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Mirrors > Home > MPE Home > Th. List > eupth2lem3 | Structured version Visualization version GIF version |
Description: Lemma for eupth2 27416. (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 27389 | . . . 4 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | |
7 | wlkcl 26745 | . . . 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 12718 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (♯‘𝐹) ∈ ℕ0 ∧ (𝑁 + 1) ≤ (♯‘𝐹)) → 𝑁 ∈ (0..^(♯‘𝐹))) | |
11 | 4, 8, 9, 10 | syl3anc 1475 | . 2 ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) |
12 | eupth2.u | . 2 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
13 | eupthistrl 27388 | . . 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 6335 | . . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))〉) |
17 | fvex 6342 | . . . . . 6 ⊢ (Vtx‘𝐺) ∈ V | |
18 | 1, 17 | eqeltri 2845 | . . . . 5 ⊢ 𝑉 ∈ V |
19 | fvex 6342 | . . . . . . 7 ⊢ (iEdg‘𝐺) ∈ V | |
20 | 2, 19 | eqeltri 2845 | . . . . . 6 ⊢ 𝐼 ∈ V |
21 | 20 | resex 5584 | . . . . 5 ⊢ (𝐼 ↾ (𝐹 “ (0..^𝑁))) ∈ V |
22 | 18, 21 | opvtxfvi 26109 | . . . 4 ⊢ (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))〉) = 𝑉 |
23 | 16, 22 | eqtri 2792 | . . 3 ⊢ (Vtx‘𝐻) = 𝑉 |
24 | 23 | a1i 11 | . 2 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) |
25 | snex 5036 | . . . 4 ⊢ {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉} ∈ V | |
26 | 18, 25 | opvtxfvi 26109 | . . 3 ⊢ (Vtx‘〈𝑉, {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}〉) = 𝑉 |
27 | 26 | a1i 11 | . 2 ⊢ (𝜑 → (Vtx‘〈𝑉, {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}〉) = 𝑉) |
28 | eupth2.x | . . . . 5 ⊢ 𝑋 = 〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))〉 | |
29 | 28 | fveq2i 6335 | . . . 4 ⊢ (Vtx‘𝑋) = (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))〉) |
30 | 20 | resex 5584 | . . . . 5 ⊢ (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1)))) ∈ V |
31 | 18, 30 | opvtxfvi 26109 | . . . 4 ⊢ (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))〉) = 𝑉 |
32 | 29, 31 | eqtri 2792 | . . 3 ⊢ (Vtx‘𝑋) = 𝑉 |
33 | 32 | a1i 11 | . 2 ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉) |
34 | 15 | fveq2i 6335 | . . . 4 ⊢ (iEdg‘𝐻) = (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))〉) |
35 | 18, 21 | opiedgfvi 26110 | . . . 4 ⊢ (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))〉) = (𝐼 ↾ (𝐹 “ (0..^𝑁))) |
36 | 34, 35 | eqtri 2792 | . . 3 ⊢ (iEdg‘𝐻) = (𝐼 ↾ (𝐹 “ (0..^𝑁))) |
37 | 36 | a1i 11 | . 2 ⊢ (𝜑 → (iEdg‘𝐻) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) |
38 | 18, 25 | opiedgfvi 26110 | . . 3 ⊢ (iEdg‘〈𝑉, {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}〉) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉} |
39 | 38 | a1i 11 | . 2 ⊢ (𝜑 → (iEdg‘〈𝑉, {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}〉) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) |
40 | 28 | fveq2i 6335 | . . . 4 ⊢ (iEdg‘𝑋) = (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))〉) |
41 | 18, 30 | opiedgfvi 26110 | . . . 4 ⊢ (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))〉) = (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1)))) |
42 | 40, 41 | eqtri 2792 | . . 3 ⊢ (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1)))) |
