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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 27550 | . 2 ⊢ (𝑊 ∈ (2 WWalksN 𝐺) → (2 ∈ ℕ0 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (2 + 1))) | |
2 | elwwlks2s3.v | . . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | 2 | wrdeqi 13877 | . . . . . 6 ⊢ Word 𝑉 = Word (Vtx‘𝐺) |
4 | 3 | eleq2i 2904 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑉 ↔ 𝑊 ∈ Word (Vtx‘𝐺)) |
5 | df-3 11690 | . . . . . 6 ⊢ 3 = (2 + 1) | |
6 | 5 | eqeq2i 2834 | . . . . 5 ⊢ ((♯‘𝑊) = 3 ↔ (♯‘𝑊) = (2 + 1)) |
7 | 4, 6 | anbi12i 626 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3) ↔ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (2 + 1))) |
8 | wrdl3s3 14316 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 𝑊 = 〈“𝑎𝑏𝑐”〉) | |
9 | 7, 8 | sylbb1 238 | . . 3 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (2 + 1)) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 𝑊 = 〈“𝑎𝑏𝑐”〉) |
10 | 9 | 3adant1 1122 | . 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 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∃wrex 3139 ‘cfv 6349 (class class class)co 7145 1c1 10527 + caddc 10529 2c2 11681 3c3 11682 ℕ0cn0 11886 ♯chash 13680 Word cword 13851 〈“cs3 14194 Vtxcvtx 26709 WWalksN cwwlksn 27532 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 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 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-oadd 8097 df-er 8279 df-map 8398 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-card 9357 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11628 df-2 11689 df-3 11690 df-n0 11887 df-z 11971 df-uz 12233 df-fz 12883 df-fzo 13024 df-hash 13681 df-word 13852 df-concat 13913 df-s1 13940 df-s2 14200 df-s3 14201 df-wwlks 27536 df-wwlksn 27537 |
This theorem is referenced by: midwwlks2s3 27659 |
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