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Theorem eupth2lem3 30180

Description: Lemma for eupth2 30183. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)

**Hypotheses**
Ref	Expression
eupth2.v	⊢ 𝑉 = (Vtx‘𝐺)
eupth2.i	⊢ 𝐼 = (iEdg‘𝐺)
eupth2.g	⊢ (𝜑 → 𝐺 ∈ UPGraph)
eupth2.f	⊢ (𝜑 → Fun 𝐼)
eupth2.p	⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃)
eupth2.h	⊢ 𝐻 = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))⟩
eupth2.x	⊢ 𝑋 = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))⟩
eupth2.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
eupth2.l	⊢ (𝜑 → (𝑁 + 1) ≤ (♯‘𝐹))
eupth2.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
eupth2.o	⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐻)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑁), ∅, {(𝑃‘0), (𝑃‘𝑁)}))

**Assertion**
Ref	Expression
eupth2lem3	⊢ (𝜑 → (¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑈) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))})))

Distinct variable groups: 𝑥,𝐻 𝑥,𝑈 𝑥,𝑉 Allowed substitution hints: 𝜑(𝑥) 𝑃(𝑥) 𝐹(𝑥) 𝐺(𝑥) 𝐼(𝑥) 𝑁(𝑥) 𝑋(𝑥)

Proof of Theorem eupth2lem3 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eupth2.v	. 2 ⊢ 𝑉 = (Vtx‘𝐺)
2		eupth2.i	. 2 ⊢ 𝐼 = (iEdg‘𝐺)
3		eupth2.f	. 2 ⊢ (𝜑 → Fun 𝐼)
4		eupth2.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
5		eupth2.p	. . . 4 ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃)
6		eupthiswlk 30156	. . . 4 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
7		wlkcl 29561	. . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ₀)
8	5, 6, 7	3syl 18	. . 3 ⊢ (𝜑 → (♯‘𝐹) ∈ ℕ₀)
9		eupth2.l	. . 3 ⊢ (𝜑 → (𝑁 + 1) ≤ (♯‘𝐹))
10		nn0p1elfzo 13605	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ₀ ∧ (𝑁 + 1) ≤ (♯‘𝐹)) → 𝑁 ∈ (0..^(♯‘𝐹)))
11	4, 8, 9, 10	syl3anc 1373	. 2 ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))
12		eupth2.u	. 2 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
13		eupthistrl 30155	. . 3 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃)
14	5, 13	syl 17	. 2 ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)
15		eupth2.h	. . . . 5 ⊢ 𝐻 = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))⟩
16	15	fveq2i 6825	. . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))⟩)
17	1	fvexi 6836	. . . . 5 ⊢ 𝑉 ∈ V
18	2	fvexi 6836	. . . . . 6 ⊢ 𝐼 ∈ V
19	18	resex 5980	. . . . 5 ⊢ (𝐼 ↾ (𝐹 “ (0..^𝑁))) ∈ V
20	17, 19	opvtxfvi 28954	. . . 4 ⊢ (Vtx‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))⟩) = 𝑉
21	16, 20	eqtri 2752	. . 3 ⊢ (Vtx‘𝐻) = 𝑉
22	21	a1i 11	. 2 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)
23		snex 5375	. . . 4 ⊢ {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩} ∈ V
24	17, 23	opvtxfvi 28954	. . 3 ⊢ (Vtx‘⟨𝑉, {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩}⟩) = 𝑉
25	24	a1i 11	. 2 ⊢ (𝜑 → (Vtx‘⟨𝑉, {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩}⟩) = 𝑉)
26		eupth2.x	. . . . 5 ⊢ 𝑋 = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))⟩
27	26	fveq2i 6825	. . . 4 ⊢ (Vtx‘𝑋) = (Vtx‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))⟩)
28	18	resex 5980	. . . . 5 ⊢ (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1)))) ∈ V
29	17, 28	opvtxfvi 28954	. . . 4 ⊢ (Vtx‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))⟩) = 𝑉
30	27, 29	eqtri 2752	. . 3 ⊢ (Vtx‘𝑋) = 𝑉
31	30	a1i 11	. 2 ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉)
32	15	fveq2i 6825	. . . 4 ⊢ (iEdg‘𝐻) = (iEdg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))⟩)
33	17, 19	opiedgfvi 28955	. . . 4 ⊢ (iEdg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))⟩) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))
34	32, 33	eqtri 2752	. . 3 ⊢ (iEdg‘𝐻) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))
35	34	a1i 11	. 2 ⊢ (𝜑 → (iEdg‘𝐻) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))
