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Mirrors > Home > ILE Home > Th. List > hashfzo | GIF version |
Description: Cardinality of a half-open set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
Ref | Expression |
---|---|
hashfzo | ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (♯‘(𝐴..^𝐵)) = (𝐵 − 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzo0 10151 | . . . . . 6 ⊢ (𝐴..^𝐴) = ∅ | |
2 | 1 | fveq2i 5514 | . . . . 5 ⊢ (♯‘(𝐴..^𝐴)) = (♯‘∅) |
3 | hash0 10757 | . . . . 5 ⊢ (♯‘∅) = 0 | |
4 | 2, 3 | eqtri 2198 | . . . 4 ⊢ (♯‘(𝐴..^𝐴)) = 0 |
5 | eluzel2 9519 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐴 ∈ ℤ) | |
6 | 5 | zcnd 9362 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐴 ∈ ℂ) |
7 | 6 | subidd 8243 | . . . 4 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐴 − 𝐴) = 0) |
8 | 4, 7 | eqtr4id 2229 | . . 3 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (♯‘(𝐴..^𝐴)) = (𝐴 − 𝐴)) |
9 | oveq2 5877 | . . . . 5 ⊢ (𝐵 = 𝐴 → (𝐴..^𝐵) = (𝐴..^𝐴)) | |
10 | 9 | fveq2d 5515 | . . . 4 ⊢ (𝐵 = 𝐴 → (♯‘(𝐴..^𝐵)) = (♯‘(𝐴..^𝐴))) |
11 | oveq1 5876 | . . . 4 ⊢ (𝐵 = 𝐴 → (𝐵 − 𝐴) = (𝐴 − 𝐴)) | |
12 | 10, 11 | eqeq12d 2192 | . . 3 ⊢ (𝐵 = 𝐴 → ((♯‘(𝐴..^𝐵)) = (𝐵 − 𝐴) ↔ (♯‘(𝐴..^𝐴)) = (𝐴 − 𝐴))) |
13 | 8, 12 | syl5ibrcom 157 | . 2 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐵 = 𝐴 → (♯‘(𝐴..^𝐵)) = (𝐵 − 𝐴))) |
14 | eluzelz 9523 | . . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐵 ∈ ℤ) | |
15 | fzoval 10131 | . . . . . . 7 ⊢ (𝐵 ∈ ℤ → (𝐴..^𝐵) = (𝐴...(𝐵 − 1))) | |
16 | 14, 15 | syl 14 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐴..^𝐵) = (𝐴...(𝐵 − 1))) |
17 | 16 | fveq2d 5515 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (♯‘(𝐴..^𝐵)) = (♯‘(𝐴...(𝐵 − 1)))) |
18 | 17 | adantr 276 | . . . 4 ⊢ ((𝐵 ∈ (ℤ≥‘𝐴) ∧ (𝐵 − 1) ∈ (ℤ≥‘𝐴)) → (♯‘(𝐴..^𝐵)) = (♯‘(𝐴...(𝐵 − 1)))) |
19 | hashfz 10782 | . . . . 5 ⊢ ((𝐵 − 1) ∈ (ℤ≥‘𝐴) → (♯‘(𝐴...(𝐵 − 1))) = (((𝐵 − 1) − 𝐴) + 1)) | |
20 | 14 | zcnd 9362 | . . . . . . . 8 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐵 ∈ ℂ) |
21 | 1cnd 7961 | . . . . . . . 8 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 1 ∈ ℂ) | |
22 | 20, 21, 6 | sub32d 8287 | . . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → ((𝐵 − 1) − 𝐴) = ((𝐵 − 𝐴) − 1)) |
23 | 22 | oveq1d 5884 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (((𝐵 − 1) − 𝐴) + 1) = (((𝐵 − 𝐴) − 1) + 1)) |
24 | 20, 6 | subcld 8255 | . . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐵 − 𝐴) ∈ ℂ) |
25 | ax-1cn 7892 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
26 | npcan 8153 | . . . . . . 7 ⊢ (((𝐵 − 𝐴) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝐵 − 𝐴) − 1) + 1) = (𝐵 − 𝐴)) | |
27 | 24, 25, 26 | sylancl 413 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (((𝐵 − 𝐴) − 1) + 1) = (𝐵 − 𝐴)) |
28 | 23, 27 | eqtrd 2210 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (((𝐵 − 1) − 𝐴) + 1) = (𝐵 − 𝐴)) |
29 | 19, 28 | sylan9eqr 2232 | . . . 4 ⊢ ((𝐵 ∈ (ℤ≥‘𝐴) ∧ (𝐵 − 1) ∈ (ℤ≥‘𝐴)) → (♯‘(𝐴...(𝐵 − 1))) = (𝐵 − 𝐴)) |
30 | 18, 29 | eqtrd 2210 | . . 3 ⊢ ((𝐵 ∈ (ℤ≥‘𝐴) ∧ (𝐵 − 1) ∈ (ℤ≥‘𝐴)) → (♯‘(𝐴..^𝐵)) = (𝐵 − 𝐴)) |
31 | 30 | ex 115 | . 2 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → ((𝐵 − 1) ∈ (ℤ≥‘𝐴) → (♯‘(𝐴..^𝐵)) = (𝐵 − 𝐴))) |
32 | uzm1 9544 | . 2 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐵 = 𝐴 ∨ (𝐵 − 1) ∈ (ℤ≥‘𝐴))) | |
33 | 13, 31, 32 | mpjaod 718 | 1 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (♯‘(𝐴..^𝐵)) = (𝐵 − 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ∅c0 3422 ‘cfv 5212 (class class class)co 5869 ℂcc 7797 0cc0 7799 1c1 7800 + caddc 7802 − cmin 8115 ℤcz 9239 ℤ≥cuz 9514 ...cfz 9992 ..^cfzo 10125 ♯chash 10736 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 ax-cnex 7890 ax-resscn 7891 ax-1cn 7892 ax-1re 7893 ax-icn 7894 ax-addcl 7895 ax-addrcl 7896 ax-mulcl 7897 ax-addcom 7899 ax-addass 7901 ax-distr 7903 ax-i2m1 7904 ax-0lt1 7905 ax-0id 7907 ax-rnegex 7908 ax-cnre 7910 ax-pre-ltirr 7911 ax-pre-ltwlin 7912 ax-pre-lttrn 7913 ax-pre-apti 7914 ax-pre-ltadd 7915 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-iord 4363 df-on 4365 df-ilim 4366 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-frec 6386 df-1o 6411 df-er 6529 df-en 6735 df-dom 6736 df-fin 6737 df-pnf 7981 df-mnf 7982 df-xr 7983 df-ltxr 7984 df-le 7985 df-sub 8117 df-neg 8118 df-inn 8906 df-n0 9163 df-z 9240 df-uz 9515 df-fz 9993 df-fzo 10126 df-ihash 10737 |
This theorem is referenced by: hashfzo0 10784 |
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