| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > wlk2v2e | Structured version Visualization version GIF version | ||
| Description: In a graph with two vertices and one edge connecting these two vertices, to go from one vertex to the other and back to the first vertex via the same/only edge is a walk. Notice that 𝐺 is a simple graph (without loops) only if 𝑋 ≠ 𝑌. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 8-Jan-2021.) |
| Ref | Expression |
|---|---|
| wlk2v2e.i | ⊢ 𝐼 = 〈“{𝑋, 𝑌}”〉 |
| wlk2v2e.f | ⊢ 𝐹 = 〈“00”〉 |
| wlk2v2e.x | ⊢ 𝑋 ∈ V |
| wlk2v2e.y | ⊢ 𝑌 ∈ V |
| wlk2v2e.p | ⊢ 𝑃 = 〈“𝑋𝑌𝑋”〉 |
| wlk2v2e.g | ⊢ 𝐺 = 〈{𝑋, 𝑌}, 𝐼〉 |
| Ref | Expression |
|---|---|
| wlk2v2e | ⊢ 𝐹(Walks‘𝐺)𝑃 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | wlk2v2e.g | . . . . 5 ⊢ 𝐺 = 〈{𝑋, 𝑌}, 𝐼〉 | |
| 2 | wlk2v2e.i | . . . . . 6 ⊢ 𝐼 = 〈“{𝑋, 𝑌}”〉 | |
| 3 | 2 | opeq2i 4877 | . . . . 5 ⊢ 〈{𝑋, 𝑌}, 𝐼〉 = 〈{𝑋, 𝑌}, 〈“{𝑋, 𝑌}”〉〉 |
| 4 | 1, 3 | eqtri 2765 | . . . 4 ⊢ 𝐺 = 〈{𝑋, 𝑌}, 〈“{𝑋, 𝑌}”〉〉 |
| 5 | wlk2v2e.x | . . . . 5 ⊢ 𝑋 ∈ V | |
| 6 | wlk2v2e.y | . . . . 5 ⊢ 𝑌 ∈ V | |
| 7 | uspgr2v1e2w 29268 | . . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 〈{𝑋, 𝑌}, 〈“{𝑋, 𝑌}”〉〉 ∈ USPGraph) | |
| 8 | 5, 6, 7 | mp2an 692 | . . . 4 ⊢ 〈{𝑋, 𝑌}, 〈“{𝑋, 𝑌}”〉〉 ∈ USPGraph |
| 9 | 4, 8 | eqeltri 2837 | . . 3 ⊢ 𝐺 ∈ USPGraph |
| 10 | uspgrupgr 29195 | . . 3 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph) | |
| 11 | 9, 10 | ax-mp 5 | . 2 ⊢ 𝐺 ∈ UPGraph |
| 12 | wlk2v2e.f | . . . . 5 ⊢ 𝐹 = 〈“00”〉 | |
| 13 | 2, 12 | wlk2v2elem1 30174 | . . . 4 ⊢ 𝐹 ∈ Word dom 𝐼 |
| 14 | wlk2v2e.p | . . . . . . . 8 ⊢ 𝑃 = 〈“𝑋𝑌𝑋”〉 | |
| 15 | 5 | prid1 4762 | . . . . . . . . 9 ⊢ 𝑋 ∈ {𝑋, 𝑌} |
| 16 | 6 | prid2 4763 | . . . . . . . . 9 ⊢ 𝑌 ∈ {𝑋, 𝑌} |
| 17 | s3cl 14918 | . . . . . . . . 9 ⊢ ((𝑋 ∈ {𝑋, 𝑌} ∧ 𝑌 ∈ {𝑋, 𝑌} ∧ 𝑋 ∈ {𝑋, 𝑌}) → 〈“𝑋𝑌𝑋”〉 ∈ Word {𝑋, 𝑌}) | |
| 18 | 15, 16, 15, 17 | mp3an 1463 | . . . . . . . 8 ⊢ 〈“𝑋𝑌𝑋”〉 ∈ Word {𝑋, 𝑌} |
| 19 | 14, 18 | eqeltri 2837 | . . . . . . 7 ⊢ 𝑃 ∈ Word {𝑋, 𝑌} |
| 20 | wrdf 14557 | . . . . . . 7 ⊢ (𝑃 ∈ Word {𝑋, 𝑌} → 𝑃:(0..^(♯‘𝑃))⟶{𝑋, 𝑌}) | |
| 21 | 19, 20 | ax-mp 5 | . . . . . 6 ⊢ 𝑃:(0..^(♯‘𝑃))⟶{𝑋, 𝑌} |
| 22 | 14 | fveq2i 6909 | . . . . . . . . 9 ⊢ (♯‘𝑃) = (♯‘〈“𝑋𝑌𝑋”〉) |
| 23 | s3len 14933 | . . . . . . . . 9 ⊢ (♯‘〈“𝑋𝑌𝑋”〉) = 3 | |
| 24 | 22, 23 | eqtr2i 2766 | . . . . . . . 8 ⊢ 3 = (♯‘𝑃) |
| 25 | 24 | oveq2i 7442 | . . . . . . 7 ⊢ (0..^3) = (0..^(♯‘𝑃)) |
| 26 | 25 | feq2i 6728 | . . . . . 6 ⊢ (𝑃:(0..^3)⟶{𝑋, 𝑌} ↔ 𝑃:(0..^(♯‘𝑃))⟶{𝑋, 𝑌}) |
| 27 | 21, 26 | mpbir 231 | . . . . 5 ⊢ 𝑃:(0..^3)⟶{𝑋, 𝑌} |
| 28 | 12 | fveq2i 6909 | . . . . . . . . 9 ⊢ (♯‘𝐹) = (♯‘〈“00”〉) |
| 29 | s2len 14928 | . . . . . . . . 9 ⊢ (♯‘〈“00”〉) = 2 | |
| 30 | 28, 29 | eqtri 2765 | . . . . . . . 8 ⊢ (♯‘𝐹) = 2 |
| 31 | 30 | oveq2i 7442 | . . . . . . 7 ⊢ (0...(♯‘𝐹)) = (0...2) |
