| 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 30255: 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 2737 | . 2 ⊢ (iEdg‘𝑋) = (iEdg‘𝑋) | |
| 2 | eqid 2737 | . 2 ⊢ (iEdg‘𝑌) = (iEdg‘𝑌) | |
| 3 | eqid 2737 | . 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 2780 | . 2 ⊢ (𝜑 → (Vtx‘𝑌) = (Vtx‘𝑋)) |
| 7 | trlsegvdeg.vz | . . 3 ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉) | |
| 8 | 7, 5 | eqtr4d 2780 | . 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 30242 | . . . 4 ⊢ (𝜑 → dom (iEdg‘𝑋) = ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼)) |
| 19 | 9, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17 | trlsegvdeglem5 30243 | . . . 4 ⊢ (𝜑 → dom (iEdg‘𝑌) = {(𝐹‘𝑁)}) |
| 20 | 18, 19 | ineq12d 4221 | . . 3 ⊢ (𝜑 → (dom (iEdg‘𝑋) ∩ dom (iEdg‘𝑌)) = (((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∩ {(𝐹‘𝑁)})) |
| 21 | fzonel 13713 | . . . . . . 7 ⊢ ¬ 𝑁 ∈ (0..^𝑁) | |
| 22 | 10 | trlf1 29716 | . . . . . . . . 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 13714 | . . . . . . . . 9 ⊢ (𝑁 ∈ (0..^(♯‘𝐹)) → (♯‘𝐹) ∈ (ℤ≥‘𝑁)) | |
| 25 | fzoss2 13727 | . . . . . . . . 9 ⊢ ((♯‘𝐹) ∈ (ℤ≥‘𝑁) → (0..^𝑁) ⊆ (0..^(♯‘𝐹))) | |
| 26 | 12, 24, 25 | 3syl 18 | . . . . . . . 8 ⊢ (𝜑 → (0..^𝑁) ⊆ (0..^(♯‘𝐹))) |
| 27 | f1elima 7283 | . . . . . . . 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 1373 | . . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ↔ 𝑁 ∈ (0..^𝑁))) |
| 29 | 21, 28 | mtbiri 327 | . . . . . 6 ⊢ (𝜑 → ¬ (𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁))) |
| 30 | 29 | orcd 874 | . . . . 5 ⊢ (𝜑 → (¬ (𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ∨ ¬ (𝐹‘𝑁) ∈ dom 𝐼)) |
| 31 | ianor 984 | . . . . . 6 ⊢ (¬ ((𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ∧ (𝐹‘𝑁) ∈ dom 𝐼) ↔ (¬ (𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ∨ ¬ (𝐹‘𝑁) ∈ dom 𝐼)) | |
| 32 | elin 3967 | . . . . . 6 ⊢ ((𝐹‘𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ↔ ((𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ∧ (𝐹‘𝑁) ∈ dom 𝐼)) | |
| 33 | 31, 32 | xchnxbir 333 | . . . . 5 ⊢ (¬ (𝐹‘𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ↔ (¬ (𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ∨ ¬ (𝐹‘𝑁) ∈ dom 𝐼)) |
| 34 | 30, 33 | sylibr 234 | . . . 4 ⊢ (𝜑 → ¬ (𝐹‘𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼)) |
| 35 | disjsn 4711 | . . . 4 ⊢ ((((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∩ {(𝐹‘𝑁)}) = ∅ ↔ ¬ (𝐹‘𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼)) | |
| 36 | 34, 35 | sylibr 234 | . . 3 ⊢ (𝜑 → (((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∩ {(𝐹‘𝑁)}) = ∅) |
| 37 | 20, 36 | eqtrd 2777 | . 2 ⊢ (𝜑 → (dom (iEdg‘𝑋) ∩ dom (iEdg‘𝑌)) = ∅) |
| 38 | 9, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17 | trlsegvdeglem2 30240 | . 2 ⊢ (𝜑 → Fun (iEdg‘𝑋)) |
| 39 | 9, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17 | trlsegvdeglem3 30241 | . 2 ⊢ (𝜑 → Fun (iEdg‘𝑌)) |
| 40 | 13, 5 | eleqtrrd 2844 | . 2 ⊢ (𝜑 → 𝑈 ∈ (Vtx‘𝑋)) |
| 41 | f1f 6804 | . . . . 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 14484 | . . 3 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉})) |
| 44 | 15, 16 | uneq12d 4169 | . . 3 ⊢ (𝜑 → ((iEdg‘𝑋) ∪ (iEdg‘𝑌)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉})) |
| 45 | 43, 17, 44 | 3eqtr4d 2787 | . 2 ⊢ (𝜑 → (iEdg‘𝑍) = ((iEdg‘𝑋) ∪ (iEdg‘𝑌))) |
| 46 | 9, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17 | trlsegvdeglem6 30244 | . 2 ⊢ (𝜑 → dom (iEdg‘𝑋) ∈ Fin) |
| 47 | 9, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17 | trlsegvdeglem7 30245 | . 2 ⊢ (𝜑 → dom (iEdg‘𝑌) ∈ Fin) |
| 48 | 1, 2, 3, 6, 8, 37, 38, 39, 40, 45, 46, 47 | vtxdfiun 29500 | 1 ⊢ (𝜑 → ((VtxDeg‘𝑍)‘𝑈) = (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈))) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∪ cun 3949 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 {csn 4626 〈cop 4632 class class class wbr 5143 dom cdm 5685 ↾ cres 5687 “ cima 5688 Fun wfun 6555 ⟶wf 6557 –1-1→wf1 6558 ‘cfv 6561 (class class class)co 7431 0cc0 11155 + caddc 11158 ℤ≥cuz 12878 ...cfz 13547 ..^cfzo 13694 ♯chash 14369 Vtxcvtx 29013 iEdgciedg 29014 VtxDegcvtxdg 29483 Trailsctrls 29708 |
| 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-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-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-xadd 13155 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-vtxdg 29484 df-wlks 29617 df-trls 29710 |
| This theorem is referenced by: eupth2lem3lem7 30253 |
| Copyright terms: Public domain | W3C validator |