Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > trlsegvdeg | Structured version Visualization version GIF version |
Description: Formerly part of proof of eupath2lem3 26506: 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 1-walk, and a subgraph 𝑋 with the vertices 𝑉 of 𝐺 and the edges being the edges of the 1-walk except the last one, and a subgraph 𝑌 with the vertices 𝑉 of 𝐺 and one edges being the last edge of the 1-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 1-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 2610 | . 2 ⊢ (iEdg‘𝑋) = (iEdg‘𝑋) | |
2 | eqid 2610 | . 2 ⊢ (iEdg‘𝑌) = (iEdg‘𝑌) | |
3 | eqid 2610 | . 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 2647 | . 2 ⊢ (𝜑 → (Vtx‘𝑌) = (Vtx‘𝑋)) |
7 | trlsegvdeg.vz | . . 3 ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉) | |
8 | 7, 5 | eqtr4d 2647 | . 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 41391 | . . . 4 ⊢ (𝜑 → dom (iEdg‘𝑋) = ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼)) |
19 | 9, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17 | trlsegvdeglem5 41392 | . . . 4 ⊢ (𝜑 → dom (iEdg‘𝑌) = {(𝐹‘𝑁)}) |
20 | 18, 19 | ineq12d 3777 | . . 3 ⊢ (𝜑 → (dom (iEdg‘𝑋) ∩ dom (iEdg‘𝑌)) = (((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∩ {(𝐹‘𝑁)})) |
21 | fzonel 12352 | . . . . . . 7 ⊢ ¬ 𝑁 ∈ (0..^𝑁) | |
22 | 10 | trlf1 40906 | . . . . . . . . 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 12353 | . . . . . . . . 9 ⊢ (𝑁 ∈ (0..^(#‘𝐹)) → (#‘𝐹) ∈ (ℤ≥‘𝑁)) | |
25 | fzoss2 12365 | . . . . . . . . 9 ⊢ ((#‘𝐹) ∈ (ℤ≥‘𝑁) → (0..^𝑁) ⊆ (0..^(#‘𝐹))) | |
26 | 12, 24, 25 | 3syl 18 | . . . . . . . 8 ⊢ (𝜑 → (0..^𝑁) ⊆ (0..^(#‘𝐹))) |
27 | f1elima 6421 | . . . . . . . 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 1318 | . . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ↔ 𝑁 ∈ (0..^𝑁))) |
29 | 21, 28 | mtbiri 316 | . . . . . 6 ⊢ (𝜑 → ¬ (𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁))) |
30 | 29 | orcd 406 | . . . . 5 ⊢ (𝜑 → (¬ (𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ∨ ¬ (𝐹‘𝑁) ∈ dom 𝐼)) |
31 | ianor 508 | . . . . . 6 ⊢ (¬ ((𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ∧ (𝐹‘𝑁) ∈ dom 𝐼) ↔ (¬ (𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ∨ ¬ (𝐹‘𝑁) ∈ dom 𝐼)) | |
32 | elin 3758 | . . . . . 6 ⊢ ((𝐹‘𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ↔ ((𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ∧ (𝐹‘𝑁) ∈ dom 𝐼)) | |
33 | 31, 32 | xchnxbir 322 | . . . . 5 ⊢ (¬ (𝐹‘𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ↔ (¬ (𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ∨ ¬ (𝐹‘𝑁) ∈ dom 𝐼)) |
34 | 30, 33 | sylibr 223 | . . . 4 ⊢ (𝜑 → ¬ (𝐹‘𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼)) |
35 | disjsn 4192 | . . . 4 ⊢ ((((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∩ {(𝐹‘𝑁)}) = ∅ ↔ ¬ (𝐹‘𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼)) | |
36 | 34, 35 | sylibr 223 | . . 3 ⊢ (𝜑 → (((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∩ {(𝐹‘𝑁)}) = ∅) |
37 | 20, 36 | eqtrd 2644 | . 2 ⊢ (𝜑 → (dom (iEdg‘𝑋) ∩ dom (iEdg‘𝑌)) = ∅) |
38 | 9, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17 | trlsegvdeglem2 41389 | . 2 ⊢ (𝜑 → Fun (iEdg‘𝑋)) |
39 | 9, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17 | trlsegvdeglem3 41390 | . 2 ⊢ (𝜑 → Fun (iEdg‘𝑌)) |
40 | 13, 5 | eleqtrrd 2691 | . 2 ⊢ (𝜑 → 𝑈 ∈ (Vtx‘𝑋)) |
41 | f1f 6014 | . . . . 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 40368 | . . 3 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉})) |
44 | 15, 16 | uneq12d 3730 | . . 3 ⊢ (𝜑 → ((iEdg‘𝑋) ∪ (iEdg‘𝑌)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉})) |
45 | 43, 17, 44 | 3eqtr4d 2654 | . 2 ⊢ (𝜑 → (iEdg‘𝑍) = ((iEdg‘𝑋) ∪ (iEdg‘𝑌))) |
46 | 9, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17 | trlsegvdeglem6 41393 | . 2 ⊢ (𝜑 → dom (iEdg‘𝑋) ∈ Fin) |
47 | 9, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17 | trlsegvdeglem7 41394 | . 2 ⊢ (𝜑 → dom (iEdg‘𝑌) ∈ Fin) |
48 | 1, 2, 3, 6, 8, 37, 38, 39, 40, 45, 46, 47 | vtxdfiun 40697 | 1 ⊢ (𝜑 → ((VtxDeg‘𝑍)‘𝑈) = (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 {csn 4125 〈cop 4131 class class class wbr 4583 dom cdm 5038 ↾ cres 5040 “ cima 5041 Fun wfun 5798 ⟶wf 5800 –1-1→wf1 5801 ‘cfv 5804 (class class class)co 6549 0cc0 9815 + caddc 9818 ℤ≥cuz 11563 ...cfz 12197 ..^cfzo 12334 #chash 12979 Vtxcvtx 25673 iEdgciedg 25674 VtxDegcvtxdg 40681 TrailSctrls 40899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ifp 1007 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-xadd 11823 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-vtxdg 40682 df-1wlks 40800 df-trls 40901 |
This theorem is referenced by: eupth2lem3lem7 41402 |
Copyright terms: Public domain | W3C validator |