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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 27641 | . . 3 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) |
3 | 2 | adantl 485 | . 2 ⊢ ((𝐵 ∈ 𝑈 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) |
4 | simprl 770 | . . 3 ⊢ (((𝐵 ∈ 𝑈 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐴 ∈ 𝑉) | |
5 | wwlknon 27643 | . . . . . 6 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ (〈“𝐴𝐵𝐶”〉 ∈ (2 WWalksN 𝐺) ∧ (〈“𝐴𝐵𝐶”〉‘0) = 𝐴 ∧ (〈“𝐴𝐵𝐶”〉‘2) = 𝐶)) | |
6 | wwlknbp1 27630 | . . . . . . . 8 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ (2 WWalksN 𝐺) → (2 ∈ ℕ0 ∧ 〈“𝐴𝐵𝐶”〉 ∈ Word (Vtx‘𝐺) ∧ (♯‘〈“𝐴𝐵𝐶”〉) = (2 + 1))) | |
7 | s3fv1 14245 | . . . . . . . . . . . . 13 ⊢ (𝐵 ∈ 𝑈 → (〈“𝐴𝐵𝐶”〉‘1) = 𝐵) | |
8 | 7 | eqcomd 2804 | . . . . . . . . . . . 12 ⊢ (𝐵 ∈ 𝑈 → 𝐵 = (〈“𝐴𝐵𝐶”〉‘1)) |
9 | 8 | adantl 485 | . . . . . . . . . . 11 ⊢ ((〈“𝐴𝐵𝐶”〉 ∈ Word (Vtx‘𝐺) ∧ 𝐵 ∈ 𝑈) → 𝐵 = (〈“𝐴𝐵𝐶”〉‘1)) |
10 | 1 | eqcomi 2807 | . . . . . . . . . . . . . . . 16 ⊢ (Vtx‘𝐺) = 𝑉 |
11 | 10 | wrdeqi 13880 | . . . . . . . . . . . . . . 15 ⊢ Word (Vtx‘𝐺) = Word 𝑉 |
12 | 11 | eleq2i 2881 | . . . . . . . . . . . . . 14 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ Word (Vtx‘𝐺) ↔ 〈“𝐴𝐵𝐶”〉 ∈ Word 𝑉) |
13 | 12 | biimpi 219 | . . . . . . . . . . . . 13 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ Word (Vtx‘𝐺) → 〈“𝐴𝐵𝐶”〉 ∈ Word 𝑉) |
14 | 1ex 10626 | . . . . . . . . . . . . . . 15 ⊢ 1 ∈ V | |
15 | 14 | tpid2 4666 | . . . . . . . . . . . . . 14 ⊢ 1 ∈ {0, 1, 2} |
16 | s3len 14247 | . . . . . . . . . . . . . . . 16 ⊢ (♯‘〈“𝐴𝐵𝐶”〉) = 3 | |
17 | 16 | oveq2i 7146 | . . . . . . . . . . . . . . 15 ⊢ (0..^(♯‘〈“𝐴𝐵𝐶”〉)) = (0..^3) |
18 | fzo0to3tp 13118 | . . . . . . . . . . . . . . 15 ⊢ (0..^3) = {0, 1, 2} | |
19 | 17, 18 | eqtri 2821 | . . . . . . . . . . . . . 14 ⊢ (0..^(♯‘〈“𝐴𝐵𝐶”〉)) = {0, 1, 2} |
20 | 15, 19 | eleqtrri 2889 | . . . . . . . . . . . . 13 ⊢ 1 ∈ (0..^(♯‘〈“𝐴𝐵𝐶”〉)) |
21 | wrdsymbcl 13870 | . . . . . . . . . . . . 13 ⊢ ((〈“𝐴𝐵𝐶”〉 ∈ Word 𝑉 ∧ 1 ∈ (0..^(♯‘〈“𝐴𝐵𝐶”〉))) → (〈“𝐴𝐵𝐶”〉‘1) ∈ 𝑉) | |
22 | 13, 20, 21 | sylancl 589 | . . . . . . . . . . . 12 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ Word (Vtx‘𝐺) → (〈“𝐴𝐵𝐶”〉‘1) ∈ 𝑉) |
23 | 22 | adantr 484 | . . . . . . . . . . 11 ⊢ ((〈“𝐴𝐵𝐶”〉 ∈ Word (Vtx‘𝐺) ∧ 𝐵 ∈ 𝑈) → (〈“𝐴𝐵𝐶”〉‘1) ∈ 𝑉) |
24 | 9, 23 | eqeltrd 2890 | . . . . . . . . . 10 ⊢ ((〈“𝐴𝐵𝐶”〉 ∈ Word (Vtx‘𝐺) ∧ 𝐵 ∈ 𝑈) → 𝐵 ∈ 𝑉) |
25 | 24 | ex 416 | . . . . . . . . 9 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ Word (Vtx‘𝐺) → (𝐵 ∈ 𝑈 → 𝐵 ∈ 𝑉)) |
26 | 25 | 3ad2ant2 1131 | . . . . . . . 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 1130 | . . . . . 6 ⊢ ((〈“𝐴𝐵𝐶”〉 ∈ (2 WWalksN 𝐺) ∧ (〈“𝐴𝐵𝐶”〉‘0) = 𝐴 ∧ (〈“𝐴𝐵𝐶”〉‘2) = 𝐶) → (𝐵 ∈ 𝑈 → 𝐵 ∈ 𝑉)) |
29 | 5, 28 | sylbi 220 | . . . . 5 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝐵 ∈ 𝑈 → 𝐵 ∈ 𝑉)) |
30 | 29 | impcom 411 | . . . 4 ⊢ ((𝐵 ∈ 𝑈 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → 𝐵 ∈ 𝑉) |
31 | 30 | adantr 484 | . . 3 ⊢ (((𝐵 ∈ 𝑈 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐵 ∈ 𝑉) |
32 | simprr 772 | . . 3 ⊢ (((𝐵 ∈ 𝑈 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐶 ∈ 𝑉) | |
33 | 4, 31, 32 | 3jca 1125 | . 2 ⊢ (((𝐵 ∈ 𝑈 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) |
34 | 3, 33 | mpdan 686 | 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 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 {ctp 4529 ‘cfv 6324 (class class class)co 7135 0cc0 10526 1c1 10527 + caddc 10529 2c2 11680 3c3 11681 ℕ0cn0 11885 ..^cfzo 13028 ♯chash 13686 Word cword 13857 〈“cs3 14195 Vtxcvtx 26789 WWalksN cwwlksn 27612 WWalksNOn cwwlksnon 27613 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-hash 13687 df-word 13858 df-concat 13914 df-s1 13941 df-s2 14201 df-s3 14202 df-wwlks 27616 df-wwlksn 27617 df-wwlksnon 27618 |
This theorem is referenced by: frgr2wwlkeqm 28116 |
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