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Mirrors > Home > MPE Home > Th. List > s2elclwwlknon2 | Structured version Visualization version GIF version |
Description: Sufficient conditions of a doubleton word to represent a closed walk on vertex 𝑋 of length 2. (Contributed by AV, 11-May-2022.) |
Ref | Expression |
---|---|
clwwlknon2.c | ⊢ 𝐶 = (ClWWalksNOn‘𝐺) |
clwwlknon2x.v | ⊢ 𝑉 = (Vtx‘𝐺) |
clwwlknon2x.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
s2elclwwlknon2 | ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ {𝑋, 𝑌} ∈ 𝐸) → 〈“𝑋𝑌”〉 ∈ (𝑋𝐶2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | s2cl 14239 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 〈“𝑋𝑌”〉 ∈ Word 𝑉) | |
2 | 1 | 3adant3 1128 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ {𝑋, 𝑌} ∈ 𝐸) → 〈“𝑋𝑌”〉 ∈ Word 𝑉) |
3 | s2len 14250 | . . . 4 ⊢ (♯‘〈“𝑋𝑌”〉) = 2 | |
4 | 3 | a1i 11 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ {𝑋, 𝑌} ∈ 𝐸) → (♯‘〈“𝑋𝑌”〉) = 2) |
5 | s2fv0 14248 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → (〈“𝑋𝑌”〉‘0) = 𝑋) | |
6 | 5 | adantr 483 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (〈“𝑋𝑌”〉‘0) = 𝑋) |
7 | s2fv1 14249 | . . . . . . . 8 ⊢ (𝑌 ∈ 𝑉 → (〈“𝑋𝑌”〉‘1) = 𝑌) | |
8 | 7 | adantl 484 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (〈“𝑋𝑌”〉‘1) = 𝑌) |
9 | 6, 8 | preq12d 4676 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {(〈“𝑋𝑌”〉‘0), (〈“𝑋𝑌”〉‘1)} = {𝑋, 𝑌}) |
10 | 9 | eqcomd 2827 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {𝑋, 𝑌} = {(〈“𝑋𝑌”〉‘0), (〈“𝑋𝑌”〉‘1)}) |
11 | 10 | eleq1d 2897 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ({𝑋, 𝑌} ∈ 𝐸 ↔ {(〈“𝑋𝑌”〉‘0), (〈“𝑋𝑌”〉‘1)} ∈ 𝐸)) |
12 | 11 | biimp3a 1465 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ {𝑋, 𝑌} ∈ 𝐸) → {(〈“𝑋𝑌”〉‘0), (〈“𝑋𝑌”〉‘1)} ∈ 𝐸) |
13 | 6 | 3adant3 1128 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ {𝑋, 𝑌} ∈ 𝐸) → (〈“𝑋𝑌”〉‘0) = 𝑋) |
14 | 4, 12, 13 | 3jca 1124 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ {𝑋, 𝑌} ∈ 𝐸) → ((♯‘〈“𝑋𝑌”〉) = 2 ∧ {(〈“𝑋𝑌”〉‘0), (〈“𝑋𝑌”〉‘1)} ∈ 𝐸 ∧ (〈“𝑋𝑌”〉‘0) = 𝑋)) |
15 | fveqeq2 6678 | . . . 4 ⊢ (𝑤 = 〈“𝑋𝑌”〉 → ((♯‘𝑤) = 2 ↔ (♯‘〈“𝑋𝑌”〉) = 2)) | |
16 | fveq1 6668 | . . . . . 6 ⊢ (𝑤 = 〈“𝑋𝑌”〉 → (𝑤‘0) = (〈“𝑋𝑌”〉‘0)) | |
17 | fveq1 6668 | . . . . . 6 ⊢ (𝑤 = 〈“𝑋𝑌”〉 → (𝑤‘1) = (〈“𝑋𝑌”〉‘1)) | |
18 | 16, 17 | preq12d 4676 | . . . . 5 ⊢ (𝑤 = 〈“𝑋𝑌”〉 → {(𝑤‘0), (𝑤‘1)} = {(〈“𝑋𝑌”〉‘0), (〈“𝑋𝑌”〉‘1)}) |
19 | 18 | eleq1d 2897 | . . . 4 ⊢ (𝑤 = 〈“𝑋𝑌”〉 → ({(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ↔ {(〈“𝑋𝑌”〉‘0), (〈“𝑋𝑌”〉‘1)} ∈ 𝐸)) |
20 | 16 | eqeq1d 2823 | . . . 4 ⊢ (𝑤 = 〈“𝑋𝑌”〉 → ((𝑤‘0) = 𝑋 ↔ (〈“𝑋𝑌”〉‘0) = 𝑋)) |
21 | 15, 19, 20 | 3anbi123d 1432 | . . 3 ⊢ (𝑤 = 〈“𝑋𝑌”〉 → (((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋) ↔ ((♯‘〈“𝑋𝑌”〉) = 2 ∧ {(〈“𝑋𝑌”〉‘0), (〈“𝑋𝑌”〉‘1)} ∈ 𝐸 ∧ (〈“𝑋𝑌”〉‘0) = 𝑋))) |
22 | clwwlknon2.c | . . . 4 ⊢ 𝐶 = (ClWWalksNOn‘𝐺) | |
23 | clwwlknon2x.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
24 | clwwlknon2x.e | . . . 4 ⊢ 𝐸 = (Edg‘𝐺) | |
25 | 22, 23, 24 | clwwlknon2x 27881 | . . 3 ⊢ (𝑋𝐶2) = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋)} |
26 | 21, 25 | elrab2 3682 | . 2 ⊢ (〈“𝑋𝑌”〉 ∈ (𝑋𝐶2) ↔ (〈“𝑋𝑌”〉 ∈ Word 𝑉 ∧ ((♯‘〈“𝑋𝑌”〉) = 2 ∧ {(〈“𝑋𝑌”〉‘0), (〈“𝑋𝑌”〉‘1)} ∈ 𝐸 ∧ (〈“𝑋𝑌”〉‘0) = 𝑋))) |
27 | 2, 14, 26 | sylanbrc 585 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ {𝑋, 𝑌} ∈ 𝐸) → 〈“𝑋𝑌”〉 ∈ (𝑋𝐶2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 {cpr 4568 ‘cfv 6354 (class class class)co 7155 0cc0 10536 1c1 10537 2c2 11691 ♯chash 13689 Word cword 13860 〈“cs2 14202 Vtxcvtx 26780 Edgcedg 26831 ClWWalksNOncclwwlknon 27865 |
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 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
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 3496 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 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-n0 11897 df-xnn0 11967 df-z 11981 df-uz 12243 df-fz 12892 df-fzo 13033 df-hash 13690 df-word 13861 df-lsw 13914 df-concat 13922 df-s1 13949 df-s2 14209 df-clwwlk 27759 df-clwwlkn 27802 df-clwwlknon 27866 |
This theorem is referenced by: 2clwwlk2clwwlklem 28124 |
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