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Mirrors > Home > MPE Home > Th. List > hashwrdn | 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 sysmbols is the number of these symbols raised to the power of the length. (Contributed by Alexander van der Vekens, 25-Mar-2018.) |
Ref | Expression |
---|---|
hashwrdn | ⊢ ((𝑉 ∈ Fin ∧ 𝑁 ∈ ℕ0) → (♯‘{𝑤 ∈ Word 𝑉 ∣ (♯‘𝑤) = 𝑁}) = ((♯‘𝑉)↑𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wrdnval 13604 | . . 3 ⊢ ((𝑉 ∈ Fin ∧ 𝑁 ∈ ℕ0) → {𝑤 ∈ Word 𝑉 ∣ (♯‘𝑤) = 𝑁} = (𝑉 ↑𝑚 (0..^𝑁))) | |
2 | 1 | fveq2d 6436 | . 2 ⊢ ((𝑉 ∈ Fin ∧ 𝑁 ∈ ℕ0) → (♯‘{𝑤 ∈ Word 𝑉 ∣ (♯‘𝑤) = 𝑁}) = (♯‘(𝑉 ↑𝑚 (0..^𝑁)))) |
3 | fzofi 13067 | . . . 4 ⊢ (0..^𝑁) ∈ Fin | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (0..^𝑁) ∈ Fin) |
5 | hashmap 13510 | . . 3 ⊢ ((𝑉 ∈ Fin ∧ (0..^𝑁) ∈ Fin) → (♯‘(𝑉 ↑𝑚 (0..^𝑁))) = ((♯‘𝑉)↑(♯‘(0..^𝑁)))) | |
6 | 4, 5 | sylan2 588 | . 2 ⊢ ((𝑉 ∈ Fin ∧ 𝑁 ∈ ℕ0) → (♯‘(𝑉 ↑𝑚 (0..^𝑁))) = ((♯‘𝑉)↑(♯‘(0..^𝑁)))) |
7 | hashfzo0 13505 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (♯‘(0..^𝑁)) = 𝑁) | |
8 | 7 | adantl 475 | . . 3 ⊢ ((𝑉 ∈ Fin ∧ 𝑁 ∈ ℕ0) → (♯‘(0..^𝑁)) = 𝑁) |
9 | 8 | oveq2d 6920 | . 2 ⊢ ((𝑉 ∈ Fin ∧ 𝑁 ∈ ℕ0) → ((♯‘𝑉)↑(♯‘(0..^𝑁))) = ((♯‘𝑉)↑𝑁)) |
10 | 2, 6, 9 | 3eqtrd 2864 | 1 ⊢ ((𝑉 ∈ Fin ∧ 𝑁 ∈ ℕ0) → (♯‘{𝑤 ∈ Word 𝑉 ∣ (♯‘𝑤) = 𝑁}) = ((♯‘𝑉)↑𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 {crab 3120 ‘cfv 6122 (class class class)co 6904 ↑𝑚 cmap 8121 Fincfn 8221 0cc0 10251 ℕ0cn0 11617 ..^cfzo 12759 ↑cexp 13153 ♯chash 13409 Word cword 13573 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-int 4697 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-om 7326 df-1st 7427 df-2nd 7428 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-1o 7825 df-2o 7826 df-oadd 7829 df-er 8008 df-map 8123 df-pm 8124 df-en 8222 df-dom 8223 df-sdom 8224 df-fin 8225 df-card 9077 df-cda 9304 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-nn 11350 df-n0 11618 df-z 11704 df-uz 11968 df-fz 12619 df-fzo 12760 df-seq 13095 df-exp 13154 df-hash 13410 df-word 13574 |
This theorem is referenced by: wrdnfi 13607 |
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