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Mirrors > Home > MPE Home > Th. List > clwwlkn | Structured version Visualization version GIF version |
Description: The set of closed walks of a fixed length 𝑁 as words over the set of vertices in a graph 𝐺. (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.) (Revised by AV, 22-Mar-2022.) |
Ref | Expression |
---|---|
clwwlkn | ⊢ (𝑁 ClWWalksN 𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6896 | . . . . 5 ⊢ (𝑔 = 𝐺 → (ClWWalks‘𝑔) = (ClWWalks‘𝐺)) | |
2 | 1 | adantl 480 | . . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (ClWWalks‘𝑔) = (ClWWalks‘𝐺)) |
3 | eqeq2 2737 | . . . . 5 ⊢ (𝑛 = 𝑁 → ((♯‘𝑤) = 𝑛 ↔ (♯‘𝑤) = 𝑁)) | |
4 | 3 | adantr 479 | . . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → ((♯‘𝑤) = 𝑛 ↔ (♯‘𝑤) = 𝑁)) |
5 | 2, 4 | rabeqbidv 3436 | . . 3 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → {𝑤 ∈ (ClWWalks‘𝑔) ∣ (♯‘𝑤) = 𝑛} = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁}) |
6 | df-clwwlkn 29907 | . . 3 ⊢ ClWWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (ClWWalks‘𝑔) ∣ (♯‘𝑤) = 𝑛}) | |
7 | fvex 6909 | . . . 4 ⊢ (ClWWalks‘𝐺) ∈ V | |
8 | 7 | rabex 5335 | . . 3 ⊢ {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁} ∈ V |
9 | 5, 6, 8 | ovmpoa 7576 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝑁 ClWWalksN 𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁}) |
10 | 6 | mpondm0 7661 | . . 3 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝑁 ClWWalksN 𝐺) = ∅) |
11 | eqid 2725 | . . . . . . . . . . 11 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
12 | 11 | clwwlkbp 29867 | . . . . . . . . . 10 ⊢ (𝑤 ∈ (ClWWalks‘𝐺) → (𝐺 ∈ V ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑤 ≠ ∅)) |
13 | 12 | simp2d 1140 | . . . . . . . . 9 ⊢ (𝑤 ∈ (ClWWalks‘𝐺) → 𝑤 ∈ Word (Vtx‘𝐺)) |
14 | lencl 14519 | . . . . . . . . 9 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) → (♯‘𝑤) ∈ ℕ0) | |
15 | 13, 14 | syl 17 | . . . . . . . 8 ⊢ (𝑤 ∈ (ClWWalks‘𝐺) → (♯‘𝑤) ∈ ℕ0) |
16 | eleq1 2813 | . . . . . . . 8 ⊢ ((♯‘𝑤) = 𝑁 → ((♯‘𝑤) ∈ ℕ0 ↔ 𝑁 ∈ ℕ0)) | |
17 | 15, 16 | syl5ibcom 244 | . . . . . . 7 ⊢ (𝑤 ∈ (ClWWalks‘𝐺) → ((♯‘𝑤) = 𝑁 → 𝑁 ∈ ℕ0)) |
18 | 17 | con3rr3 155 | . . . . . 6 ⊢ (¬ 𝑁 ∈ ℕ0 → (𝑤 ∈ (ClWWalks‘𝐺) → ¬ (♯‘𝑤) = 𝑁)) |
19 | 18 | ralrimiv 3134 | . . . . 5 ⊢ (¬ 𝑁 ∈ ℕ0 → ∀𝑤 ∈ (ClWWalks‘𝐺) ¬ (♯‘𝑤) = 𝑁) |
20 | ral0 4514 | . . . . . 6 ⊢ ∀𝑤 ∈ ∅ ¬ (♯‘𝑤) = 𝑁 | |
21 | fvprc 6888 | . . . . . . 7 ⊢ (¬ 𝐺 ∈ V → (ClWWalks‘𝐺) = ∅) | |
22 | 21 | raleqdv 3314 | . . . . . 6 ⊢ (¬ 𝐺 ∈ V → (∀𝑤 ∈ (ClWWalks‘𝐺) ¬ (♯‘𝑤) = 𝑁 ↔ ∀𝑤 ∈ ∅ ¬ (♯‘𝑤) = 𝑁)) |
23 | 20, 22 | mpbiri 257 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → ∀𝑤 ∈ (ClWWalks‘𝐺) ¬ (♯‘𝑤) = 𝑁) |
24 | 19, 23 | jaoi 855 | . . . 4 ⊢ ((¬ 𝑁 ∈ ℕ0 ∨ ¬ 𝐺 ∈ V) → ∀𝑤 ∈ (ClWWalks‘𝐺) ¬ (♯‘𝑤) = 𝑁) |
25 | ianor 979 | . . . 4 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ↔ (¬ 𝑁 ∈ ℕ0 ∨ ¬ 𝐺 ∈ V)) | |
26 | rabeq0 4386 | . . . 4 ⊢ ({𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁} = ∅ ↔ ∀𝑤 ∈ (ClWWalks‘𝐺) ¬ (♯‘𝑤) = 𝑁) | |
27 | 24, 25, 26 | 3imtr4i 291 | . . 3 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁} = ∅) |
28 | 10, 27 | eqtr4d 2768 | . 2 ⊢ (¬ (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝑁 ClWWalksN 𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁}) |
29 | 9, 28 | pm2.61i 182 | 1 ⊢ (𝑁 ClWWalksN 𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∀wral 3050 {crab 3418 Vcvv 3461 ∅c0 4322 ‘cfv 6549 (class class class)co 7419 ℕ0cn0 12505 ♯chash 14325 Word cword 14500 Vtxcvtx 28881 ClWWalkscclwwlk 29863 ClWWalksN cclwwlkn 29906 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-n0 12506 df-z 12592 df-uz 12856 df-fz 13520 df-fzo 13663 df-hash 14326 df-word 14501 df-clwwlk 29864 df-clwwlkn 29907 |
This theorem is referenced by: isclwwlkn 29909 clwwlkn0 29910 clwwlknfi 29927 clwlknf1oclwwlkn 29966 |
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