![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > trlsegvdeg | Structured version Visualization version GIF version |
Description: Formerly part of proof of eupth2lem3 28021: If a trail in a graph 𝐺 induces a subgraph 𝑍 with the vertices 𝑉 of 𝐺 and the edges being the edges of the walk, and a subgraph 𝑋 with the vertices 𝑉 of 𝐺 and the edges being the edges of the walk except the last one, and a subgraph 𝑌 with the vertices 𝑉 of 𝐺 and one edges being the last edge of the walk, then the vertex degree of any vertex 𝑈 of 𝐺 within 𝑍 is the sum of the vertex degree of 𝑈 within 𝑋 and the vertex degree of 𝑈 within 𝑌. Note that this theorem would not hold for arbitrary walks (if the last edge was identical with a previous edge, the degree of the vertices incident with this edge would not be increased because of this edge). (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 20-Feb-2021.) |
Ref | Expression |
---|---|
trlsegvdeg.v | ⊢ 𝑉 = (Vtx‘𝐺) |
trlsegvdeg.i | ⊢ 𝐼 = (iEdg‘𝐺) |
trlsegvdeg.f | ⊢ (𝜑 → Fun 𝐼) |
trlsegvdeg.n | ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) |
trlsegvdeg.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
trlsegvdeg.w | ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) |
trlsegvdeg.vx | ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉) |
trlsegvdeg.vy | ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉) |
trlsegvdeg.vz | ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉) |
trlsegvdeg.ix | ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) |
trlsegvdeg.iy | ⊢ (𝜑 → (iEdg‘𝑌) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) |
trlsegvdeg.iz | ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) |
Ref | Expression |
---|---|
trlsegvdeg | ⊢ (𝜑 → ((VtxDeg‘𝑍)‘𝑈) = (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . 2 ⊢ (iEdg‘𝑋) = (iEdg‘𝑋) | |
2 | eqid 2798 | . 2 ⊢ (iEdg‘𝑌) = (iEdg‘𝑌) | |
3 | eqid 2798 | . 2 ⊢ (Vtx‘𝑋) = (Vtx‘𝑋) | |
4 | trlsegvdeg.vy | . . 3 ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉) | |
5 | trlsegvdeg.vx | . . 3 ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉) | |
6 | 4, 5 | eqtr4d 2836 | . 2 ⊢ (𝜑 → (Vtx‘𝑌) = (Vtx‘𝑋)) |
7 | trlsegvdeg.vz | . . 3 ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉) | |
8 | 7, 5 | eqtr4d 2836 | . 2 ⊢ (𝜑 → (Vtx‘𝑍) = (Vtx‘𝑋)) |
9 | trlsegvdeg.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
10 | trlsegvdeg.i | . . . . 5 ⊢ 𝐼 = (iEdg‘𝐺) | |
11 | trlsegvdeg.f | . . . . 5 ⊢ (𝜑 → Fun 𝐼) | |
12 | trlsegvdeg.n | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) | |
13 | trlsegvdeg.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
14 | trlsegvdeg.w | . . . . 5 ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) | |
15 | trlsegvdeg.ix | . . . . 5 ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) | |
16 | trlsegvdeg.iy | . . . . 5 ⊢ (𝜑 → (iEdg‘𝑌) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) | |
17 | trlsegvdeg.iz | . . . . 5 ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) | |
18 | 9, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17 | trlsegvdeglem4 28008 | . . . 4 ⊢ (𝜑 → dom (iEdg‘𝑋) = ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼)) |
19 | 9, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17 | trlsegvdeglem5 28009 | . . . 4 ⊢ (𝜑 → dom (iEdg‘𝑌) = {(𝐹‘𝑁)}) |
20 | 18, 19 | ineq12d 4140 | . . 3 ⊢ (𝜑 → (dom (iEdg‘𝑋) ∩ dom (iEdg‘𝑌)) = (((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∩ {(𝐹‘𝑁)})) |
21 | fzonel 13046 | . . . . . . 7 ⊢ ¬ 𝑁 ∈ (0..^𝑁) | |
22 | 10 | trlf1 27488 | . . . . . . . . 9 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼) |
23 | 14, 22 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼) |
