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Mirrors > Home > ILE Home > Th. List > hashfz1 | GIF version |
Description: The set (1...𝑁) has 𝑁 elements. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.) |
Ref | Expression |
---|---|
hashfz1 | ⊢ (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0zd 8964 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ ℤ) | |
2 | eqid 2113 | . . . . . 6 ⊢ frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) | |
3 | 1, 2 | frec2uzf1od 10066 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0):ω–1-1-onto→(ℤ≥‘0)) |
4 | f1ocnv 5334 | . . . . 5 ⊢ (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0):ω–1-1-onto→(ℤ≥‘0) → ◡frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0):(ℤ≥‘0)–1-1-onto→ω) | |
5 | f1of 5321 | . . . . 5 ⊢ (◡frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0):(ℤ≥‘0)–1-1-onto→ω → ◡frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0):(ℤ≥‘0)⟶ω) | |
6 | 3, 4, 5 | 3syl 17 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ◡frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0):(ℤ≥‘0)⟶ω) |
7 | elnn0uz 9259 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (ℤ≥‘0)) | |
8 | 7 | biimpi 119 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (ℤ≥‘0)) |
9 | 6, 8 | ffvelrnd 5508 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (◡frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘𝑁) ∈ ω) |
10 | 2 | frecfzennn 10086 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) ≈ (◡frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘𝑁)) |
11 | 10 | ensymd 6629 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (◡frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘𝑁) ≈ (1...𝑁)) |
12 | hashennn 10413 | . . 3 ⊢ (((◡frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘𝑁) ∈ ω ∧ (◡frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘𝑁) ≈ (1...𝑁)) → (♯‘(1...𝑁)) = (frec((𝑦 ∈ ℤ ↦ (𝑦 + 1)), 0)‘(◡frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘𝑁))) | |
13 | 9, 11, 12 | syl2anc 406 | . 2 ⊢ (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = (frec((𝑦 ∈ ℤ ↦ (𝑦 + 1)), 0)‘(◡frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘𝑁))) |
14 | oveq1 5733 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 + 1) = (𝑦 + 1)) | |
15 | 14 | cbvmptv 3982 | . . . . . 6 ⊢ (𝑥 ∈ ℤ ↦ (𝑥 + 1)) = (𝑦 ∈ ℤ ↦ (𝑦 + 1)) |
16 | freceq1 6241 | . . . . . 6 ⊢ ((𝑥 ∈ ℤ ↦ (𝑥 + 1)) = (𝑦 ∈ ℤ ↦ (𝑦 + 1)) → frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) = frec((𝑦 ∈ ℤ ↦ (𝑦 + 1)), 0)) | |
17 | 15, 16 | ax-mp 7 | . . . . 5 ⊢ frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) = frec((𝑦 ∈ ℤ ↦ (𝑦 + 1)), 0) |
18 | 17 | fveq1i 5374 | . . . 4 ⊢ (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘(◡frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘𝑁)) = (frec((𝑦 ∈ ℤ ↦ (𝑦 + 1)), 0)‘(◡frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘𝑁)) |
19 | f1ocnvfv2 5631 | . . . 4 ⊢ ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0):ω–1-1-onto→(ℤ≥‘0) ∧ 𝑁 ∈ (ℤ≥‘0)) → (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘(◡frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘𝑁)) = 𝑁) | |
20 | 18, 19 | syl5eqr 2159 | . . 3 ⊢ ((frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0):ω–1-1-onto→(ℤ≥‘0) ∧ 𝑁 ∈ (ℤ≥‘0)) → (frec((𝑦 ∈ ℤ ↦ (𝑦 + 1)), 0)‘(◡frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘𝑁)) = 𝑁) |
21 | 3, 8, 20 | syl2anc 406 | . 2 ⊢ (𝑁 ∈ ℕ0 → (frec((𝑦 ∈ ℤ ↦ (𝑦 + 1)), 0)‘(◡frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘𝑁)) = 𝑁) |
22 | 13, 21 | eqtrd 2145 | 1 ⊢ (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1312 ∈ wcel 1461 class class class wbr 3893 ↦ cmpt 3947 ωcom 4462 ◡ccnv 4496 ⟶wf 5075 –1-1-onto→wf1o 5078 ‘cfv 5079 (class class class)co 5726 freccfrec 6239 ≈ cen 6584 0cc0 7541 1c1 7542 + caddc 7544 ℕ0cn0 8875 ℤcz 8952 ℤ≥cuz 9222 ...cfz 9677 ♯chash 10408 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-coll 4001 ax-sep 4004 ax-nul 4012 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 ax-iinf 4460 ax-cnex 7630 ax-resscn 7631 ax-1cn 7632 ax-1re 7633 ax-icn 7634 ax-addcl 7635 ax-addrcl 7636 ax-mulcl 7637 ax-addcom 7639 ax-addass 7641 ax-distr 7643 ax-i2m1 7644 ax-0lt1 7645 ax-0id 7647 ax-rnegex 7648 ax-cnre 7650 ax-pre-ltirr 7651 ax-pre-ltwlin 7652 ax-pre-lttrn 7653 ax-pre-apti 7654 ax-pre-ltadd 7655 |
This theorem depends on definitions: df-bi 116 df-dc 803 df-3or 944 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-nel 2376 df-ral 2393 df-rex 2394 df-reu 2395 df-rab 2397 df-v 2657 df-sbc 2877 df-csb 2970 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-nul 3328 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-int 3736 df-iun 3779 df-br 3894 df-opab 3948 df-mpt 3949 df-tr 3985 df-id 4173 df-iord 4246 df-on 4248 df-ilim 4249 df-suc 4251 df-iom 4463 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-ima 4510 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-f1 5084 df-fo 5085 df-f1o 5086 df-fv 5087 df-riota 5682 df-ov 5729 df-oprab 5730 df-mpo 5731 df-recs 6154 df-frec 6240 df-1o 6265 df-er 6381 df-en 6587 df-dom 6588 df-fin 6589 df-pnf 7720 df-mnf 7721 df-xr 7722 df-ltxr 7723 df-le 7724 df-sub 7852 df-neg 7853 df-inn 8625 df-n0 8876 df-z 8953 df-uz 9223 df-fz 9678 df-ihash 10409 |
This theorem is referenced by: fz1eqb 10424 isfinite4im 10426 fihasheq0 10427 hashsng 10431 fseq1hash 10434 hashfz 10454 isummolemnm 11034 summodclem2a 11036 summodc 11038 zsumdc 11039 fsum3 11042 mertenslemi1 11190 phicl2 11729 phibnd 11732 hashdvds 11736 phiprmpw 11737 |
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