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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 29875 | . . 3 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) |
| 3 | 2 | adantl 481 | . 2 ⊢ ((𝐵 ∈ 𝑈 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) |
| 4 | simprl 771 | . . 3 ⊢ (((𝐵 ∈ 𝑈 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐴 ∈ 𝑉) | |
| 5 | wwlknon 29877 | . . . . . 6 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ (〈“𝐴𝐵𝐶”〉 ∈ (2 WWalksN 𝐺) ∧ (〈“𝐴𝐵𝐶”〉‘0) = 𝐴 ∧ (〈“𝐴𝐵𝐶”〉‘2) = 𝐶)) | |
| 6 | wwlknbp1 29864 | . . . . . . . 8 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ (2 WWalksN 𝐺) → (2 ∈ ℕ0 ∧ 〈“𝐴𝐵𝐶”〉 ∈ Word (Vtx‘𝐺) ∧ (♯‘〈“𝐴𝐵𝐶”〉) = (2 + 1))) | |
| 7 | s3fv1 14931 | . . . . . . . . . . . . 13 ⊢ (𝐵 ∈ 𝑈 → (〈“𝐴𝐵𝐶”〉‘1) = 𝐵) | |
| 8 | 7 | eqcomd 2743 | . . . . . . . . . . . 12 ⊢ (𝐵 ∈ 𝑈 → 𝐵 = (〈“𝐴𝐵𝐶”〉‘1)) |
| 9 | 8 | adantl 481 | . . . . . . . . . . 11 ⊢ ((〈“𝐴𝐵𝐶”〉 ∈ Word (Vtx‘𝐺) ∧ 𝐵 ∈ 𝑈) → 𝐵 = (〈“𝐴𝐵𝐶”〉‘1)) |
| 10 | 1 | eqcomi 2746 | . . . . . . . . . . . . . . . 16 ⊢ (Vtx‘𝐺) = 𝑉 |
| 11 | 10 | wrdeqi 14575 | . . . . . . . . . . . . . . 15 ⊢ Word (Vtx‘𝐺) = Word 𝑉 |
| 12 | 11 | eleq2i 2833 | . . . . . . . . . . . . . 14 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ Word (Vtx‘𝐺) ↔ 〈“𝐴𝐵𝐶”〉 ∈ Word 𝑉) |
| 13 | 12 | biimpi 216 | . . . . . . . . . . . . 13 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ Word (Vtx‘𝐺) → 〈“𝐴𝐵𝐶”〉 ∈ Word 𝑉) |
| 14 | 1ex 11257 | . . . . . . . . . . . . . . 15 ⊢ 1 ∈ V | |
| 15 | 14 | tpid2 4770 | . . . . . . . . . . . . . 14 ⊢ 1 ∈ {0, 1, 2} |
| 16 | s3len 14933 | . . . . . . . . . . . . . . . 16 ⊢ (♯‘〈“𝐴𝐵𝐶”〉) = 3 | |
| 17 | 16 | oveq2i 7442 | . . . . . . . . . . . . . . 15 ⊢ (0..^(♯‘〈“𝐴𝐵𝐶”〉)) = (0..^3) |
| 18 | fzo0to3tp 13791 | . . . . . . . . . . . . . . 15 ⊢ (0..^3) = {0, 1, 2} | |
| 19 | 17, 18 | eqtri 2765 | . . . . . . . . . . . . . 14 ⊢ (0..^(♯‘〈“𝐴𝐵𝐶”〉)) = {0, 1, 2} |
| 20 | 15, 19 | eleqtrri 2840 | . . . . . . . . . . . . 13 ⊢ 1 ∈ (0..^(♯‘〈“𝐴𝐵𝐶”〉)) |
| 21 | wrdsymbcl 14565 | . . . . . . . . . . . . 13 ⊢ ((〈“𝐴𝐵𝐶”〉 ∈ Word 𝑉 ∧ 1 ∈ (0..^(♯‘〈“𝐴𝐵𝐶”〉))) → (〈“𝐴𝐵𝐶”〉‘1) ∈ 𝑉) | |
| 22 | 13, 20, 21 | sylancl 586 | . . . . . . . . . . . 12 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ Word (Vtx‘𝐺) → (〈“𝐴𝐵𝐶”〉‘1) ∈ 𝑉) |
| 23 | 22 | adantr 480 | . . . . . . . . . . 11 ⊢ ((〈“𝐴𝐵𝐶”〉 ∈ Word (Vtx‘𝐺) ∧ 𝐵 ∈ 𝑈) → (〈“𝐴𝐵𝐶”〉‘1) ∈ 𝑉) |
| 24 | 9, 23 | eqeltrd 2841 | . . . . . . . . . 10 ⊢ ((〈“𝐴𝐵𝐶”〉 ∈ Word (Vtx‘𝐺) ∧ 𝐵 ∈ 𝑈) → 𝐵 ∈ 𝑉) |
| 25 | 24 | ex 412 | . . . . . . . . 9 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ Word (Vtx‘𝐺) → (𝐵 ∈ 𝑈 → 𝐵 ∈ 𝑉)) |
| 26 | 25 | 3ad2ant2 1135 | . . . . . . . 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 1134 | . . . . . 6 ⊢ ((〈“𝐴𝐵𝐶”〉 ∈ (2 WWalksN 𝐺) ∧ (〈“𝐴𝐵𝐶”〉‘0) = 𝐴 ∧ (〈“𝐴𝐵𝐶”〉‘2) = 𝐶) → (𝐵 ∈ 𝑈 → 𝐵 ∈ 𝑉)) |
| 29 | 5, 28 | sylbi 217 | . . . . 5 ⊢ (〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝐵 ∈ 𝑈 → 𝐵 ∈ 𝑉)) |
| 30 | 29 | impcom 407 | . . . 4 ⊢ ((𝐵 ∈ 𝑈 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → 𝐵 ∈ 𝑉) |
| 31 | 30 | adantr 480 | . . 3 ⊢ (((𝐵 ∈ 𝑈 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐵 ∈ 𝑉) |
| 32 | simprr 773 | . . 3 ⊢ (((𝐵 ∈ 𝑈 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐶 ∈ 𝑉) | |
| 33 | 4, 31, 32 | 3jca 1129 | . 2 ⊢ (((𝐵 ∈ 𝑈 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) |
| 34 | 3, 33 | mpdan 687 | 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 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 {ctp 4630 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 2c2 12321 3c3 12322 ℕ0cn0 12526 ..^cfzo 13694 ♯chash 14369 Word cword 14552 〈“cs3 14881 Vtxcvtx 29013 WWalksN cwwlksn 29846 WWalksNOn cwwlksnon 29847 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-concat 14609 df-s1 14634 df-s2 14887 df-s3 14888 df-wwlks 29850 df-wwlksn 29851 df-wwlksnon 29852 |
| This theorem is referenced by: frgr2wwlkeqm 30350 |
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