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Theorem eupth2lem3 30324

Description: Lemma for eupth2 30327. (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 30300	. . . 4 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
7		wlkcl 29702	. . . 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 13651	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ₀ ∧ (𝑁 + 1) ≤ (♯‘𝐹)) → 𝑁 ∈ (0..^(♯‘𝐹)))
11	4, 8, 9, 10	syl3anc 1374	. 2 ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))
12		eupth2.u	. 2 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
13		eupthistrl 30299	. . 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 6838	. . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))⟩)
17	1	fvexi 6849	. . . . 5 ⊢ 𝑉 ∈ V
18	2	fvexi 6849	. . . . . 6 ⊢ 𝐼 ∈ V
19	18	resex 5989	. . . . 5 ⊢ (𝐼 ↾ (𝐹 “ (0..^𝑁))) ∈ V
20	17, 19	opvtxfvi 29095	. . . 4 ⊢ (Vtx‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))⟩) = 𝑉
21	16, 20	eqtri 2760	. . 3 ⊢ (Vtx‘𝐻) = 𝑉
22	21	a1i 11	. 2 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)
23		snex 5377	. . . 4 ⊢ {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩} ∈ V
24	17, 23	opvtxfvi 29095	. . 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 6838	. . . 4 ⊢ (Vtx‘𝑋) = (Vtx‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))⟩)
28	18	resex 5989	. . . . 5 ⊢ (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1)))) ∈ V
29	17, 28	opvtxfvi 29095	. . . 4 ⊢ (Vtx‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))⟩) = 𝑉
30	27, 29	eqtri 2760	. . 3 ⊢ (Vtx‘𝑋) = 𝑉
31	30	a1i 11	. 2 ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉)
32	15	fveq2i 6838	. . . 4 ⊢ (iEdg‘𝐻) = (iEdg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))⟩)
33	17, 19	opiedgfvi 29096	. . . 4 ⊢ (iEdg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))⟩) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))
34	32, 33	eqtri 2760	. . 3 ⊢ (iEdg‘𝐻) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))
35	34	a1i 11	. 2 ⊢ (𝜑 → (iEdg‘𝐻) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))
36	17, 23	opiedgfvi 29096	. . 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 6838	. . . 4 ⊢ (iEdg‘𝑋) = (iEdg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))⟩)
39	17, 28	opiedgfvi 29096	. . . 4 ⊢ (iEdg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))⟩) = (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))
40	38, 39	eqtri 2760	. . 3 ⊢ (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))
41	4	nn0zd 12543	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
42		fzval3 13683	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → (0...𝑁) = (0..^(𝑁 + 1)))
43	42	eqcomd 2743	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (0..^(𝑁 + 1)) = (0...𝑁))
44	41, 43	syl 17	. . . . 5 ⊢ (𝜑 → (0..^(𝑁 + 1)) = (0...𝑁))
45	44	imaeq2d 6020	. . . 4 ⊢ (𝜑 → (𝐹 “ (0..^(𝑁 + 1))) = (𝐹 “ (0...𝑁)))
46	45	reseq2d 5939	. . 3 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1)))) = (𝐼 ↾ (𝐹 “ (0...𝑁))))
47	40, 46	eqtrid 2784	. 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 6840	. . . 4 ⊢ (𝑘 = 𝑁 → (𝐼‘(𝐹‘𝑘)) = (𝐼‘(𝐹‘𝑁)))
50		fveq2 6835	. . . . 5 ⊢ (𝑘 = 𝑁 → (𝑃‘𝑘) = (𝑃‘𝑁))
51		fvoveq1 7384	. . . . 5 ⊢ (𝑘 = 𝑁 → (𝑃‘(𝑘 + 1)) = (𝑃‘(𝑁 + 1)))
52	50, 51	preq12d 4686	. . . 4 ⊢ (𝑘 = 𝑁 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))})
53	49, 52	eqeq12d 2753	. . 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 29728	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃) → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
57	54, 55, 56	syl2anc 585	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
58	53, 57, 11	rspcdva 3566	. 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 30322	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 1542 ∈ wcel 2114 ∀wral 3052 {crab 3390 ∅c0 4274 ifcif 4467 {csn 4568 {cpr 4570 ⟨cop 4574 class class class wbr 5086 ↾ cres 5627 “ cima 5628 Fun wfun 6487 ‘cfv 6493 (class class class)co 7361 0cc0 11032 1c1 11033 + caddc 11035 ≤ cle 11174 2c2 12230 ℕ₀cn0 12431 ℤcz 12518 ...cfz 13455 ..^cfzo 13602 ♯chash 14286 ∥ cdvds 16215 Vtxcvtx 29082 iEdgciedg 29083 UPGraphcupgr 29166 VtxDegcvtxdg 29552 Walkscwlks 29683 Trailsctrls 29775 EulerPathsceupth 30285

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109 ax-pre-sup 11110

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ifp 1064 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7812 df-1st 7936 df-2nd 7937 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-oadd 8403 df-er 8637 df-map 8769 df-pm 8770 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-sup 9349 df-inf 9350 df-dju 9819 df-card 9857 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-div 11802 df-nn 12169 df-2 12238 df-3 12239 df-n0 12432 df-xnn0 12505 df-z 12519 df-uz 12783 df-rp 12937 df-xadd 13058 df-fz 13456 df-fzo 13603 df-seq 13958 df-exp 14018 df-hash 14287 df-word 14470 df-cj 15055 df-re 15056 df-im 15057 df-sqrt 15191 df-abs 15192 df-dvds 16216 df-vtx 29084 df-iedg 29085 df-edg 29134 df-uhgr 29144 df-ushgr 29145 df-upgr 29168 df-uspgr 29236 df-vtxdg 29553 df-wlks 29686 df-trls 29777 df-eupth 30286

This theorem is referenced by: eupth2lems 30326

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