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| Mirrors > Home > MPE Home > Th. List > numclwwlk1lem2 | Structured version Visualization version GIF version | ||
| Description: The set of double loops of length 𝑁 on vertex 𝑋 and the set of closed walks of length less by 2 on 𝑋 combined with the neighbors of 𝑋 are equinumerous. (Contributed by Alexander van der Vekens, 6-Jul-2018.) (Revised by AV, 29-May-2021.) (Revised by AV, 31-Jul-2022.) (Proof shortened by AV, 3-Nov-2022.) |
| Ref | Expression |
|---|---|
| extwwlkfab.v | ⊢ 𝑉 = (Vtx‘𝐺) |
| extwwlkfab.c | ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) |
| extwwlkfab.f | ⊢ 𝐹 = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) |
| Ref | Expression |
|---|---|
| numclwwlk1lem2 | ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) → (𝑋𝐶𝑁) ≈ (𝐹 × (𝐺 NeighbVtx 𝑋))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | extwwlkfab.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 2 | extwwlkfab.c | . . 3 ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) | |
| 3 | extwwlkfab.f | . . 3 ⊢ 𝐹 = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) | |
| 4 | oveq1 7438 | . . . . 5 ⊢ (𝑥 = 𝑢 → (𝑥 prefix (𝑁 − 2)) = (𝑢 prefix (𝑁 − 2))) | |
| 5 | fveq1 6905 | . . . . 5 ⊢ (𝑥 = 𝑢 → (𝑥‘(𝑁 − 1)) = (𝑢‘(𝑁 − 1))) | |
| 6 | 4, 5 | opeq12d 4881 | . . . 4 ⊢ (𝑥 = 𝑢 → 〈(𝑥 prefix (𝑁 − 2)), (𝑥‘(𝑁 − 1))〉 = 〈(𝑢 prefix (𝑁 − 2)), (𝑢‘(𝑁 − 1))〉) |
| 7 | 6 | cbvmptv 5255 | . . 3 ⊢ (𝑥 ∈ (𝑋𝐶𝑁) ↦ 〈(𝑥 prefix (𝑁 − 2)), (𝑥‘(𝑁 − 1))〉) = (𝑢 ∈ (𝑋𝐶𝑁) ↦ 〈(𝑢 prefix (𝑁 − 2)), (𝑢‘(𝑁 − 1))〉) |
| 8 | 1, 2, 3, 7 | numclwwlk1lem2f1o 30378 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) → (𝑥 ∈ (𝑋𝐶𝑁) ↦ 〈(𝑥 prefix (𝑁 − 2)), (𝑥‘(𝑁 − 1))〉):(𝑋𝐶𝑁)–1-1-onto→(𝐹 × (𝐺 NeighbVtx 𝑋))) |
| 9 | ovex 7464 | . . 3 ⊢ (𝑋𝐶𝑁) ∈ V | |
| 10 | 9 | f1oen 9013 | . 2 ⊢ ((𝑥 ∈ (𝑋𝐶𝑁) ↦ 〈(𝑥 prefix (𝑁 − 2)), (𝑥‘(𝑁 − 1))〉):(𝑋𝐶𝑁)–1-1-onto→(𝐹 × (𝐺 NeighbVtx 𝑋)) → (𝑋𝐶𝑁) ≈ (𝐹 × (𝐺 NeighbVtx 𝑋))) |
| 11 | 8, 10 | syl 17 | 1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) → (𝑋𝐶𝑁) ≈ (𝐹 × (𝐺 NeighbVtx 𝑋))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 {crab 3436 〈cop 4632 class class class wbr 5143 ↦ cmpt 5225 × cxp 5683 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ≈ cen 8982 1c1 11156 − cmin 11492 2c2 12321 3c3 12322 ℤ≥cuz 12878 prefix cpfx 14708 Vtxcvtx 29013 USGraphcusgr 29166 NeighbVtx cnbgr 29349 ClWWalksNOncclwwlknon 30106 |
| 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-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-2o 8507 df-oadd 8510 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 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-xnn0 12600 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-lsw 14601 df-concat 14609 df-s1 14634 df-substr 14679 df-pfx 14709 df-s2 14887 df-edg 29065 df-upgr 29099 df-umgr 29100 df-usgr 29168 df-nbgr 29350 df-wwlks 29850 df-wwlksn 29851 df-clwwlk 30001 df-clwwlkn 30044 df-clwwlknon 30107 |
| This theorem is referenced by: numclwwlk1 30380 |
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