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Mirrors > Home > MPE Home > Th. List > eupth2lem3lem1 | Structured version Visualization version GIF version |
Description: Lemma for eupth2lem3 27769. (Contributed by AV, 21-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 |
---|---|
eupth2lem3lem1 | ⊢ (𝜑 → ((VtxDeg‘𝑋)‘𝑈) ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trlsegvdeg.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
2 | trlsegvdeg.vx | . . . . 5 ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉) | |
3 | 1, 2 | eleqtrrd 2869 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ (Vtx‘𝑋)) |
4 | 3 | elfvexd 6536 | . . 3 ⊢ (𝜑 → 𝑋 ∈ V) |
5 | trlsegvdeg.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
6 | trlsegvdeg.i | . . . 4 ⊢ 𝐼 = (iEdg‘𝐺) | |
7 | trlsegvdeg.f | . . . 4 ⊢ (𝜑 → Fun 𝐼) | |
8 | trlsegvdeg.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) | |
9 | trlsegvdeg.w | . . . 4 ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) | |
10 | trlsegvdeg.vy | . . . 4 ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉) | |
11 | trlsegvdeg.vz | . . . 4 ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉) | |
12 | trlsegvdeg.ix | . . . 4 ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) | |
13 | trlsegvdeg.iy | . . . 4 ⊢ (𝜑 → (iEdg‘𝑌) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) | |
14 | trlsegvdeg.iz | . . . 4 ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) | |
15 | 5, 6, 7, 8, 1, 9, 2, 10, 11, 12, 13, 14 | trlsegvdeglem6 27758 | . . 3 ⊢ (𝜑 → dom (iEdg‘𝑋) ∈ Fin) |
16 | eqid 2778 | . . . 4 ⊢ (Vtx‘𝑋) = (Vtx‘𝑋) | |
17 | eqid 2778 | . . . 4 ⊢ (iEdg‘𝑋) = (iEdg‘𝑋) | |
18 | eqid 2778 | . . . 4 ⊢ dom (iEdg‘𝑋) = dom (iEdg‘𝑋) | |
19 | 16, 17, 18 | vtxdgfisf 26964 | . . 3 ⊢ ((𝑋 ∈ V ∧ dom (iEdg‘𝑋) ∈ Fin) → (VtxDeg‘𝑋):(Vtx‘𝑋)⟶ℕ0) |
20 | 4, 15, 19 | syl2anc 576 | . 2 ⊢ (𝜑 → (VtxDeg‘𝑋):(Vtx‘𝑋)⟶ℕ0) |
21 | 20, 3 | ffvelrnd 6679 | 1 ⊢ (𝜑 → ((VtxDeg‘𝑋)‘𝑈) ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 Vcvv 3415 {csn 4442 〈cop 4448 class class class wbr 4930 dom cdm 5408 ↾ cres 5410 “ cima 5411 Fun wfun 6184 ⟶wf 6186 ‘cfv 6190 (class class class)co 6978 Fincfn 8308 0cc0 10337 ℕ0cn0 11710 ...cfz 12711 ..^cfzo 12852 ♯chash 13508 Vtxcvtx 26487 iEdgciedg 26488 VtxDegcvtxdg 26953 Trailsctrls 27181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-ifp 1044 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-int 4751 df-iun 4795 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-om 7399 df-1st 7503 df-2nd 7504 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-1o 7907 df-er 8091 df-map 8210 df-en 8309 df-dom 8310 df-sdom 8311 df-fin 8312 df-card 9164 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-nn 11442 df-n0 11711 df-xnn0 11783 df-z 11797 df-uz 12062 df-xadd 12328 df-fz 12712 df-fzo 12853 df-hash 13509 df-word 13676 df-vtxdg 26954 df-wlks 27087 df-trls 27183 |
This theorem is referenced by: eupth2lem3lem3 27763 eupth2lem3lem4 27764 eupth2lem3lem6 27766 |
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