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Mirrors > Home > MPE Home > Th. List > wwlks2onv | Structured version Visualization version GIF version |
Description: If a length 3 string represents a walk of length 2, its components are vertices. (Contributed by Alexander van der Vekens, 19-Feb-2018.) (Proof shortened by AV, 14-Mar-2022.) |
Ref | Expression |
---|---|
wwlks2onv.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
wwlks2onv | ⊢ ((𝐵 ∈ 𝑈 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wwlks2onv.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 1 | wwlksonvtx 27221 | . . 3 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) |
3 | 2 | adantl 475 | . 2 ⊢ ((𝐵 ∈ 𝑈 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) |
4 | simprl 761 | . . 3 ⊢ (((𝐵 ∈ 𝑈 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐴 ∈ 𝑉) | |
5 | wwlknon 27223 | . . . . . 6 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ (〈“𝐴𝐵𝐶”〉 ∈ (2 WWalksN 𝐺) ∧ (〈“𝐴𝐵𝐶”〉‘0) = 𝐴 ∧ (〈“𝐴𝐵𝐶”〉‘2) = 𝐶)) | |
6 | wwlknbp1 27210 | . . . . . . . 8 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ (2 WWalksN 𝐺) → (2 ∈ ℕ0 ∧ 〈“𝐴𝐵𝐶”〉 ∈ Word (Vtx‘𝐺) ∧ (♯‘〈“𝐴𝐵𝐶”〉) = (2 + 1))) | |
7 | s3fv1 14049 | . . . . . . . . . . . . 13 ⊢ (𝐵 ∈ 𝑈 → (〈“𝐴𝐵𝐶”〉‘1) = 𝐵) | |
8 | 7 | eqcomd 2784 | . . . . . . . . . . . 12 ⊢ (𝐵 ∈ 𝑈 → 𝐵 = (〈“𝐴𝐵𝐶”〉‘1)) |
9 | 8 | adantl 475 | . . . . . . . . . . 11 ⊢ ((〈“𝐴𝐵𝐶”〉 ∈ Word (Vtx‘𝐺) ∧ 𝐵 ∈ 𝑈) → 𝐵 = (〈“𝐴𝐵𝐶”〉‘1)) |
10 | 1 | eqcomi 2787 | . . . . . . . . . . . . . . . 16 ⊢ (Vtx‘𝐺) = 𝑉 |
11 | 10 | wrdeqi 13631 | . . . . . . . . . . . . . . 15 ⊢ Word (Vtx‘𝐺) = Word 𝑉 |
12 | 11 | eleq2i 2851 | . . . . . . . . . . . . . 14 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ Word (Vtx‘𝐺) ↔ 〈“𝐴𝐵𝐶”〉 ∈ Word 𝑉) |
13 | 12 | biimpi 208 | . . . . . . . . . . . . 13 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ Word (Vtx‘𝐺) → 〈“𝐴𝐵𝐶”〉 ∈ Word 𝑉) |
14 | 1ex 10374 | . . . . . . . . . . . . . . 15 ⊢ 1 ∈ V | |
15 | 14 | tpid2 4537 | . . . . . . . . . . . . . 14 ⊢ 1 ∈ {0, 1, 2} |
16 | s3len 14051 | . . . . . . . . . . . . . . . 16 ⊢ (♯‘〈“𝐴𝐵𝐶”〉) = 3 | |
17 | 16 | oveq2i 6935 | . . . . . . . . . . . . . . 15 ⊢ (0..^(♯‘〈“𝐴𝐵𝐶”〉)) = (0..^3) |
18 | fzo0to3tp 12878 | . . . . . . . . . . . . . . 15 ⊢ (0..^3) = {0, 1, 2} | |
19 | 17, 18 | eqtri 2802 | . . . . . . . . . . . . . 14 ⊢ (0..^(♯‘〈“𝐴𝐵𝐶”〉)) = {0, 1, 2} |