36	17, 23	opiedgfvi 28955	. . 3 ⊢ (iEdg‘⟨𝑉, {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩}⟩) = {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩}
37	36	a1i 11	. 2 ⊢ (𝜑 → (iEdg‘⟨𝑉, {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩}⟩) = {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩})
38	26	fveq2i 6825	. . . 4 ⊢ (iEdg‘𝑋) = (iEdg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))⟩)
39	17, 28	opiedgfvi 28955	. . . 4 ⊢ (iEdg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))⟩) = (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))
40	38, 39	eqtri 2752	. . 3 ⊢ (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))
41	4	nn0zd 12497	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
42		fzval3 13637	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → (0...𝑁) = (0..^(𝑁 + 1)))
43	42	eqcomd 2735	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (0..^(𝑁 + 1)) = (0...𝑁))
44	41, 43	syl 17	. . . . 5 ⊢ (𝜑 → (0..^(𝑁 + 1)) = (0...𝑁))
45	44	imaeq2d 6011	. . . 4 ⊢ (𝜑 → (𝐹 “ (0..^(𝑁 + 1))) = (𝐹 “ (0...𝑁)))
46	45	reseq2d 5930	. . 3 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1)))) = (𝐼 ↾ (𝐹 “ (0...𝑁))))
47	40, 46	eqtrid 2776	. 2 ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0...𝑁))))
48		eupth2.o	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐻)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑁), ∅, {(𝑃‘0), (𝑃‘𝑁)}))
49		2fveq3 6827	. . . 4 ⊢ (𝑘 = 𝑁 → (𝐼‘(𝐹‘𝑘)) = (𝐼‘(𝐹‘𝑁)))
50		fveq2 6822	. . . . 5 ⊢ (𝑘 = 𝑁 → (𝑃‘𝑘) = (𝑃‘𝑁))
51		fvoveq1 7372	. . . . 5 ⊢ (𝑘 = 𝑁 → (𝑃‘(𝑘 + 1)) = (𝑃‘(𝑁 + 1)))
52	50, 51	preq12d 4693	. . . 4 ⊢ (𝑘 = 𝑁 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))})
53	49, 52	eqeq12d 2745	. . 3 ⊢ (𝑘 = 𝑁 → ((𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))}))
54		eupth2.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ UPGraph)
55	5, 6	syl 17	. . . 4 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃)
56	2	upgrwlkedg 29587	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃) → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
57	54, 55, 56	syl2anc 584	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
58	53, 57, 11	rspcdva 3578	. 2 ⊢ (𝜑 → (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))})
59	1, 2, 3, 11, 12, 14, 22, 25, 31, 35, 37, 47, 48, 58	eupth2lem3lem7 30178	1 ⊢ (𝜑 → (¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑈) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))})))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2109 ∀wral 3044 {crab 3394 ∅c0 4284 ifcif 4476 {csn 4577 {cpr 4579 ⟨cop 4583 class class class wbr 5092 ↾ cres 5621 “ cima 5622 Fun wfun 6476 ‘cfv 6482 (class class class)co 7349 0cc0 11009 1c1 11010 + caddc 11012 ≤ cle 11150 2c2 12183 ℕ₀cn0 12384 ℤcz 12471 ...cfz 13410 ..^cfzo 13557 ♯chash 14237 ∥ cdvds 16163 Vtxcvtx 28941 iEdgciedg 28942 UPGraphcupgr 29025 VtxDegcvtxdg 29411 Walkscwlks 29542 Trailsctrls 29634 EulerPathsceupth 30141

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087

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 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-oadd 8392 df-er 8625 df-map 8755 df-pm 8756 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-sup 9332 df-inf 9333 df-dju 9797 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-n0 12385 df-xnn0 12458 df-z 12472 df-uz 12736 df-rp 12894 df-xadd 13015 df-fz 13411 df-fzo 13558 df-seq 13909 df-exp 13969 df-hash 14238 df-word 14421 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-dvds 16164 df-vtx 28943 df-iedg 28944 df-edg 28993 df-uhgr 29003 df-ushgr 29004 df-upgr 29027 df-uspgr 29095 df-vtxdg 29412 df-wlks 29545 df-trls 29636 df-eupth 30142

This theorem is referenced by: eupth2lems 30182

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