eupth2lem3 - Metamath Proof Explorer

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Theorem eupth2lem3 30894

Description: Lemma for eupth2 30897. (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 30870	. . . 4 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
7		wlkcl 30253	. . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ₀)
8	5, 6, 7	3syl 19	. . 3 ⊢ (𝜑 → (♯‘𝐹) ∈ ℕ₀)
9		eupth2.l	. . 3 ⊢ (𝜑 → (𝑁 + 1) ≤ (♯‘𝐹))
10		nn0p1elfzo 13877	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ₀ ∧ (𝑁 + 1) ≤ (♯‘𝐹)) → 𝑁 ∈ (0..^(♯‘𝐹)))
11	4, 8, 9, 10	syl3anc 1399	. 2 ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))
12		eupth2.u	. 2 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
13		eupthistrl 30869	. . 3 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃)
14	5, 13	syl 18	. 2 ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)
15		eupth2.h	. . . . 5 ⊢ 𝐻 = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))⟩
16	15	fveq2i 6919	. . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))⟩)
17	1	fvexi 6930	. . . . 5 ⊢ 𝑉 ∈ V
18	2	fvexi 6930	. . . . . 6 ⊢ 𝐼 ∈ V
19	18	resex 6036	. . . . 5 ⊢ (𝐼 ↾ (𝐹 “ (0..^𝑁))) ∈ V
20	17, 19	opvtxfvi 29644	. . . 4 ⊢ (Vtx‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))⟩) = 𝑉
21	16, 20	eqtri 2786	. . 3 ⊢ (Vtx‘𝐻) = 𝑉
22	21	a1i 11	. 2 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)
23		snex 5404	. . . 4 ⊢ {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩} ∈ V
24	17, 23	opvtxfvi 29644	. . 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 6919	. . . 4 ⊢ (Vtx‘𝑋) = (Vtx‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))⟩)
28	18	resex 6036	. . . . 5 ⊢ (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1)))) ∈ V
29	17, 28	opvtxfvi 29644	. . . 4 ⊢ (Vtx‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))⟩) = 𝑉
30	27, 29	eqtri 2786	. . 3 ⊢ (Vtx‘𝑋) = 𝑉
31	30	a1i 11	. 2 ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉)
32	15	fveq2i 6919	. . . 4 ⊢ (iEdg‘𝐻) = (iEdg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))⟩)
33	17, 19	opiedgfvi 29645	. . . 4 ⊢ (iEdg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))⟩) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))
34	32, 33	eqtri 2786	. . 3 ⊢ (iEdg‘𝐻) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))
35	34	a1i 11	. 2 ⊢ (𝜑 → (iEdg‘𝐻) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))
36	17, 23	opiedgfvi 29645	. . 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 6919	. . . 4 ⊢ (iEdg‘𝑋) = (iEdg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))⟩)
39	17, 28	opiedgfvi 29645	. . . 4 ⊢ (iEdg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))⟩) = (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))
40	38, 39	eqtri 2786	. . 3 ⊢ (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))
41	4	nn0zd 12758	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
42		fzval3 13909	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → (0...𝑁) = (0..^(𝑁 + 1)))
43	42	eqcomd 2769	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (0..^(𝑁 + 1)) = (0...𝑁))
44	41, 43	syl 18	. . . . 5 ⊢ (𝜑 → (0..^(𝑁 + 1)) = (0...𝑁))
45	44	imaeq2d 6070	. . . 4 ⊢ (𝜑 → (𝐹 “ (0..^(𝑁 + 1))) = (𝐹 “ (0...𝑁)))
46	45	reseq2d 5986	. . 3 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1)))) = (𝐼 ↾ (𝐹 “ (0...𝑁))))
47	40, 46	eqtrid 2810	. 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 6921	. . . 4 ⊢ (𝑘 = 𝑁 → (𝐼‘(𝐹‘𝑘)) = (𝐼‘(𝐹‘𝑁)))
50		fveq2 6916	. . . . 5 ⊢ (𝑘 = 𝑁 → (𝑃‘𝑘) = (𝑃‘𝑁))
51		fvoveq1 7474	. . . . 5 ⊢ (𝑘 = 𝑁 → (𝑃‘(𝑘 + 1)) = (𝑃‘(𝑁 + 1)))
52	50, 51	preq12d 4704	. . . 4 ⊢ (𝑘 = 𝑁 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))})
53	49, 52	eqeq12d 2779	. . 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 18	. . . 4 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃)
56	2	upgrwlkedg 30279	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃) → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
57	54, 55, 56	syl2anc 596	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
58	53, 57, 11	rspcdva 3581	. 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 30892	1 ⊢ (𝜑 → (¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑈) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))})))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 = wceq 1570 ∈ wcel 2147 ∀wral 3080 {crab 3416 ∅c0 4281 ifcif 4485 {csn 4588 {cpr 4590 ⟨cop 4840 class class class wbr 5108 ↾ cres 5660 “ cima 5661 Fun wfun 6560 ‘cfv 6566 (class class class)co 7451 0cc0 11239 1c1 11240 + caddc 11242 ≤ cle 11383 2c2 12437 ℕ₀cn0 12646 ℤcz 12733 ...cfz 13679 ..^cfzo 13828 ♯chash 14515 ∥ cdvds 16466 Vtxcvtx 29631 iEdgciedg 29632 UPGraphcupgr 29715 VtxDegcvtxdg 30103 Walkscwlks 30234 Trailsctrls 30330 EulerPathsceupth 30855

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-ifp 1079 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-1st 8035 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-oadd 8518 df-er 8755 df-map 8887 df-pm 8888 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-sup 9471 df-inf 9472 df-dju 10019 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-n0 12647 df-xnn0 12720 df-z 12734 df-uz 13006 df-rp 13161 df-xadd 13282 df-fz 13680 df-fzo 13829 df-seq 14185 df-exp 14245 df-hash 14516 df-word 14700 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-dvds 16467 df-vtx 29633 df-iedg 29634 df-edg 29683 df-uhgr 29693 df-ushgr 29694 df-upgr 29717 df-uspgr 29788 df-vtxdg 30104 df-wlks 30237 df-trls 30332 df-eupth 30856

This theorem is used by: eupth2lems 30896

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