ( ZZ>= ` A ) -> DECID A = B )
5 exmiddc 831 . . 3 |- (DECID A = B -> ( A = B \/ -. A = B ) )
64, 5syl 14 . 2 |- ( B e. ( ZZ>= ` A ) -> ( A = B \/ -. A = B ) )
7 hash0 10731 . . . . 5 |- (♯ ` (/) ) = 0
8 eluzelre 9497 . . . . . . . 8 |- ( B e. ( ZZ>= ` A ) -> B e. RR )
98ltp1d 8846 . . . . . . 7 |- ( B e. ( ZZ>= ` A ) -> B < ( B + 1 ) )
10 peano2z 9248 . . . . . . . . 9 |- ( B e. ZZ -> ( B + 1 ) e. ZZ )
1110ancri 322 . . . . . . . 8 |- ( B e. ZZ -> ( ( B + 1 ) e. ZZ /\ B e. ZZ ) )
12 fzn 9998 . . . . . . . 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 5500 . . . . 5 |- ( B e. ( ZZ>= ` A ) -> (♯ ` ( ( B + 1 ) ... B ) ) = (♯ ` (/) ) )
162zcnd 9335 . . . . . 6 |- ( B e. ( ZZ>= ` A ) -> B e. CC )
1716subidd 8218 . . . . 5 |- ( B e. ( ZZ>= ` A ) -> ( B - B ) = 0 )
187, 15, 173eqtr4a 2229 . . . 4 |- ( B e. ( ZZ>= ` A ) -> (♯ ` ( ( B + 1 ) ... B ) ) = ( B - B ) )
19 oveq1 5860 . . . . . . 7 |- ( A = B -> ( A + 1 ) = ( B + 1 ) )
2019oveq1d 5868 . . . . . 6 |- ( A = B -> ( ( A + 1 ) ... B ) = ( ( B + 1 ) ... B ) )
2120fveq2d 5500 . . . . 5 |- ( A = B -> (♯ ` ( ( A + 1 ) ... B ) ) = (♯ ` ( ( B + 1 ) ... B ) ) )
22 oveq2 5861 . . . . 5 |- ( A = B -> ( B - A ) = ( B - B ) )
2321, 22eqeq12d 2185 . . . 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 9520 . . . . . . . 8 |- ( B e. ( ZZ>= ` A ) -> ( B = A \/ B e. ( ZZ>= ` ( A + 1 ) ) ) )
26 pm2.24 616 . . . . . . . . . 10 |- ( A = B -> ( -. A = B -> B e. ( ZZ>= ` ( A + 1 ) ) ) )
2726eqcoms 2173 . . . . . . . . 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 711 . . . . . . . 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 10756 . . . . . 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 9335 . . . . . . 7 |- ( B e. ( ZZ>= ` A ) -> A e. CC )
35 1cnd 7936 . . . . . . 7 |- ( B e. ( ZZ>= ` A ) -> 1 e. CC )
3616, 34, 35nppcan2d 8256 . . . . . 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 2203 . . . 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 711 . 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 703  DECID wdc 829   = wceq 1348   e. wcel 2141   (/)c0 3414   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   0cc0 7774   1c1 7775   + caddc 7777   < clt 7954   - cmin 8090   ZZcz 9212   ZZ>=cuz 9487   ...cfz 9965  ♯chash 10709
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-1o 6395  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-fz 9966  df-ihash 10710
This theorem is referenced by: (None)
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