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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 12902 | . . . . 5 ⊢ (𝐿 ∈ (0...(♯‘𝑊)) → (♯‘𝑊) ∈ (ℤ≥‘𝐿)) | |
4 | fzoss2 13062 | . . . . 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 13865 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑆 → dom 𝑊 = (0..^(♯‘𝑊))) | |
8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊))) |
9 | 5, 8 | sseqtrrd 4001 | . . 3 ⊢ (𝜑 → (0..^𝐿) ⊆ dom 𝑊) |
10 | wrdf 13863 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑆 → 𝑊:(0..^(♯‘𝑊))⟶𝑆) | |
11 | 6, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑊:(0..^(♯‘𝑊))⟶𝑆) |
12 | 11, 5 | fssresd 6538 | . . 3 ⊢ (𝜑 → (𝑊 ↾ (0..^𝐿)):(0..^𝐿)⟶𝑆) |
13 | f1resf1 6576 | . . 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 1366 | . 2 ⊢ (𝜑 → (𝑊 ↾ (0..^𝐿)):(0..^𝐿)–1-1→𝑆) |
15 | pfxres 14034 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑆 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (𝑊 prefix 𝐿) = (𝑊 ↾ (0..^𝐿))) | |
16 | 6, 2, 15 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝑊 prefix 𝐿) = (𝑊 ↾ (0..^𝐿))) |
17 | pfxfn 14036 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑆 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (𝑊 prefix 𝐿) Fn (0..^𝐿)) | |
18 | 6, 2, 17 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝑊 prefix 𝐿) Fn (0..^𝐿)) |
19 | 18 | fndmd 6449 | . . 3 ⊢ (𝜑 → dom (𝑊 prefix 𝐿) = (0..^𝐿)) |
20 | eqidd 2821 | . . 3 ⊢ (𝜑 → 𝑆 = 𝑆) | |
21 | 16, 19, 20 | f1eq123d 6601 | . 2 ⊢ (𝜑 → ((𝑊 prefix 𝐿):dom (𝑊 prefix 𝐿)–1-1→𝑆 ↔ (𝑊 ↾ (0..^𝐿)):(0..^𝐿)–1-1→𝑆)) |
22 | 14, 21 | mpbird 259 | 1 ⊢ (𝜑 → (𝑊 prefix 𝐿):dom (𝑊 prefix 𝐿)–1-1→𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ⊆ wss 3929 dom cdm 5548 ↾ cres 5550 Fn wfn 6343 ⟶wf 6344 –1-1→wf1 6345 ‘cfv 6348 (class class class)co 7149 0cc0 10530 ℤ≥cuz 12237 ...cfz 12889 ..^cfzo 13030 ♯chash 13687 Word cword 13858 prefix cpfx 14025 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12890 df-fzo 13031 df-hash 13688 df-word 13859 df-substr 13996 df-pfx 14026 |
This theorem is referenced by: cycpmco2f1 30785 |
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