Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wrdred1hash | Structured version Visualization version GIF version |
Description: The length of a word truncated by a symbol. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 29-Jan-2021.) |
Ref | Expression |
---|---|
wrdred1hash | ⊢ ((𝐹 ∈ Word 𝑆 ∧ 1 ≤ (#‘𝐹)) → (#‘(𝐹 ↾ (0..^((#‘𝐹) − 1)))) = ((#‘𝐹) − 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lencl 13179 | . . 3 ⊢ (𝐹 ∈ Word 𝑆 → (#‘𝐹) ∈ ℕ0) | |
2 | wrdf 13165 | . . . 4 ⊢ (𝐹 ∈ Word 𝑆 → 𝐹:(0..^(#‘𝐹))⟶𝑆) | |
3 | ffn 5958 | . . . 4 ⊢ (𝐹:(0..^(#‘𝐹))⟶𝑆 → 𝐹 Fn (0..^(#‘𝐹))) | |
4 | nn0z 11277 | . . . . . . . . . . 11 ⊢ ((#‘𝐹) ∈ ℕ0 → (#‘𝐹) ∈ ℤ) | |
5 | fzossrbm1 12366 | . . . . . . . . . . 11 ⊢ ((#‘𝐹) ∈ ℤ → (0..^((#‘𝐹) − 1)) ⊆ (0..^(#‘𝐹))) | |
6 | 4, 5 | syl 17 | . . . . . . . . . 10 ⊢ ((#‘𝐹) ∈ ℕ0 → (0..^((#‘𝐹) − 1)) ⊆ (0..^(#‘𝐹))) |
7 | 6 | adantr 480 | . . . . . . . . 9 ⊢ (((#‘𝐹) ∈ ℕ0 ∧ 1 ≤ (#‘𝐹)) → (0..^((#‘𝐹) − 1)) ⊆ (0..^(#‘𝐹))) |
8 | 7 | adantl 481 | . . . . . . . 8 ⊢ ((𝐹 Fn (0..^(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ0 ∧ 1 ≤ (#‘𝐹))) → (0..^((#‘𝐹) − 1)) ⊆ (0..^(#‘𝐹))) |
9 | fnssresb 5917 | . . . . . . . . 9 ⊢ (𝐹 Fn (0..^(#‘𝐹)) → ((𝐹 ↾ (0..^((#‘𝐹) − 1))) Fn (0..^((#‘𝐹) − 1)) ↔ (0..^((#‘𝐹) − 1)) ⊆ (0..^(#‘𝐹)))) | |
10 | 9 | adantr 480 | . . . . . . . 8 ⊢ ((𝐹 Fn (0..^(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ0 ∧ 1 ≤ (#‘𝐹))) → ((𝐹 ↾ (0..^((#‘𝐹) − 1))) Fn (0..^((#‘𝐹) − 1)) ↔ (0..^((#‘𝐹) − 1)) ⊆ (0..^(#‘𝐹)))) |
11 | 8, 10 | mpbird 246 | . . . . . . 7 ⊢ ((𝐹 Fn (0..^(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ0 ∧ 1 ≤ (#‘𝐹))) → (𝐹 ↾ (0..^((#‘𝐹) − 1))) Fn (0..^((#‘𝐹) − 1))) |
12 | hashfn 13025 | . . . . . . 7 ⊢ ((𝐹 ↾ (0..^((#‘𝐹) − 1))) Fn (0..^((#‘𝐹) − 1)) → (#‘(𝐹 ↾ (0..^((#‘𝐹) − 1)))) = (#‘(0..^((#‘𝐹) − 1)))) | |
13 | 11, 12 | syl 17 | . . . . . 6 ⊢ ((𝐹 Fn (0..^(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ0 ∧ 1 ≤ (#‘𝐹))) → (#‘(𝐹 ↾ (0..^((#‘𝐹) − 1)))) = (#‘(0..^((#‘𝐹) − 1)))) |
14 | 1nn0 11185 | . . . . . . . . 9 ⊢ 1 ∈ ℕ0 | |
15 | nn0sub2 11315 | . . . . . . . . 9 ⊢ ((1 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ0 ∧ 1 ≤ (#‘𝐹)) → ((#‘𝐹) − 1) ∈ ℕ0) | |
16 | 14, 15 | mp3an1 1403 | . . . . . . . 8 ⊢ (((#‘𝐹) ∈ ℕ0 ∧ 1 ≤ (#‘𝐹)) → ((#‘𝐹) − 1) ∈ ℕ0) |
17 | hashfzo0 13077 | . . . . . . . 8 ⊢ (((#‘𝐹) − 1) ∈ ℕ0 → (#‘(0..^((#‘𝐹) − 1))) = ((#‘𝐹) − 1)) | |
18 | 16, 17 | syl 17 | . . . . . . 7 ⊢ (((#‘𝐹) ∈ ℕ0 ∧ 1 ≤ (#‘𝐹)) → (#‘(0..^((#‘𝐹) − 1))) = ((#‘𝐹) − 1)) |
19 | 18 | adantl 481 | . . . . . 6 ⊢ ((𝐹 Fn (0..^(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ0 ∧ 1 ≤ (#‘𝐹))) → (#‘(0..^((#‘𝐹) − 1))) = ((#‘𝐹) − 1)) |
20 | 13, 19 | eqtrd 2644 | . . . . 5 ⊢ ((𝐹 Fn (0..^(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ0 ∧ 1 ≤ (#‘𝐹))) → (#‘(𝐹 ↾ (0..^((#‘𝐹) − 1)))) = ((#‘𝐹) − 1)) |
21 | 20 | ex 449 | . . . 4 ⊢ (𝐹 Fn (0..^(#‘𝐹)) → (((#‘𝐹) ∈ ℕ0 ∧ 1 ≤ (#‘𝐹)) → (#‘(𝐹 ↾ (0..^((#‘𝐹) − 1)))) = ((#‘𝐹) − 1))) |
22 | 2, 3, 21 | 3syl 18 | . . 3 ⊢ (𝐹 ∈ Word 𝑆 → (((#‘𝐹) ∈ ℕ0 ∧ 1 ≤ (#‘𝐹)) → (#‘(𝐹 ↾ (0..^((#‘𝐹) − 1)))) = ((#‘𝐹) − 1))) |
23 | 1, 22 | mpand 707 | . 2 ⊢ (𝐹 ∈ Word 𝑆 → (1 ≤ (#‘𝐹) → (#‘(𝐹 ↾ (0..^((#‘𝐹) − 1)))) = ((#‘𝐹) − 1))) |
24 | 23 | imp 444 | 1 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 1 ≤ (#‘𝐹)) → (#‘(𝐹 ↾ (0..^((#‘𝐹) − 1)))) = ((#‘𝐹) − 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 class class class wbr 4583 ↾ cres 5040 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 ≤ cle 9954 − cmin 10145 ℕ0cn0 11169 ℤcz 11254 ..^cfzo 12334 #chash 12979 Word cword 13146 |
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-rep 4699 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-oadd 7451 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-fzo 12335 df-hash 12980 df-word 13154 |
This theorem is referenced by: red1wlklem 40880 red1wlk 40881 |
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