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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 32671 | . . . 4 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → ran (𝑊 substr 〈𝑀, 𝑁〉) = (𝑊 “ (𝑀..^𝑁))) | |
5 | 1, 2, 3, 4 | syl3anc 1369 | . . 3 ⊢ (𝜑 → ran (𝑊 substr 〈𝑀, 𝑁〉) = (𝑊 “ (𝑀..^𝑁))) |
6 | elfzuz 13524 | . . . . . 6 ⊢ (𝑁 ∈ (0...(♯‘𝑊)) → 𝑁 ∈ (ℤ≥‘0)) | |
7 | fzss1 13567 | . . . . . 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 3980 | . . . 4 ⊢ (𝜑 → 𝑂 ∈ (0...𝑃)) |
11 | fzss1 13567 | . . . . . 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 3980 | . . . 4 ⊢ (𝜑 → 𝑃 ∈ (0...(♯‘𝑊))) |
15 | swrdrn3 32671 | . . . 4 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝑂 ∈ (0...𝑃) ∧ 𝑃 ∈ (0...(♯‘𝑊))) → ran (𝑊 substr 〈𝑂, 𝑃〉) = (𝑊 “ (𝑂..^𝑃))) | |
16 | 1, 10, 14, 15 | syl3anc 1369 | . . 3 ⊢ (𝜑 → ran (𝑊 substr 〈𝑂, 𝑃〉) = (𝑊 “ (𝑂..^𝑃))) |
17 | 5, 16 | ineq12d 4210 | . 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 6548 | . . . 4 ⊢ (𝑊:dom 𝑊–1-1→𝐷 ↔ (𝑊:dom 𝑊⟶𝐷 ∧ Fun ◡𝑊)) | |
20 | 19 | simprbi 496 | . . 3 ⊢ (𝑊:dom 𝑊–1-1→𝐷 → Fun ◡𝑊) |
21 | imain 6633 | . . 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 13524 | . . . . . . . . 9 ⊢ (𝑂 ∈ (𝑁...𝑃) → 𝑂 ∈ (ℤ≥‘𝑁)) | |
24 | fzoss1 13686 | . . . . . . . . 9 ⊢ (𝑂 ∈ (ℤ≥‘𝑁) → (𝑂..^𝑃) ⊆ (𝑁..^𝑃)) | |
25 | 9, 23, 24 | 3syl 18 | . . . . . . . 8 ⊢ (𝜑 → (𝑂..^𝑃) ⊆ (𝑁..^𝑃)) |
26 | elfzuz3 13525 | . . . . . . . . 9 ⊢ (𝑃 ∈ (𝑁...(♯‘𝑊)) → (♯‘𝑊) ∈ (ℤ≥‘𝑃)) | |
27 | fzoss2 13687 | . . . . . . . . 9 ⊢ ((♯‘𝑊) ∈ (ℤ≥‘𝑃) → (𝑁..^𝑃) ⊆ (𝑁..^(♯‘𝑊))) | |
28 | 13, 26, 27 | 3syl 18 | . . . . . . . 8 ⊢ (𝜑 → (𝑁..^𝑃) ⊆ (𝑁..^(♯‘𝑊))) |
29 | 25, 28 | sstrd 3989 | . . . . . . 7 ⊢ (𝜑 → (𝑂..^𝑃) ⊆ (𝑁..^(♯‘𝑊))) |
30 | sslin 4231 | . . . . . . 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 13693 | . . . . . 6 ⊢ ((𝑀..^𝑁) ∩ (𝑁..^(♯‘𝑊))) = ∅ | |
33 | 31, 32 | sseqtrdi 4029 | . . . . 5 ⊢ (𝜑 → ((𝑀..^𝑁) ∩ (𝑂..^𝑃)) ⊆ ∅) |
34 | ss0 4395 | . . . . 5 ⊢ (((𝑀..^𝑁) ∩ (𝑂..^𝑃)) ⊆ ∅ → ((𝑀..^𝑁) ∩ (𝑂..^𝑃)) = ∅) | |
35 | 33, 34 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝑀..^𝑁) ∩ (𝑂..^𝑃)) = ∅) |
36 | 35 | imaeq2d 6058 | . . 3 ⊢ (𝜑 → (𝑊 “ ((𝑀..^𝑁) ∩ (𝑂..^𝑃))) = (𝑊 “ ∅)) |
37 | ima0 6075 | . . 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 1534 ∈ wcel 2099 ∩ cin 3944 ⊆ wss 3945 ∅c0 4319 〈cop 4631 ◡ccnv 5672 dom cdm 5673 ran crn 5674 “ cima 5676 Fun wfun 6537 ⟶wf 6539 –1-1→wf1 6540 ‘cfv 6543 (class class class)co 7415 0cc0 11133 ℤ≥cuz 12847 ...cfz 13511 ..^cfzo 13654 ♯chash 14316 Word cword 14491 substr csubstr 14617 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-1st 7988 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-er 8719 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-n0 12498 df-z 12584 df-uz 12848 df-fz 13512 df-fzo 13655 df-hash 14317 df-word 14492 df-substr 14618 |
This theorem is referenced by: cycpmco2f1 32840 |
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