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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 27633 | . . 3 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) |
3 | 2 | adantl 484 | . 2 ⊢ ((𝐵 ∈ 𝑈 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) |
4 | simprl 769 | . . 3 ⊢ (((𝐵 ∈ 𝑈 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐴 ∈ 𝑉) | |
5 | wwlknon 27635 | . . . . . 6 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ (〈“𝐴𝐵𝐶”〉 ∈ (2 WWalksN 𝐺) ∧ (〈“𝐴𝐵𝐶”〉‘0) = 𝐴 ∧ (〈“𝐴𝐵𝐶”〉‘2) = 𝐶)) | |
6 | wwlknbp1 27622 | . . . . . . . 8 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ (2 WWalksN 𝐺) → (2 ∈ ℕ0 ∧ 〈“𝐴𝐵𝐶”〉 ∈ Word (Vtx‘𝐺) ∧ (♯‘〈“𝐴𝐵𝐶”〉) = (2 + 1))) | |
7 | s3fv1 14254 | . . . . . . . . . . . . 13 ⊢ (𝐵 ∈ 𝑈 → (〈“𝐴𝐵𝐶”〉‘1) = 𝐵) | |
8 | 7 | eqcomd 2827 | . . . . . . . . . . . 12 ⊢ (𝐵 ∈ 𝑈 → 𝐵 = (〈“𝐴𝐵𝐶”〉‘1)) |
9 | 8 | adantl 484 | . . . . . . . . . . 11 ⊢ ((〈“𝐴𝐵𝐶”〉 ∈ Word (Vtx‘𝐺) ∧ 𝐵 ∈ 𝑈) → 𝐵 = (〈“𝐴𝐵𝐶”〉‘1)) |
10 | 1 | eqcomi 2830 | . . . . . . . . . . . . . . . 16 ⊢ (Vtx‘𝐺) = 𝑉 |
11 | 10 | wrdeqi 13887 | . . . . . . . . . . . . . . 15 ⊢ Word (Vtx‘𝐺) = Word 𝑉 |
12 | 11 | eleq2i 2904 | . . . . . . . . . . . . . 14 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ Word (Vtx‘𝐺) ↔ 〈“𝐴𝐵𝐶”〉 ∈ Word 𝑉) |
13 | 12 | biimpi 218 | . . . . . . . . . . . . 13 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ Word (Vtx‘𝐺) → 〈“𝐴𝐵𝐶”〉 ∈ Word 𝑉) |
14 | 1ex 10637 | . . . . . . . . . . . . . . 15 ⊢ 1 ∈ V | |
15 | 14 | tpid2 4706 | . . . . . . . . . . . . . 14 ⊢ 1 ∈ {0, 1, 2} |
16 | s3len 14256 | . . . . . . . . . . . . . . . 16 ⊢ (♯‘〈“𝐴𝐵𝐶”〉) = 3 | |
17 | 16 | oveq2i 7167 | . . . . . . . . . . . . . . 15 ⊢ (0..^(♯‘〈“𝐴𝐵𝐶”〉)) = (0..^3) |
18 | fzo0to3tp 13124 | . . . . . . . . . . . . . . 15 ⊢ (0..^3) = {0, 1, 2} | |
19 | 17, 18 | eqtri 2844 | . . . . . . . . . . . . . 14 ⊢ (0..^(♯‘〈“𝐴𝐵𝐶”〉)) = {0, 1, 2} |
20 | 15, 19 | eleqtrri 2912 | . . . . . . . . . . . . 13 ⊢ 1 ∈ (0..^(♯‘〈“𝐴𝐵𝐶”〉)) |
21 | wrdsymbcl 13876 | . . . . . . . . . . . . 13 ⊢ ((〈“𝐴𝐵𝐶”〉 ∈ Word 𝑉 ∧ 1 ∈ (0..^(♯‘〈“𝐴𝐵𝐶”〉))) → (〈“𝐴𝐵𝐶”〉‘1) ∈ 𝑉) | |
22 | 13, 20, 21 | sylancl 588 | . . . . . . . . . . . 12 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ Word (Vtx‘𝐺) → (〈“𝐴𝐵𝐶”〉‘1) ∈ 𝑉) |
23 | 22 | adantr 483 | . . . . . . . . . . 11 ⊢ ((〈“𝐴𝐵𝐶”〉 ∈ Word (Vtx‘𝐺) ∧ 𝐵 ∈ 𝑈) → (〈“𝐴𝐵𝐶”〉‘1) ∈ 𝑉) |
24 | 9, 23 | eqeltrd 2913 | . . . . . . . . . 10 ⊢ ((〈“𝐴𝐵𝐶”〉 ∈ Word (Vtx‘𝐺) ∧ 𝐵 ∈ 𝑈) → 𝐵 ∈ 𝑉) |
25 | 24 | ex 415 | . . . . . . . . 9 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ Word (Vtx‘𝐺) → (𝐵 ∈ 𝑈 → 𝐵 ∈ 𝑉)) |
26 | 25 | 3ad2ant2 1130 | . . . . . . . 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 1129 | . . . . . 6 ⊢ ((〈“𝐴𝐵𝐶”〉 ∈ (2 WWalksN 𝐺) ∧ (〈“𝐴𝐵𝐶”〉‘0) = 𝐴 ∧ (〈“𝐴𝐵𝐶”〉‘2) = 𝐶) → (𝐵 ∈ 𝑈 → 𝐵 ∈ 𝑉)) |
29 | 5, 28 | sylbi 219 | . . . . 5 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝐵 ∈ 𝑈 → 𝐵 ∈ 𝑉)) |
30 | 29 | impcom 410 | . . . 4 ⊢ ((𝐵 ∈ 𝑈 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → 𝐵 ∈ 𝑉) |
31 | 30 | adantr 483 | . . 3 ⊢ (((𝐵 ∈ 𝑈 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐵 ∈ 𝑉) |
32 | simprr 771 | . . 3 ⊢ (((𝐵 ∈ 𝑈 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐶 ∈ 𝑉) | |
33 | 4, 31, 32 | 3jca 1124 | . 2 ⊢ (((𝐵 ∈ 𝑈 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) |
34 | 3, 33 | mpdan 685 | 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 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 {ctp 4571 ‘cfv 6355 (class class class)co 7156 0cc0 10537 1c1 10538 + caddc 10540 2c2 11693 3c3 11694 ℕ0cn0 11898 ..^cfzo 13034 ♯chash 13691 Word cword 13862 〈“cs3 14204 Vtxcvtx 26781 WWalksN cwwlksn 27604 WWalksNOn cwwlksnon 27605 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-hash 13692 df-word 13863 df-concat 13923 df-s1 13950 df-s2 14210 df-s3 14211 df-wwlks 27608 df-wwlksn 27609 df-wwlksnon 27610 |
This theorem is referenced by: frgr2wwlkeqm 28110 |
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