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Mathbox for Thierry Arnoux |
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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 31865 | . . . 4 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → ran (𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑊 “ (𝑀..^𝑁))) | |
5 | 1, 2, 3, 4 | syl3anc 1372 | . . 3 ⊢ (𝜑 → ran (𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑊 “ (𝑀..^𝑁))) |
6 | elfzuz 13446 | . . . . . 6 ⊢ (𝑁 ∈ (0...(♯‘𝑊)) → 𝑁 ∈ (ℤ≥‘0)) | |
7 | fzss1 13489 | . . . . . 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 3949 | . . . 4 ⊢ (𝜑 → 𝑂 ∈ (0...𝑃)) |
11 | fzss1 13489 | . . . . . 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 3949 | . . . 4 ⊢ (𝜑 → 𝑃 ∈ (0...(♯‘𝑊))) |
15 | swrdrn3 31865 | . . . 4 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝑂 ∈ (0...𝑃) ∧ 𝑃 ∈ (0...(♯‘𝑊))) → ran (𝑊 substr ⟨𝑂, 𝑃⟩) = (𝑊 “ (𝑂..^𝑃))) | |
16 | 1, 10, 14, 15 | syl3anc 1372 | . . 3 ⊢ (𝜑 → ran (𝑊 substr ⟨𝑂, 𝑃⟩) = (𝑊 “ (𝑂..^𝑃))) |
17 | 5, 16 | ineq12d 4177 | . 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 6505 | . . . 4 ⊢ (𝑊:dom 𝑊–1-1→𝐷 ↔ (𝑊:dom 𝑊⟶𝐷 ∧ Fun ◡𝑊)) | |
20 | 19 | simprbi 498 | . . 3 ⊢ (𝑊:dom 𝑊–1-1→𝐷 → Fun ◡𝑊) |
21 | imain 6590 | . . 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 13446 | . . . . . . . . 9 ⊢ (𝑂 ∈ (𝑁...𝑃) → 𝑂 ∈ (ℤ≥‘𝑁)) | |
24 | fzoss1 13608 | . . . . . . . . 9 ⊢ (𝑂 ∈ (ℤ≥‘𝑁) → (𝑂..^𝑃) ⊆ (𝑁..^𝑃)) | |
25 | 9, 23, 24 | 3syl 18 | . . . . . . . 8 ⊢ (𝜑 → (𝑂..^𝑃) ⊆ (𝑁..^𝑃)) |
26 | elfzuz3 13447 | . . . . . . . . 9 ⊢ (𝑃 ∈ (𝑁...(♯‘𝑊)) → (♯‘𝑊) ∈ (ℤ≥‘𝑃)) | |
27 | fzoss2 13609 | . . . . . . . . 9 ⊢ ((♯‘𝑊) ∈ (ℤ≥‘𝑃) → (𝑁..^𝑃) ⊆ (𝑁..^(♯‘𝑊))) | |
28 | 13, 26, 27 | 3syl 18 | . . . . . . . 8 ⊢ (𝜑 → (𝑁..^𝑃) ⊆ (𝑁..^(♯‘𝑊))) |
29 | 25, 28 | sstrd 3958 | . . . . . . 7 ⊢ (𝜑 → (𝑂..^𝑃) ⊆ (𝑁..^(♯‘𝑊))) |
30 | sslin 4198 | . . . . . . 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 13615 | . . . . . 6 ⊢ ((𝑀..^𝑁) ∩ (𝑁..^(♯‘𝑊))) = ∅ | |
33 | 31, 32 | sseqtrdi 3998 | . . . . 5 ⊢ (𝜑 → ((𝑀..^𝑁) ∩ (𝑂..^𝑃)) ⊆ ∅) |
34 | ss0 4362 | . . . . 5 ⊢ (((𝑀..^𝑁) ∩ (𝑂..^𝑃)) ⊆ ∅ → ((𝑀..^𝑁) ∩ (𝑂..^𝑃)) = ∅) | |
35 | 33, 34 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝑀..^𝑁) ∩ (𝑂..^𝑃)) = ∅) |
36 | 35 | imaeq2d 6017 | . . 3 ⊢ (𝜑 → (𝑊 “ ((𝑀..^𝑁) ∩ (𝑂..^𝑃))) = (𝑊 “ ∅)) |
37 | ima0 6033 | . . 3 ⊢ (𝑊 “ ∅) = ∅ | |
38 | 36, 37 | eqtrdi 2789 | . 2 ⊢ (𝜑 → (𝑊 “ ((𝑀..^𝑁) ∩ (𝑂..^𝑃))) = ∅) |
39 | 17, 22, 38 | 3eqtr2d 2779 | 1 ⊢ (𝜑 → (ran (𝑊 substr ⟨𝑀, 𝑁⟩) ∩ ran (𝑊 substr ⟨𝑂, 𝑃⟩)) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∩ cin 3913 ⊆ wss 3914 ∅c0 4286 ⟨cop 4596 ◡ccnv 5636 dom cdm 5637 ran crn 5638 “ cima 5640 Fun wfun 6494 ⟶wf 6496 –1-1→wf1 6497 ‘cfv 6500 (class class class)co 7361 0cc0 11059 ℤ≥cuz 12771 ...cfz 13433 ..^cfzo 13576 ♯chash 14239 Word cword 14411 substr csubstr 14537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 df-fzo 13577 df-hash 14240 df-word 14412 df-substr 14538 |
This theorem is referenced by: cycpmco2f1 32029 |
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