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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pfxf1 | Structured version Visualization version GIF version |
Description: Condition for a prefix to be injective. (Contributed by Thierry Arnoux, 13-Dec-2023.) |
Ref | Expression |
---|---|
pfxf1.1 | ⊢ (𝜑 → 𝑊 ∈ Word 𝑆) |
pfxf1.2 | ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝑆) |
pfxf1.3 | ⊢ (𝜑 → 𝐿 ∈ (0...(♯‘𝑊))) |
Ref | Expression |
---|---|
pfxf1 | ⊢ (𝜑 → (𝑊 prefix 𝐿):dom (𝑊 prefix 𝐿)–1-1→𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pfxf1.2 | . . 3 ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝑆) | |
2 | pfxf1.3 | . . . . 5 ⊢ (𝜑 → 𝐿 ∈ (0...(♯‘𝑊))) | |
3 | elfzuz3 13557 | . . . . 5 ⊢ (𝐿 ∈ (0...(♯‘𝑊)) → (♯‘𝑊) ∈ (ℤ≥‘𝐿)) | |
4 | fzoss2 13723 | . . . . 5 ⊢ ((♯‘𝑊) ∈ (ℤ≥‘𝐿) → (0..^𝐿) ⊆ (0..^(♯‘𝑊))) | |
5 | 2, 3, 4 | 3syl 18 | . . . 4 ⊢ (𝜑 → (0..^𝐿) ⊆ (0..^(♯‘𝑊))) |
6 | pfxf1.1 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ Word 𝑆) | |
7 | wrddm 14555 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑆 → dom 𝑊 = (0..^(♯‘𝑊))) | |
8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊))) |
9 | 5, 8 | sseqtrrd 4020 | . . 3 ⊢ (𝜑 → (0..^𝐿) ⊆ dom 𝑊) |
10 | wrdf 14553 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑆 → 𝑊:(0..^(♯‘𝑊))⟶𝑆) | |
11 | 6, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑊:(0..^(♯‘𝑊))⟶𝑆) |
12 | 11, 5 | fssresd 6773 | . . 3 ⊢ (𝜑 → (𝑊 ↾ (0..^𝐿)):(0..^𝐿)⟶𝑆) |
13 | f1resf1 6810 | . . 3 ⊢ ((𝑊:dom 𝑊–1-1→𝑆 ∧ (0..^𝐿) ⊆ dom 𝑊 ∧ (𝑊 ↾ (0..^𝐿)):(0..^𝐿)⟶𝑆) → (𝑊 ↾ (0..^𝐿)):(0..^𝐿)–1-1→𝑆) | |
14 | 1, 9, 12, 13 | syl3anc 1373 | . 2 ⊢ (𝜑 → (𝑊 ↾ (0..^𝐿)):(0..^𝐿)–1-1→𝑆) |
15 | pfxres 14713 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑆 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (𝑊 prefix 𝐿) = (𝑊 ↾ (0..^𝐿))) | |
16 | 6, 2, 15 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝑊 prefix 𝐿) = (𝑊 ↾ (0..^𝐿))) |
17 | pfxfn 14715 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑆 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (𝑊 prefix 𝐿) Fn (0..^𝐿)) | |
18 | 6, 2, 17 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝑊 prefix 𝐿) Fn (0..^𝐿)) |
19 | 18 | fndmd 6671 | . . 3 ⊢ (𝜑 → dom (𝑊 prefix 𝐿) = (0..^𝐿)) |
20 | eqidd 2737 | . . 3 ⊢ (𝜑 → 𝑆 = 𝑆) | |
21 | 16, 19, 20 | f1eq123d 6838 | . 2 ⊢ (𝜑 → ((𝑊 prefix 𝐿):dom (𝑊 prefix 𝐿)–1-1→𝑆 ↔ (𝑊 ↾ (0..^𝐿)):(0..^𝐿)–1-1→𝑆)) |
22 | 14, 21 | mpbird 257 | 1 ⊢ (𝜑 → (𝑊 prefix 𝐿):dom (𝑊 prefix 𝐿)–1-1→𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ⊆ wss 3950 dom cdm 5683 ↾ cres 5685 Fn wfn 6554 ⟶wf 6555 –1-1→wf1 6556 ‘cfv 6559 (class class class)co 7429 0cc0 11151 ℤ≥cuz 12874 ...cfz 13543 ..^cfzo 13690 ♯chash 14365 Word cword 14548 prefix cpfx 14704 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-cnex 11207 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-int 4945 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-om 7884 df-1st 8010 df-2nd 8011 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-1o 8502 df-er 8741 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-card 9975 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-nn 12263 df-n0 12523 df-z 12610 df-uz 12875 df-fz 13544 df-fzo 13691 df-hash 14366 df-word 14549 df-substr 14675 df-pfx 14705 |
This theorem is referenced by: cycpmco2f1 33129 |
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