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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 9059 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ ℤ) | |
2 | eqid 2137 | . . . . . 6 ⊢ frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) | |
3 | 1, 2 | frec2uzf1od 10172 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0):ω–1-1-onto→(ℤ≥‘0)) |
4 | f1ocnv 5373 | . . . . 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 5360 | . . . . 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 9356 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (ℤ≥‘0)) | |
8 | 7 | biimpi 119 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (ℤ≥‘0)) |
9 | 6, 8 | ffvelrnd 5549 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (◡frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘𝑁) ∈ ω) |
10 | 2 | frecfzennn 10192 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) ≈ (◡frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘𝑁)) |
11 | 10 | ensymd 6670 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (◡frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘𝑁) ≈ (1...𝑁)) |
12 | hashennn 10519 | . . 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 408 | . 2 ⊢ (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = (frec((𝑦 ∈ ℤ ↦ (𝑦 + 1)), 0)‘(◡frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘𝑁))) |
14 | oveq1 5774 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 + 1) = (𝑦 + 1)) | |
15 | 14 | cbvmptv 4019 | . . . . . 6 ⊢ (𝑥 ∈ ℤ ↦ (𝑥 + 1)) = (𝑦 ∈ ℤ ↦ (𝑦 + 1)) |
16 | freceq1 6282 | . . . . . 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 5 | . . . . 5 ⊢ frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) = frec((𝑦 ∈ ℤ ↦ (𝑦 + 1)), 0) |
18 | 17 | fveq1i 5415 | . . . 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 5672 | . . . 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 2184 | . . 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 408 | . 2 ⊢ (𝑁 ∈ ℕ0 → (frec((𝑦 ∈ ℤ ↦ (𝑦 + 1)), 0)‘(◡frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)‘𝑁)) = 𝑁) |
22 | 13, 21 | eqtrd 2170 | 1 ⊢ (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 class class class wbr 3924 ↦ cmpt 3984 ωcom 4499 ◡ccnv 4533 ⟶wf 5114 –1-1-onto→wf1o 5117 ‘cfv 5118 (class class class)co 5767 freccfrec 6280 ≈ cen 6625 0cc0 7613 1c1 7614 + caddc 7616 ℕ0cn0 8970 ℤcz 9047 ℤ≥cuz 9319 ...cfz 9783 ♯chash 10514 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-recs 6195 df-frec 6281 df-1o 6306 df-er 6422 df-en 6628 df-dom 6629 df-fin 6630 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-inn 8714 df-n0 8971 df-z 9048 df-uz 9320 df-fz 9784 df-ihash 10515 |
This theorem is referenced by: fz1eqb 10530 isfinite4im 10532 fihasheq0 10533 hashsng 10537 fseq1hash 10540 hashfz 10560 isummolemnm 11141 summodclem2a 11143 summodc 11145 zsumdc 11146 fsum3 11149 mertenslemi1 11297 phicl2 11879 phibnd 11882 hashdvds 11886 phiprmpw 11887 |
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