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Mirrors > Home > MPE Home > Th. List > hashfz | Structured version Visualization version GIF version |
Description: Value of the numeric cardinality of a nonempty integer range. (Contributed by Stefan O'Rear, 12-Sep-2014.) (Proof shortened by Mario Carneiro, 15-Apr-2015.) |
Ref | Expression |
---|---|
hashfz | ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (♯‘(𝐴...𝐵)) = ((𝐵 − 𝐴) + 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzel2 12858 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐴 ∈ ℤ) | |
2 | eluzelz 12863 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐵 ∈ ℤ) | |
3 | 1z 12623 | . . . . . 6 ⊢ 1 ∈ ℤ | |
4 | zsubcl 12635 | . . . . . 6 ⊢ ((1 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (1 − 𝐴) ∈ ℤ) | |
5 | 3, 1, 4 | sylancr 586 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (1 − 𝐴) ∈ ℤ) |
6 | fzen 13551 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ (1 − 𝐴) ∈ ℤ) → (𝐴...𝐵) ≈ ((𝐴 + (1 − 𝐴))...(𝐵 + (1 − 𝐴)))) | |
7 | 1, 2, 5, 6 | syl3anc 1369 | . . . 4 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐴...𝐵) ≈ ((𝐴 + (1 − 𝐴))...(𝐵 + (1 − 𝐴)))) |
8 | 1 | zcnd 12698 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐴 ∈ ℂ) |
9 | ax-1cn 11197 | . . . . . 6 ⊢ 1 ∈ ℂ | |
10 | pncan3 11499 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 + (1 − 𝐴)) = 1) | |
11 | 8, 9, 10 | sylancl 585 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐴 + (1 − 𝐴)) = 1) |
12 | 1cnd 11240 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 1 ∈ ℂ) | |
13 | 2 | zcnd 12698 | . . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐵 ∈ ℂ) |
14 | 13, 8 | subcld 11602 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐵 − 𝐴) ∈ ℂ) |
15 | 13, 12, 8 | addsub12d 11625 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐵 + (1 − 𝐴)) = (1 + (𝐵 − 𝐴))) |
16 | 12, 14, 15 | comraddd 11459 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐵 + (1 − 𝐴)) = ((𝐵 − 𝐴) + 1)) |
17 | 11, 16 | oveq12d 7438 | . . . 4 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → ((𝐴 + (1 − 𝐴))...(𝐵 + (1 − 𝐴))) = (1...((𝐵 − 𝐴) + 1))) |
18 | 7, 17 | breqtrd 5174 | . . 3 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐴...𝐵) ≈ (1...((𝐵 − 𝐴) + 1))) |
19 | hasheni 14340 | . . 3 ⊢ ((𝐴...𝐵) ≈ (1...((𝐵 − 𝐴) + 1)) → (♯‘(𝐴...𝐵)) = (♯‘(1...((𝐵 − 𝐴) + 1)))) | |
20 | 18, 19 | syl 17 | . 2 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (♯‘(𝐴...𝐵)) = (♯‘(1...((𝐵 − 𝐴) + 1)))) |
21 | uznn0sub 12892 | . . 3 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐵 − 𝐴) ∈ ℕ0) | |
22 | peano2nn0 12543 | . . 3 ⊢ ((𝐵 − 𝐴) ∈ ℕ0 → ((𝐵 − 𝐴) + 1) ∈ ℕ0) | |
23 | hashfz1 14338 | . . 3 ⊢ (((𝐵 − 𝐴) + 1) ∈ ℕ0 → (♯‘(1...((𝐵 − 𝐴) + 1))) = ((𝐵 − 𝐴) + 1)) | |
24 | 21, 22, 23 | 3syl 18 | . 2 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (♯‘(1...((𝐵 − 𝐴) + 1))) = ((𝐵 − 𝐴) + 1)) |
25 | 20, 24 | eqtrd 2768 | 1 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (♯‘(𝐴...𝐵)) = ((𝐵 − 𝐴) + 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 class class class wbr 5148 ‘cfv 6548 (class class class)co 7420 ≈ cen 8961 ℂcc 11137 1c1 11140 + caddc 11142 − cmin 11475 ℕ0cn0 12503 ℤcz 12589 ℤ≥cuz 12853 ...cfz 13517 ♯chash 14322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-n0 12504 df-z 12590 df-uz 12854 df-fz 13518 df-hash 14323 |
This theorem is referenced by: fzsdom2 14420 hashfzo 14421 hashfzp1 14423 hashfz0 14424 0sgmppw 27144 logfaclbnd 27168 gausslemma2dlem5 27317 ballotlem2 34108 subfacp1lem5 34794 fzisoeu 44682 stoweidlem11 45399 stoweidlem26 45414 |
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