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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 11568 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐴 ∈ ℤ) | |
2 | eluzelz 11573 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐵 ∈ ℤ) | |
3 | 1z 11284 | . . . . . 6 ⊢ 1 ∈ ℤ | |
4 | zsubcl 11296 | . . . . . 6 ⊢ ((1 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (1 − 𝐴) ∈ ℤ) | |
5 | 3, 1, 4 | sylancr 694 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (1 − 𝐴) ∈ ℤ) |
6 | fzen 12229 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ (1 − 𝐴) ∈ ℤ) → (𝐴...𝐵) ≈ ((𝐴 + (1 − 𝐴))...(𝐵 + (1 − 𝐴)))) | |
7 | 1, 2, 5, 6 | syl3anc 1318 | . . . 4 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐴...𝐵) ≈ ((𝐴 + (1 − 𝐴))...(𝐵 + (1 − 𝐴)))) |
8 | 1 | zcnd 11359 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐴 ∈ ℂ) |
9 | ax-1cn 9873 | . . . . . 6 ⊢ 1 ∈ ℂ | |
10 | pncan3 10168 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 + (1 − 𝐴)) = 1) | |
11 | 8, 9, 10 | sylancl 693 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐴 + (1 − 𝐴)) = 1) |
12 | 2 | zcnd 11359 | . . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐵 ∈ ℂ) |
13 | 1cnd 9935 | . . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 1 ∈ ℂ) | |
14 | 12, 13, 8 | addsub12d 10294 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐵 + (1 − 𝐴)) = (1 + (𝐵 − 𝐴))) |
15 | 12, 8 | subcld 10271 | . . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐵 − 𝐴) ∈ ℂ) |
16 | addcom 10101 | . . . . . . 7 ⊢ ((1 ∈ ℂ ∧ (𝐵 − 𝐴) ∈ ℂ) → (1 + (𝐵 − 𝐴)) = ((𝐵 − 𝐴) + 1)) | |
17 | 9, 15, 16 | sylancr 694 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (1 + (𝐵 − 𝐴)) = ((𝐵 − 𝐴) + 1)) |
18 | 14, 17 | eqtrd 2644 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐵 + (1 − 𝐴)) = ((𝐵 − 𝐴) + 1)) |
19 | 11, 18 | oveq12d 6567 | . . . 4 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → ((𝐴 + (1 − 𝐴))...(𝐵 + (1 − 𝐴))) = (1...((𝐵 − 𝐴) + 1))) |
20 | 7, 19 | breqtrd 4609 | . . 3 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐴...𝐵) ≈ (1...((𝐵 − 𝐴) + 1))) |
21 | hasheni 12998 | . . 3 ⊢ ((𝐴...𝐵) ≈ (1...((𝐵 − 𝐴) + 1)) → (#‘(𝐴...𝐵)) = (#‘(1...((𝐵 − 𝐴) + 1)))) | |
22 | 20, 21 | syl 17 | . 2 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (#‘(𝐴...𝐵)) = (#‘(1...((𝐵 − 𝐴) + 1)))) |
23 | uznn0sub 11595 | . . 3 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐵 − 𝐴) ∈ ℕ0) | |
24 | peano2nn0 11210 | . . 3 ⊢ ((𝐵 − 𝐴) ∈ ℕ0 → ((𝐵 − 𝐴) + 1) ∈ ℕ0) | |
25 | hashfz1 12996 | . . 3 ⊢ (((𝐵 − 𝐴) + 1) ∈ ℕ0 → (#‘(1...((𝐵 − 𝐴) + 1))) = ((𝐵 − 𝐴) + 1)) | |
26 | 23, 24, 25 | 3syl 18 | . 2 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (#‘(1...((𝐵 − 𝐴) + 1))) = ((𝐵 − 𝐴) + 1)) |
27 | 22, 26 | eqtrd 2644 | 1 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (#‘(𝐴...𝐵)) = ((𝐵 − 𝐴) + 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ≈ cen 7838 ℂcc 9813 1c1 9816 + caddc 9818 − cmin 10145 ℕ0cn0 11169 ℤcz 11254 ℤ≥cuz 11563 ...cfz 12197 #chash 12979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-hash 12980 |
This theorem is referenced by: fzsdom2 13075 hashfzo 13076 hashfzp1 13078 hashfz0 13079 0sgmppw 24723 logfaclbnd 24747 gausslemma2dlem5 24896 ballotlem2 29877 subfacp1lem5 30420 fzisoeu 38455 stoweidlem11 38904 stoweidlem26 38919 |
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