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Mirrors > Home > MPE Home > Th. List > eupth2lem3 | Structured version Visualization version GIF version |
Description: Lemma for eupth2 28890. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.) |
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)})) |
Ref | Expression |
---|---|
eupth2lem3 | ⊢ (𝜑 → (¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑈) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))}))) |
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 28863 | . . . 4 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | |
7 | wlkcl 28270 | . . . 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 13535 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (♯‘𝐹) ∈ ℕ0 ∧ (𝑁 + 1) ≤ (♯‘𝐹)) → 𝑁 ∈ (0..^(♯‘𝐹))) | |
11 | 4, 8, 9, 10 | syl3anc 1371 | . 2 ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) |
12 | eupth2.u | . 2 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
13 | eupthistrl 28862 | . . 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 6832 | . . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))〉) |
17 | 1 | fvexi 6843 | . . . . 5 ⊢ 𝑉 ∈ V |
18 | 2 | fvexi 6843 | . . . . . 6 ⊢ 𝐼 ∈ V |
19 | 18 | resex 5975 | . . . . 5 ⊢ (𝐼 ↾ (𝐹 “ (0..^𝑁))) ∈ V |
20 | 17, 19 | opvtxfvi 27667 | . . . 4 ⊢ (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))〉) = 𝑉 |
21 | 16, 20 | eqtri 2765 | . . 3 ⊢ (Vtx‘𝐻) = 𝑉 |
22 | 21 | a1i 11 | . 2 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) |
23 | snex 5380 | . . . 4 ⊢ {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉} ∈ V | |
24 | 17, 23 | opvtxfvi 27667 | . . 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 6832 | . . . 4 ⊢ (Vtx‘𝑋) = (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))〉) |
28 | 18 | resex 5975 | . . . . 5 ⊢ (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1)))) ∈ V |
29 | 17, 28 | opvtxfvi 27667 | . . . 4 ⊢ (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))〉) = 𝑉 |
30 | 27, 29 | eqtri 2765 | . . 3 ⊢ (Vtx‘𝑋) = 𝑉 |
31 | 30 | a1i 11 | . 2 ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉) |
32 | 15 | fveq2i 6832 | . . . 4 ⊢ (iEdg‘𝐻) = (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))〉) |
33 | 17, 19 | opiedgfvi 27668 | . . . 4 ⊢ (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))〉) = (𝐼 ↾ (𝐹 “ (0..^𝑁))) |
34 | 32, 33 | eqtri 2765 | . . 3 ⊢ (iEdg‘𝐻) = (𝐼 ↾ (𝐹 “ (0..^𝑁))) |
35 | 34 | a1i 11 | . 2 ⊢ (𝜑 → (iEdg‘𝐻) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) |
36 | 17, 23 | opiedgfvi 27668 | . . 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 6832 | . . . 4 ⊢ (iEdg‘𝑋) = (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))〉) |
39 | 17, 28 | opiedgfvi 27668 | . . . 4 ⊢ (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))〉) = (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1)))) |
40 | 38, 39 | eqtri 2765 | . . 3 ⊢ (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1)))) |
41 | 4 | nn0zd 12529 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
42 | fzval3 13561 | . . . . . . 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 6003 | . . . 4 ⊢ (𝜑 → (𝐹 “ (0..^(𝑁 + 1))) = (𝐹 “ (0...𝑁))) |
46 | 45 | reseq2d 5927 | . . 3 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1)))) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) |
47 | 40, 46 | eqtrid 2789 | . 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 6834 | . . . 4 ⊢ (𝑘 = 𝑁 → (𝐼‘(𝐹‘𝑘)) = (𝐼‘(𝐹‘𝑁))) | |
50 | fveq2 6829 | . . . . 5 ⊢ (𝑘 = 𝑁 → (𝑃‘𝑘) = (𝑃‘𝑁)) | |
51 | fvoveq1 7364 | . . . . 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 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 28297 | . . . 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 3574 | . 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 28885 | 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 205 = wceq 1541 ∈ wcel 2106 ∀wral 3062 {crab 3404 ∅c0 4273 ifcif 4477 {csn 4577 {cpr 4579 〈cop 4583 class class class wbr 5096 ↾ cres 5626 “ cima 5627 Fun wfun 6477 ‘cfv 6483 (class class class)co 7341 0cc0 10976 1c1 10977 + caddc 10979 ≤ cle 11115 2c2 12133 ℕ0cn0 12338 ℤcz 12424 ...cfz 13344 ..^cfzo 13487 ♯chash 14149 ∥ cdvds 16062 Vtxcvtx 27654 iEdgciedg 27655 UPGraphcupgr 27738 VtxDegcvtxdg 28120 Walkscwlks 28251 Trailsctrls 28345 EulerPathsceupth 28848 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5233 ax-sep 5247 ax-nul 5254 ax-pow 5312 ax-pr 5376 ax-un 7654 ax-cnex 11032 ax-resscn 11033 ax-1cn 11034 ax-icn 11035 ax-addcl 11036 ax-addrcl 11037 ax-mulcl 11038 ax-mulrcl 11039 ax-mulcom 11040 ax-addass 11041 ax-mulass 11042 ax-distr 11043 ax-i2m1 11044 ax-1ne0 11045 ax-1rid 11046 ax-rnegex 11047 ax-rrecex 11048 ax-cnre 11049 ax-pre-lttri 11050 ax-pre-lttrn 11051 ax-pre-ltadd 11052 ax-pre-mulgt0 11053 ax-pre-sup 11054 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-ifp 1062 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3350 df-reu 3351 df-rab 3405 df-v 3444 df-sbc 3731 df-csb 3847 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3920 df-nul 4274 df-if 4478 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4857 df-int 4899 df-iun 4947 df-br 5097 df-opab 5159 df-mpt 5180 df-tr 5214 df-id 5522 df-eprel 5528 df-po 5536 df-so 5537 df-fr 5579 df-we 5581 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6242 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6435 df-fun 6485 df-fn 6486 df-f 6487 df-f1 6488 df-fo 6489 df-f1o 6490 df-fv 6491 df-riota 7297 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7785 df-1st 7903 df-2nd 7904 df-frecs 8171 df-wrecs 8202 df-recs 8276 df-rdg 8315 df-1o 8371 df-2o 8372 df-oadd 8375 df-er 8573 df-map 8692 df-pm 8693 df-en 8809 df-dom 8810 df-sdom 8811 df-fin 8812 df-sup 9303 df-inf 9304 df-dju 9762 df-card 9800 df-pnf 11116 df-mnf 11117 df-xr 11118 df-ltxr 11119 df-le 11120 df-sub 11312 df-neg 11313 df-div 11738 df-nn 12079 df-2 12141 df-3 12142 df-n0 12339 df-xnn0 12411 df-z 12425 df-uz 12688 df-rp 12836 df-xadd 12954 df-fz 13345 df-fzo 13488 df-seq 13827 df-exp 13888 df-hash 14150 df-word 14322 df-cj 14909 df-re 14910 df-im 14911 df-sqrt 15045 df-abs 15046 df-dvds 16063 df-vtx 27656 df-iedg 27657 df-edg 27706 df-uhgr 27716 df-ushgr 27717 df-upgr 27740 df-uspgr 27808 df-vtxdg 28121 df-wlks 28254 df-trls 28347 df-eupth 28849 |
This theorem is referenced by: eupth2lems 28889 |
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