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Theorem eupth2lem3 30265

Description: Lemma for eupth2 30268. (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 30241	. . . 4 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
7		wlkcl 29648	. . . 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 13739	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ₀ ∧ (𝑁 + 1) ≤ (♯‘𝐹)) → 𝑁 ∈ (0..^(♯‘𝐹)))
11	4, 8, 9, 10	syl3anc 1370	. 2 ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))
12		eupth2.u	. 2 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
13		eupthistrl 30240	. . 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 6910	. . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))⟩)
17	1	fvexi 6921	. . . . 5 ⊢ 𝑉 ∈ V
18	2	fvexi 6921	. . . . . 6 ⊢ 𝐼 ∈ V
19	18	resex 6049	. . . . 5 ⊢ (𝐼 ↾ (𝐹 “ (0..^𝑁))) ∈ V
20	17, 19	opvtxfvi 29041	. . . 4 ⊢ (Vtx‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))⟩) = 𝑉
21	16, 20	eqtri 2763	. . 3 ⊢ (Vtx‘𝐻) = 𝑉
22	21	a1i 11	. 2 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)
23		snex 5442	. . . 4 ⊢ {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩} ∈ V
24	17, 23	opvtxfvi 29041	. . 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 6910	. . . 4 ⊢ (Vtx‘𝑋) = (Vtx‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))⟩)
28	18	resex 6049	. . . . 5 ⊢ (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1)))) ∈ V
29	17, 28	opvtxfvi 29041	. . . 4 ⊢ (Vtx‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))⟩) = 𝑉
30	27, 29	eqtri 2763	. . 3 ⊢ (Vtx‘𝑋) = 𝑉
31	30	a1i 11	. 2 ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉)
32	15	fveq2i 6910	. . . 4 ⊢ (iEdg‘𝐻) = (iEdg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))⟩)
33	17, 19	opiedgfvi 29042	. . . 4 ⊢ (iEdg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))⟩) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))
34	32, 33	eqtri 2763	. . 3 ⊢ (iEdg‘𝐻) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))
35	34	a1i 11	. 2 ⊢ (𝜑 → (iEdg‘𝐻) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))
36	17, 23	opiedgfvi 29042	. . 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 6910	. . . 4 ⊢ (iEdg‘𝑋) = (iEdg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))⟩)
39	17, 28	opiedgfvi 29042	. . . 4 ⊢ (iEdg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))⟩) = (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))
40	38, 39	eqtri 2763	. . 3 ⊢ (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))
41	4	nn0zd 12637	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
42		fzval3 13770	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → (0...𝑁) = (0..^(𝑁 + 1)))
43	42	eqcomd 2741	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (0..^(𝑁 + 1)) = (0...𝑁))
44	41, 43	syl 17	. . . . 5 ⊢ (𝜑 → (0..^(𝑁 + 1)) = (0...𝑁))
45	44	imaeq2d 6080	. . . 4 ⊢ (𝜑 → (𝐹 “ (0..^(𝑁 + 1))) = (𝐹 “ (0...𝑁)))
46	45	reseq2d 6000	. . 3 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1)))) = (𝐼 ↾ (𝐹 “ (0...𝑁))))
47	40, 46	eqtrid 2787	. 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 6912	. . . 4 ⊢ (𝑘 = 𝑁 → (𝐼‘(𝐹‘𝑘)) = (𝐼‘(𝐹‘𝑁)))
50		fveq2 6907	. . . . 5 ⊢ (𝑘 = 𝑁 → (𝑃‘𝑘) = (𝑃‘𝑁))
51		fvoveq1 7454	. . . . 5 ⊢ (𝑘 = 𝑁 → (𝑃‘(𝑘 + 1)) = (𝑃‘(𝑁 + 1)))
52	50, 51	preq12d 4746	. . . 4 ⊢ (𝑘 = 𝑁 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))})
53	49, 52	eqeq12d 2751	. . 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 29675	. . . 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 3623	. 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 30263	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 1537 ∈ wcel 2106 ∀wral 3059 {crab 3433 ∅c0 4339 ifcif 4531 {csn 4631 {cpr 4633 ⟨cop 4637 class class class wbr 5148 ↾ cres 5691 “ cima 5692 Fun wfun 6557 ‘cfv 6563 (class class class)co 7431 0cc0 11153 1c1 11154 + caddc 11156 ≤ cle 11294 2c2 12319 ℕ₀cn0 12524 ℤcz 12611 ...cfz 13544 ..^cfzo 13691 ♯chash 14366 ∥ cdvds 16287 Vtxcvtx 29028 iEdgciedg 29029 UPGraphcupgr 29112 VtxDegcvtxdg 29498 Walkscwlks 29629 Trailsctrls 29723 EulerPathsceupth 30226

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231

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 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-rp 13033 df-xadd 13153 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-word 14550 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-dvds 16288 df-vtx 29030 df-iedg 29031 df-edg 29080 df-uhgr 29090 df-ushgr 29091 df-upgr 29114 df-uspgr 29182 df-vtxdg 29499 df-wlks 29632 df-trls 29725 df-eupth 30227

This theorem is referenced by: eupth2lems 30267

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