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Mirrors > Home > MPE Home > Th. List > wrdnfi | Structured version Visualization version GIF version |
Description: If there is only a finite number of symbols, the number of words of a fixed length over these symbols is also finite. (Contributed by Alexander van der Vekens, 25-Mar-2018.) Remove unnecessary antecedent. (Revised by JJ, 18-Nov-2022.) |
Ref | Expression |
---|---|
wrdnfi | ⊢ (𝑉 ∈ Fin → {𝑤 ∈ Word 𝑉 ∣ (♯‘𝑤) = 𝑁} ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashwrdn 14597 | . . . . 5 ⊢ ((𝑉 ∈ Fin ∧ 𝑁 ∈ ℕ0) → (♯‘{𝑤 ∈ Word 𝑉 ∣ (♯‘𝑤) = 𝑁}) = ((♯‘𝑉)↑𝑁)) | |
2 | hashcl 14407 | . . . . . 6 ⊢ (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℕ0) | |
3 | nn0expcl 14128 | . . . . . 6 ⊢ (((♯‘𝑉) ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((♯‘𝑉)↑𝑁) ∈ ℕ0) | |
4 | 2, 3 | sylan 579 | . . . . 5 ⊢ ((𝑉 ∈ Fin ∧ 𝑁 ∈ ℕ0) → ((♯‘𝑉)↑𝑁) ∈ ℕ0) |
5 | 1, 4 | eqeltrd 2844 | . . . 4 ⊢ ((𝑉 ∈ Fin ∧ 𝑁 ∈ ℕ0) → (♯‘{𝑤 ∈ Word 𝑉 ∣ (♯‘𝑤) = 𝑁}) ∈ ℕ0) |
6 | 5 | ex 412 | . . 3 ⊢ (𝑉 ∈ Fin → (𝑁 ∈ ℕ0 → (♯‘{𝑤 ∈ Word 𝑉 ∣ (♯‘𝑤) = 𝑁}) ∈ ℕ0)) |
7 | lencl 14583 | . . . . . . . . . 10 ⊢ (𝑤 ∈ Word 𝑉 → (♯‘𝑤) ∈ ℕ0) | |
8 | eleq1 2832 | . . . . . . . . . 10 ⊢ ((♯‘𝑤) = 𝑁 → ((♯‘𝑤) ∈ ℕ0 ↔ 𝑁 ∈ ℕ0)) | |
9 | 7, 8 | syl5ibcom 245 | . . . . . . . . 9 ⊢ (𝑤 ∈ Word 𝑉 → ((♯‘𝑤) = 𝑁 → 𝑁 ∈ ℕ0)) |
10 | 9 | con3rr3 155 | . . . . . . . 8 ⊢ (¬ 𝑁 ∈ ℕ0 → (𝑤 ∈ Word 𝑉 → ¬ (♯‘𝑤) = 𝑁)) |
11 | 10 | ralrimiv 3151 | . . . . . . 7 ⊢ (¬ 𝑁 ∈ ℕ0 → ∀𝑤 ∈ Word 𝑉 ¬ (♯‘𝑤) = 𝑁) |
12 | rabeq0 4411 | . . . . . . 7 ⊢ ({𝑤 ∈ Word 𝑉 ∣ (♯‘𝑤) = 𝑁} = ∅ ↔ ∀𝑤 ∈ Word 𝑉 ¬ (♯‘𝑤) = 𝑁) | |
13 | 11, 12 | sylibr 234 | . . . . . 6 ⊢ (¬ 𝑁 ∈ ℕ0 → {𝑤 ∈ Word 𝑉 ∣ (♯‘𝑤) = 𝑁} = ∅) |
14 | 13 | fveq2d 6926 | . . . . 5 ⊢ (¬ 𝑁 ∈ ℕ0 → (♯‘{𝑤 ∈ Word 𝑉 ∣ (♯‘𝑤) = 𝑁}) = (♯‘∅)) |
15 | hash0 14418 | . . . . 5 ⊢ (♯‘∅) = 0 | |
16 | 14, 15 | eqtrdi 2796 | . . . 4 ⊢ (¬ 𝑁 ∈ ℕ0 → (♯‘{𝑤 ∈ Word 𝑉 ∣ (♯‘𝑤) = 𝑁}) = 0) |
17 | 0nn0 12570 | . . . 4 ⊢ 0 ∈ ℕ0 | |
18 | 16, 17 | eqeltrdi 2852 | . . 3 ⊢ (¬ 𝑁 ∈ ℕ0 → (♯‘{𝑤 ∈ Word 𝑉 ∣ (♯‘𝑤) = 𝑁}) ∈ ℕ0) |
19 | 6, 18 | pm2.61d1 180 | . 2 ⊢ (𝑉 ∈ Fin → (♯‘{𝑤 ∈ Word 𝑉 ∣ (♯‘𝑤) = 𝑁}) ∈ ℕ0) |
20 | wrdexg 14574 | . . 3 ⊢ (𝑉 ∈ Fin → Word 𝑉 ∈ V) | |
21 | rabexg 5355 | . . 3 ⊢ (Word 𝑉 ∈ V → {𝑤 ∈ Word 𝑉 ∣ (♯‘𝑤) = 𝑁} ∈ V) | |
22 | hashclb 14409 | . . 3 ⊢ ({𝑤 ∈ Word 𝑉 ∣ (♯‘𝑤) = 𝑁} ∈ V → ({𝑤 ∈ Word 𝑉 ∣ (♯‘𝑤) = 𝑁} ∈ Fin ↔ (♯‘{𝑤 ∈ Word 𝑉 ∣ (♯‘𝑤) = 𝑁}) ∈ ℕ0)) | |
23 | 20, 21, 22 | 3syl 18 | . 2 ⊢ (𝑉 ∈ Fin → ({𝑤 ∈ Word 𝑉 ∣ (♯‘𝑤) = 𝑁} ∈ Fin ↔ (♯‘{𝑤 ∈ Word 𝑉 ∣ (♯‘𝑤) = 𝑁}) ∈ ℕ0)) |
24 | 19, 23 | mpbird 257 | 1 ⊢ (𝑉 ∈ Fin → {𝑤 ∈ Word 𝑉 ∣ (♯‘𝑤) = 𝑁} ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∀wral 3067 {crab 3443 Vcvv 3488 ∅c0 4352 ‘cfv 6575 (class class class)co 7450 Fincfn 9005 0cc0 11186 ℕ0cn0 12555 ↑cexp 14114 ♯chash 14381 Word cword 14564 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-om 7906 df-1st 8032 df-2nd 8033 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-1o 8524 df-oadd 8528 df-er 8765 df-map 8888 df-pm 8889 df-en 9006 df-dom 9007 df-sdom 9008 df-fin 9009 df-dju 9972 df-card 10010 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-nn 12296 df-n0 12556 df-z 12642 df-uz 12906 df-fz 13570 df-fzo 13714 df-seq 14055 df-exp 14115 df-hash 14382 df-word 14565 |
This theorem is referenced by: wwlksnfi 29941 clwwlknfi 30079 upwrdfi 46808 |
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