24 | elfzouz2 13047 | . . . . . . . . 9 ⊢ (𝑁 ∈ (0..^(♯‘𝐹)) → (♯‘𝐹) ∈ (ℤ≥‘𝑁)) | |
25 | fzoss2 13060 | . . . . . . . . 9 ⊢ ((♯‘𝐹) ∈ (ℤ≥‘𝑁) → (0..^𝑁) ⊆ (0..^(♯‘𝐹))) | |
26 | 12, 24, 25 | 3syl 18 | . . . . . . . 8 ⊢ (𝜑 → (0..^𝑁) ⊆ (0..^(♯‘𝐹))) |
27 | f1elima 6999 | . . . . . . . 8 ⊢ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑁 ∈ (0..^(♯‘𝐹)) ∧ (0..^𝑁) ⊆ (0..^(♯‘𝐹))) → ((𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ↔ 𝑁 ∈ (0..^𝑁))) | |
28 | 23, 12, 26, 27 | syl3anc 1368 | . . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ↔ 𝑁 ∈ (0..^𝑁))) |
29 | 21, 28 | mtbiri 330 | . . . . . 6 ⊢ (𝜑 → ¬ (𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁))) |
30 | 29 | orcd 870 | . . . . 5 ⊢ (𝜑 → (¬ (𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ∨ ¬ (𝐹‘𝑁) ∈ dom 𝐼)) |
31 | ianor 979 | . . . . . 6 ⊢ (¬ ((𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ∧ (𝐹‘𝑁) ∈ dom 𝐼) ↔ (¬ (𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ∨ ¬ (𝐹‘𝑁) ∈ dom 𝐼)) | |
32 | elin 3897 | . . . . . 6 ⊢ ((𝐹‘𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ↔ ((𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ∧ (𝐹‘𝑁) ∈ dom 𝐼)) | |
33 | 31, 32 | xchnxbir 336 | . . . . 5 ⊢ (¬ (𝐹‘𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ↔ (¬ (𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ∨ ¬ (𝐹‘𝑁) ∈ dom 𝐼)) |
34 | 30, 33 | sylibr 237 | . . . 4 ⊢ (𝜑 → ¬ (𝐹‘𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼)) |
35 | disjsn 4607 | . . . 4 ⊢ ((((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∩ {(𝐹‘𝑁)}) = ∅ ↔ ¬ (𝐹‘𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼)) | |
36 | 34, 35 | sylibr 237 | . . 3 ⊢ (𝜑 → (((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∩ {(𝐹‘𝑁)}) = ∅) |
37 | 20, 36 | eqtrd 2833 | . 2 ⊢ (𝜑 → (dom (iEdg‘𝑋) ∩ dom (iEdg‘𝑌)) = ∅) |
38 | 9, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17 | trlsegvdeglem2 28006 | . 2 ⊢ (𝜑 → Fun (iEdg‘𝑋)) |
39 | 9, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17 | trlsegvdeglem3 28007 | . 2 ⊢ (𝜑 → Fun (iEdg‘𝑌)) |
40 | 13, 5 | eleqtrrd 2893 | . 2 ⊢ (𝜑 → 𝑈 ∈ (Vtx‘𝑋)) |
41 | f1f 6549 | . . . . 5 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼) | |
42 | 14, 22, 41 | 3syl 18 | . . . 4 ⊢ (𝜑 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼) |
43 | 11, 42, 12 | resunimafz0 13799 | . . 3 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉})) |
44 | 15, 16 | uneq12d 4091 | . . 3 ⊢ (𝜑 → ((iEdg‘𝑋) ∪ (iEdg‘𝑌)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉})) |
45 | 43, 17, 44 | 3eqtr4d 2843 | . 2 ⊢ (𝜑 → (iEdg‘𝑍) = ((iEdg‘𝑋) ∪ (iEdg‘𝑌))) |
46 | 9, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17 | trlsegvdeglem6 28010 | . 2 ⊢ (𝜑 → dom (iEdg‘𝑋) ∈ Fin) |
47 | 9, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17 | trlsegvdeglem7 28011 | . 2 ⊢ (𝜑 → dom (iEdg‘𝑌) ∈ Fin) |
48 | 1, 2, 3, 6, 8, 37, 38, 39, 40, 45, 46, 47 | vtxdfiun 27272 | 1 ⊢ (𝜑 → ((VtxDeg‘𝑍)‘𝑈) = (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ∪ cun 3879 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 {csn 4525 〈cop 4531 class class class wbr 5030 dom cdm 5519 ↾ cres 5521 “ cima 5522 Fun wfun 6318 ⟶wf 6320 –1-1→wf1 6321 ‘cfv 6324 (class class class)co 7135 0cc0 10526 + caddc 10529 ℤ≥cuz 12231 ...cfz 12885 ..^cfzo 13028 ♯chash 13686 Vtxcvtx 26789 iEdgciedg 26790 VtxDegcvtxdg 27255 Trailsctrls 27480 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 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 210 df-an 400 df-or 845 df-ifp 1059 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-xadd 12496 df-fz 12886 df-fzo 13029 df-hash 13687 df-word 13858 df-vtxdg 27256 df-wlks 27389 df-trls 27482 |
This theorem is referenced by: eupth2lem3lem7 28019 |
Copyright terms: Public domain | W3C validator |