20 | 15, 19 | eleqtrri 2858 | . . . . . . . . . . . . 13 ⊢ 1 ∈ (0..^(♯‘〈“𝐴𝐵𝐶”〉)) |
21 | wrdsymbcl 13619 | . . . . . . . . . . . . 13 ⊢ ((〈“𝐴𝐵𝐶”〉 ∈ Word 𝑉 ∧ 1 ∈ (0..^(♯‘〈“𝐴𝐵𝐶”〉))) → (〈“𝐴𝐵𝐶”〉‘1) ∈ 𝑉) | |
22 | 13, 20, 21 | sylancl 580 | . . . . . . . . . . . 12 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ Word (Vtx‘𝐺) → (〈“𝐴𝐵𝐶”〉‘1) ∈ 𝑉) |
23 | 22 | adantr 474 | . . . . . . . . . . 11 ⊢ ((〈“𝐴𝐵𝐶”〉 ∈ Word (Vtx‘𝐺) ∧ 𝐵 ∈ 𝑈) → (〈“𝐴𝐵𝐶”〉‘1) ∈ 𝑉) |
24 | 9, 23 | eqeltrd 2859 | . . . . . . . . . 10 ⊢ ((〈“𝐴𝐵𝐶”〉 ∈ Word (Vtx‘𝐺) ∧ 𝐵 ∈ 𝑈) → 𝐵 ∈ 𝑉) |
25 | 24 | ex 403 | . . . . . . . . 9 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ Word (Vtx‘𝐺) → (𝐵 ∈ 𝑈 → 𝐵 ∈ 𝑉)) |
26 | 25 | 3ad2ant2 1125 | . . . . . . . 8 ⊢ ((2 ∈ ℕ0 ∧ 〈“𝐴𝐵𝐶”〉 ∈ Word (Vtx‘𝐺) ∧ (♯‘〈“𝐴𝐵𝐶”〉) = (2 + 1)) → (𝐵 ∈ 𝑈 → 𝐵 ∈ 𝑉)) |
27 | 6, 26 | syl 17 | . . . . . . 7 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ (2 WWalksN 𝐺) → (𝐵 ∈ 𝑈 → 𝐵 ∈ 𝑉)) |
28 | 27 | 3ad2ant1 1124 | . . . . . 6 ⊢ ((〈“𝐴𝐵𝐶”〉 ∈ (2 WWalksN 𝐺) ∧ (〈“𝐴𝐵𝐶”〉‘0) = 𝐴 ∧ (〈“𝐴𝐵𝐶”〉‘2) = 𝐶) → (𝐵 ∈ 𝑈 → 𝐵 ∈ 𝑉)) |
29 | 5, 28 | sylbi 209 | . . . . 5 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝐵 ∈ 𝑈 → 𝐵 ∈ 𝑉)) |
30 | 29 | impcom 398 | . . . 4 ⊢ ((𝐵 ∈ 𝑈 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → 𝐵 ∈ 𝑉) |
31 | 30 | adantr 474 | . . 3 ⊢ (((𝐵 ∈ 𝑈 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐵 ∈ 𝑉) |
32 | simprr 763 | . . 3 ⊢ (((𝐵 ∈ 𝑈 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐶 ∈ 𝑉) | |
33 | 4, 31, 32 | 3jca 1119 | . 2 ⊢ (((𝐵 ∈ 𝑈 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) |
34 | 3, 33 | mpdan 677 | 1 ⊢ ((𝐵 ∈ 𝑈 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 {ctp 4402 ‘cfv 6137 (class class class)co 6924 0cc0 10274 1c1 10275 + caddc 10277 2c2 11435 3c3 11436 ℕ0cn0 11647 ..^cfzo 12789 ♯chash 13441 Word cword 13605 〈“cs3 13999 Vtxcvtx 26361 WWalksN cwwlksn 27192 WWalksNOn cwwlksnon 27193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-map 8144 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-card 9100 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11380 df-2 11443 df-3 11444 df-n0 11648 df-z 11734 df-uz 11998 df-fz 12649 df-fzo 12790 df-hash 13442 df-word 13606 df-concat 13667 df-s1 13692 df-s2 14005 df-s3 14006 df-wwlks 27196 df-wwlksn 27197 df-wwlksnon 27198 |
This theorem is referenced by: frgr2wwlkeqm 27756 |
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