| 32 | 3z 12650 | . . . . . . . . 9 ⊢ 3 ∈ ℤ | |
| 33 | fzoval 13700 | . . . . . . . . 9 ⊢ (3 ∈ ℤ → (0..^3) = (0...(3 − 1))) | |
| 34 | 32, 33 | ax-mp 5 | . . . . . . . 8 ⊢ (0..^3) = (0...(3 − 1)) |
| 35 | 3m1e2 12394 | . . . . . . . . 9 ⊢ (3 − 1) = 2 | |
| 36 | 35 | oveq2i 7442 | . . . . . . . 8 ⊢ (0...(3 − 1)) = (0...2) |
| 37 | 34, 36 | eqtr2i 2766 | . . . . . . 7 ⊢ (0...2) = (0..^3) |
| 38 | 31, 37 | eqtri 2765 | . . . . . 6 ⊢ (0...(♯‘𝐹)) = (0..^3) |
| 39 | 38 | feq2i 6728 | . . . . 5 ⊢ (𝑃:(0...(♯‘𝐹))⟶{𝑋, 𝑌} ↔ 𝑃:(0..^3)⟶{𝑋, 𝑌}) |
| 40 | 27, 39 | mpbir 231 | . . . 4 ⊢ 𝑃:(0...(♯‘𝐹))⟶{𝑋, 𝑌} |
| 41 | 2, 12, 5, 6, 14 | wlk2v2elem2 30175 | . . . 4 ⊢ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} |
| 42 | 13, 40, 41 | 3pm3.2i 1340 | . . 3 ⊢ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶{𝑋, 𝑌} ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) |
| 43 | 1 | fveq2i 6909 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘〈{𝑋, 𝑌}, 𝐼〉) |
| 44 | prex 5437 | . . . . . 6 ⊢ {𝑋, 𝑌} ∈ V | |
| 45 | s1cli 14643 | . . . . . . 7 ⊢ 〈“{𝑋, 𝑌}”〉 ∈ Word V | |
| 46 | 2, 45 | eqeltri 2837 | . . . . . 6 ⊢ 𝐼 ∈ Word V |
| 47 | opvtxfv 29021 | . . . . . 6 ⊢ (({𝑋, 𝑌} ∈ V ∧ 𝐼 ∈ Word V) → (Vtx‘〈{𝑋, 𝑌}, 𝐼〉) = {𝑋, 𝑌}) | |
| 48 | 44, 46, 47 | mp2an 692 | . . . . 5 ⊢ (Vtx‘〈{𝑋, 𝑌}, 𝐼〉) = {𝑋, 𝑌} |
| 49 | 43, 48 | eqtr2i 2766 | . . . 4 ⊢ {𝑋, 𝑌} = (Vtx‘𝐺) |
| 50 | 1 | fveq2i 6909 | . . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘〈{𝑋, 𝑌}, 𝐼〉) |
| 51 | opiedgfv 29024 | . . . . . 6 ⊢ (({𝑋, 𝑌} ∈ V ∧ 𝐼 ∈ Word V) → (iEdg‘〈{𝑋, 𝑌}, 𝐼〉) = 𝐼) | |
| 52 | 44, 46, 51 | mp2an 692 | . . . . 5 ⊢ (iEdg‘〈{𝑋, 𝑌}, 𝐼〉) = 𝐼 |
| 53 | 50, 52 | eqtr2i 2766 | . . . 4 ⊢ 𝐼 = (iEdg‘𝐺) |
| 54 | 49, 53 | upgriswlk 29659 | . . 3 ⊢ (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶{𝑋, 𝑌} ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) |
| 55 | 42, 54 | mpbiri 258 | . 2 ⊢ (𝐺 ∈ UPGraph → 𝐹(Walks‘𝐺)𝑃) |
| 56 | 11, 55 | ax-mp 5 | 1 ⊢ 𝐹(Walks‘𝐺)𝑃 |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 {cpr 4628 〈cop 4632 class class class wbr 5143 dom cdm 5685 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 − cmin 11492 2c2 12321 3c3 12322 ℤcz 12613 ...cfz 13547 ..^cfzo 13694 ♯chash 14369 Word cword 14552 〈“cs1 14633 〈“cs2 14880 〈“cs3 14881 Vtxcvtx 29013 iEdgciedg 29014 UPGraphcupgr 29097 USPGraphcuspgr 29165 Walkscwlks 29614 |
| 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-ifp 1064 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-pm 8869 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-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-concat 14609 df-s1 14634 df-s2 14887 df-s3 14888 df-vtx 29015 df-iedg 29016 df-edg 29065 df-uhgr 29075 df-upgr 29099 df-uspgr 29167 df-wlks 29617 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |