eupth2lem3lem1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > eupth2lem3lem1		Structured version Visualization version GIF version

Theorem eupth2lem3lem1 30209

Description: Lemma for eupth2lem3 30217. (Contributed by AV, 21-Feb-2021.)

**Hypotheses**
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))))

**Assertion**
Ref	Expression
eupth2lem3lem1	⊢ (𝜑 → ((VtxDeg‘𝑋)‘𝑈) ∈ ℕ₀)

**Proof of Theorem eupth2lem3lem1**
Step	Hyp	Ref	Expression
1		trlsegvdeg.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
2		trlsegvdeg.vx	. . . . 5 ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉)
3	1, 2	eleqtrrd 2837	. . . 4 ⊢ (𝜑 → 𝑈 ∈ (Vtx‘𝑋))
4	3	elfvexd 6915	. . 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 30206	. . 3 ⊢ (𝜑 → dom (iEdg‘𝑋) ∈ Fin)
16		eqid 2735	. . . 4 ⊢ (Vtx‘𝑋) = (Vtx‘𝑋)
17		eqid 2735	. . . 4 ⊢ (iEdg‘𝑋) = (iEdg‘𝑋)
18		eqid 2735	. . . 4 ⊢ dom (iEdg‘𝑋) = dom (iEdg‘𝑋)
19	16, 17, 18	vtxdgfisf 29456	. . 3 ⊢ ((𝑋 ∈ V ∧ dom (iEdg‘𝑋) ∈ Fin) → (VtxDeg‘𝑋):(Vtx‘𝑋)⟶ℕ₀)
20	4, 15, 19	syl2anc 584	. 2 ⊢ (𝜑 → (VtxDeg‘𝑋):(Vtx‘𝑋)⟶ℕ₀)
21	20, 3	ffvelcdmd 7075	1 ⊢ (𝜑 → ((VtxDeg‘𝑋)‘𝑈) ∈ ℕ₀)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3459 {csn 4601 ⟨cop 4607 class class class wbr 5119 dom cdm 5654 ↾ cres 5656 “ cima 5657 Fun wfun 6525 ⟶wf 6527 ‘cfv 6531 (class class class)co 7405 Fincfn 8959 0cc0 11129 ℕ₀cn0 12501 ...cfz 13524 ..^cfzo 13671 ♯chash 14348 Vtxcvtx 28975 iEdgciedg 28976 VtxDegcvtxdg 29445 Trailsctrls 29670

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 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8719 df-map 8842 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-nn 12241 df-n0 12502 df-xnn0 12575 df-z 12589 df-uz 12853 df-xadd 13129 df-fz 13525 df-fzo 13672 df-hash 14349 df-word 14532 df-vtxdg 29446 df-wlks 29579 df-trls 29672

This theorem is referenced by: eupth2lem3lem3 30211 eupth2lem3lem4 30212 eupth2lem3lem6 30214

W3C validator