![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > swrdrndisj | Structured version Visualization version GIF version |
Description: Condition for the range of two subwords of an injective word to be disjoint. (Contributed by Thierry Arnoux, 13-Dec-2023.) |
Ref | Expression |
---|---|
swrdf1.w | ⊢ (𝜑 → 𝑊 ∈ Word 𝐷) |
swrdf1.m | ⊢ (𝜑 → 𝑀 ∈ (0...𝑁)) |
swrdf1.n | ⊢ (𝜑 → 𝑁 ∈ (0...(♯‘𝑊))) |
swrdf1.1 | ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷) |
swrdrndisj.1 | ⊢ (𝜑 → 𝑂 ∈ (𝑁...𝑃)) |
swrdrndisj.2 | ⊢ (𝜑 → 𝑃 ∈ (𝑁...(♯‘𝑊))) |
Ref | Expression |
---|---|
swrdrndisj | ⊢ (𝜑 → (ran (𝑊 substr ⟨𝑀, 𝑁⟩) ∩ ran (𝑊 substr ⟨𝑂, 𝑃⟩)) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | swrdf1.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ Word 𝐷) | |
2 | swrdf1.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ (0...𝑁)) | |
3 | swrdf1.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (0...(♯‘𝑊))) | |
4 | swrdrn3 32697 | . . . 4 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → ran (𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑊 “ (𝑀..^𝑁))) | |
5 | 1, 2, 3, 4 | syl3anc 1368 | . . 3 ⊢ (𝜑 → ran (𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑊 “ (𝑀..^𝑁))) |
6 | elfzuz 13537 | . . . . . 6 ⊢ (𝑁 ∈ (0...(♯‘𝑊)) → 𝑁 ∈ (ℤ≥‘0)) | |
7 | fzss1 13580 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘0) → (𝑁...𝑃) ⊆ (0...𝑃)) | |
8 | 3, 6, 7 | 3syl 18 | . . . . 5 ⊢ (𝜑 → (𝑁...𝑃) ⊆ (0...𝑃)) |
9 | swrdrndisj.1 | . . . . 5 ⊢ (𝜑 → 𝑂 ∈ (𝑁...𝑃)) | |
10 | 8, 9 | sseldd 3983 | . . . 4 ⊢ (𝜑 → 𝑂 ∈ (0...𝑃)) |
11 | fzss1 13580 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘0) → (𝑁...(♯‘𝑊)) ⊆ (0...(♯‘𝑊))) | |
12 | 3, 6, 11 | 3syl 18 | . . . . 5 ⊢ (𝜑 → (𝑁...(♯‘𝑊)) ⊆ (0...(♯‘𝑊))) |
13 | swrdrndisj.2 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ (𝑁...(♯‘𝑊))) | |
14 | 12, 13 | sseldd 3983 | . . . 4 ⊢ (𝜑 → 𝑃 ∈ (0...(♯‘𝑊))) |
15 | swrdrn3 32697 | . . . 4 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝑂 ∈ (0...𝑃) ∧ 𝑃 ∈ (0...(♯‘𝑊))) → ran (𝑊 substr ⟨𝑂, 𝑃⟩) = (𝑊 “ (𝑂..^𝑃))) | |
16 | 1, 10, 14, 15 | syl3anc 1368 | . . 3 ⊢ (𝜑 → ran (𝑊 substr ⟨𝑂, 𝑃⟩) = (𝑊 “ (𝑂..^𝑃))) |
17 | 5, 16 | ineq12d 4215 | . 2 ⊢ (𝜑 → (ran (𝑊 substr ⟨𝑀, 𝑁⟩) ∩ ran (𝑊 substr ⟨𝑂, 𝑃⟩)) = ((𝑊 “ (𝑀..^𝑁)) ∩ (𝑊 “ (𝑂..^𝑃)))) |
18 | swrdf1.1 | . . 3 ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷) | |
19 | df-f1 6558 | . . . 4 ⊢ (𝑊:dom 𝑊–1-1→𝐷 ↔ (𝑊:dom 𝑊⟶𝐷 ∧ Fun ◡𝑊)) | |
20 | 19 | simprbi 495 | . . 3 ⊢ (𝑊:dom 𝑊–1-1→𝐷 → Fun ◡𝑊) |
21 | imain 6643 | . . 3 ⊢ (Fun ◡𝑊 → (𝑊 “ ((𝑀..^𝑁) ∩ (𝑂..^𝑃))) = ((𝑊 “ (𝑀..^𝑁)) ∩ (𝑊 “ (𝑂..^𝑃)))) | |
22 | 18, 20, 21 | 3syl 18 | . 2 ⊢ (𝜑 → (𝑊 “ ((𝑀..^𝑁) ∩ (𝑂..^𝑃))) = ((𝑊 “ (𝑀..^𝑁)) ∩ (𝑊 “ (𝑂..^𝑃)))) |
23 | elfzuz 13537 | . . . . . . . . 9 ⊢ (𝑂 ∈ (𝑁...𝑃) → 𝑂 ∈ (ℤ≥‘𝑁)) | |
24 | fzoss1 13699 | . . . . . . . . 9 ⊢ (𝑂 ∈ (ℤ≥‘𝑁) → (𝑂..^𝑃) ⊆ (𝑁..^𝑃)) | |
25 | 9, 23, 24 | 3syl 18 | . . . . . . . 8 ⊢ (𝜑 → (𝑂..^𝑃) ⊆ (𝑁..^𝑃)) |
26 | elfzuz3 13538 | . . . . . . . . 9 ⊢ (𝑃 ∈ (𝑁...(♯‘𝑊)) → (♯‘𝑊) ∈ (ℤ≥‘𝑃)) | |
27 | fzoss2 13700 | . . . . . . . . 9 ⊢ ((♯‘𝑊) ∈ (ℤ≥‘𝑃) → (𝑁..^𝑃) ⊆ (𝑁..^(♯‘𝑊))) | |
28 | 13, 26, 27 | 3syl 18 | . . . . . . . 8 ⊢ (𝜑 → (𝑁..^𝑃) ⊆ (𝑁..^(♯‘𝑊))) |
29 | 25, 28 | sstrd 3992 | . . . . . . 7 ⊢ (𝜑 → (𝑂..^𝑃) ⊆ (𝑁..^(♯‘𝑊))) |
30 | sslin 4237 | . . . . . . 7 ⊢ ((𝑂..^𝑃) ⊆ (𝑁..^(♯‘𝑊)) → ((𝑀..^𝑁) ∩ (𝑂..^𝑃)) ⊆ ((𝑀..^𝑁) ∩ (𝑁..^(♯‘𝑊)))) | |
31 | 29, 30 | syl 17 | . . . . . 6 ⊢ (𝜑 → ((𝑀..^𝑁) ∩ (𝑂..^𝑃)) ⊆ ((𝑀..^𝑁) ∩ (𝑁..^(♯‘𝑊)))) |
32 | fzodisj 13706 | . . . . . 6 ⊢ ((𝑀..^𝑁) ∩ (𝑁..^(♯‘𝑊))) = ∅ | |
33 | 31, 32 | sseqtrdi 4032 | . . . . 5 ⊢ (𝜑 → ((𝑀..^𝑁) ∩ (𝑂..^𝑃)) ⊆ ∅) |
34 | ss0 4402 | . . . . 5 ⊢ (((𝑀..^𝑁) ∩ (𝑂..^𝑃)) ⊆ ∅ → ((𝑀..^𝑁) ∩ (𝑂..^𝑃)) = ∅) | |
35 | 33, 34 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝑀..^𝑁) ∩ (𝑂..^𝑃)) = ∅) |
36 | 35 | imaeq2d 6068 | . . 3 ⊢ (𝜑 → (𝑊 “ ((𝑀..^𝑁) ∩ (𝑂..^𝑃))) = (𝑊 “ ∅)) |
37 | ima0 6085 | . . 3 ⊢ (𝑊 “ ∅) = ∅ | |
38 | 36, 37 | eqtrdi 2784 | . 2 ⊢ (𝜑 → (𝑊 “ ((𝑀..^𝑁) ∩ (𝑂..^𝑃))) = ∅) |
39 | 17, 22, 38 | 3eqtr2d 2774 | 1 ⊢ (𝜑 → (ran (𝑊 substr ⟨𝑀, 𝑁⟩) ∩ ran (𝑊 substr ⟨𝑂, 𝑃⟩)) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∩ cin 3948 ⊆ wss 3949 ∅c0 4326 ⟨cop 4638 ◡ccnv 5681 dom cdm 5682 ran crn 5683 “ cima 5685 Fun wfun 6547 ⟶wf 6549 –1-1→wf1 6550 ‘cfv 6553 (class class class)co 7426 0cc0 11146 ℤ≥cuz 12860 ...cfz 13524 ..^cfzo 13667 ♯chash 14329 Word cword 14504 substr csubstr 14630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-fzo 13668 df-hash 14330 df-word 14505 df-substr 14631 |
This theorem is referenced by: cycpmco2f1 32866 |
Copyright terms: Public domain | W3C validator |