Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > elwwlks2s3 | Structured version Visualization version GIF version |
Description: A walk of length 2 as word is a length 3 string. (Contributed by AV, 18-May-2021.) |
Ref | Expression |
---|---|
elwwlks2s3.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
elwwlks2s3 | ⊢ (𝑊 ∈ (2 WWalksN 𝐺) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 𝑊 = 〈“𝑎𝑏𝑐”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wwlknbp1 27616 | . 2 ⊢ (𝑊 ∈ (2 WWalksN 𝐺) → (2 ∈ ℕ0 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (2 + 1))) | |
2 | elwwlks2s3.v | . . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | 2 | wrdeqi 13881 | . . . . . 6 ⊢ Word 𝑉 = Word (Vtx‘𝐺) |
4 | 3 | eleq2i 2904 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑉 ↔ 𝑊 ∈ Word (Vtx‘𝐺)) |
5 | df-3 11695 | . . . . . 6 ⊢ 3 = (2 + 1) | |
6 | 5 | eqeq2i 2834 | . . . . 5 ⊢ ((♯‘𝑊) = 3 ↔ (♯‘𝑊) = (2 + 1)) |
7 | 4, 6 | anbi12i 628 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3) ↔ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (2 + 1))) |
8 | wrdl3s3 14320 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 𝑊 = 〈“𝑎𝑏𝑐”〉) | |
9 | 7, 8 | sylbb1 239 | . . 3 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (2 + 1)) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 𝑊 = 〈“𝑎𝑏𝑐”〉) |
10 | 9 | 3adant1 1126 | . 2 ⊢ ((2 ∈ ℕ0 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (2 + 1)) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 𝑊 = 〈“𝑎𝑏𝑐”〉) |
11 | 1, 10 | syl 17 | 1 ⊢ (𝑊 ∈ (2 WWalksN 𝐺) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 𝑊 = 〈“𝑎𝑏𝑐”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 ‘cfv 6350 (class class class)co 7150 1c1 10532 + caddc 10534 2c2 11686 3c3 11687 ℕ0cn0 11891 ♯chash 13684 Word cword 13855 〈“cs3 14198 Vtxcvtx 26775 WWalksN cwwlksn 27598 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3497 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 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-hash 13685 df-word 13856 df-concat 13917 df-s1 13944 df-s2 14204 df-s3 14205 df-wwlks 27602 df-wwlksn 27603 |
This theorem is referenced by: midwwlks2s3 27725 |
Copyright terms: Public domain | W3C validator |