43 | 4 | nn0zd 11681 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
44 | fzval3 12744 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (0...𝑁) = (0..^(𝑁 + 1))) | |
45 | 44 | eqcomd 2776 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (0..^(𝑁 + 1)) = (0...𝑁)) |
46 | 43, 45 | syl 17 | . . . . 5 ⊢ (𝜑 → (0..^(𝑁 + 1)) = (0...𝑁)) |
47 | 46 | imaeq2d 5607 | . . . 4 ⊢ (𝜑 → (𝐹 “ (0..^(𝑁 + 1))) = (𝐹 “ (0...𝑁))) |
48 | 47 | reseq2d 5534 | . . 3 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1)))) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) |
49 | 42, 48 | syl5eq 2816 | . 2 ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) |
50 | eupth2.o | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐻)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑁), ∅, {(𝑃‘0), (𝑃‘𝑁)})) | |
51 | fveq2 6332 | . . . . 5 ⊢ (𝑘 = 𝑁 → (𝐹‘𝑘) = (𝐹‘𝑁)) | |
52 | 51 | fveq2d 6336 | . . . 4 ⊢ (𝑘 = 𝑁 → (𝐼‘(𝐹‘𝑘)) = (𝐼‘(𝐹‘𝑁))) |
53 | fveq2 6332 | . . . . 5 ⊢ (𝑘 = 𝑁 → (𝑃‘𝑘) = (𝑃‘𝑁)) | |
54 | fvoveq1 6815 | . . . . 5 ⊢ (𝑘 = 𝑁 → (𝑃‘(𝑘 + 1)) = (𝑃‘(𝑁 + 1))) | |
55 | 53, 54 | preq12d 4410 | . . . 4 ⊢ (𝑘 = 𝑁 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))}) |
56 | 52, 55 | eqeq12d 2785 | . . 3 ⊢ (𝑘 = 𝑁 → ((𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))})) |
57 | eupth2.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ UPGraph) | |
58 | 5, 6 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) |
59 | 2 | upgrwlkedg 26772 | . . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃) → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) |
60 | 57, 58, 59 | syl2anc 565 | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) |
61 | 56, 60, 11 | rspcdva 3464 | . 2 ⊢ (𝜑 → (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))}) |
62 | 1, 2, 3, 11, 12, 14, 24, 27, 33, 37, 39, 49, 50, 61 | eupth2lem3lem7 27411 | 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 196 = wceq 1630 ∈ wcel 2144 ∀wral 3060 {crab 3064 Vcvv 3349 ∅c0 4061 ifcif 4223 {csn 4314 {cpr 4316 〈cop 4320 class class class wbr 4784 ↾ cres 5251 “ cima 5252 Fun wfun 6025 ‘cfv 6031 (class class class)co 6792 0cc0 10137 1c1 10138 + caddc 10140 ≤ cle 10276 2c2 11271 ℕ0cn0 11493 ℤcz 11578 ...cfz 12532 ..^cfzo 12672 ♯chash 13320 ∥ cdvds 15188 Vtxcvtx 26094 iEdgciedg 26095 UPGraphcupgr 26195 VtxDegcvtxdg 26595 Walkscwlks 26726 Trailsctrls 26821 EulerPathsceupth 27374 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-rep 4902 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 ax-un 7095 ax-cnex 10193 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 ax-pre-mulgt0 10214 ax-pre-sup 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-ifp 1049 df-3or 1071 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-nel 3046 df-ral 3065 df-rex 3066 df-reu 3067 df-rmo 3068 df-rab 3069 df-v 3351 df-sbc 3586 df-csb 3681 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-pss 3737 df-nul 4062 df-if 4224 df-pw 4297 df-sn 4315 df-pr 4317 df-tp 4319 df-op 4321 df-uni 4573 df-int 4610 df-iun 4654 df-br 4785 df-opab 4845 df-mpt 4862 df-tr 4885 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-om 7212 df-1st 7314 df-2nd 7315 df-wrecs 7558 df-recs 7620 df-rdg 7658 df-1o 7712 df-2o 7713 df-oadd 7716 df-er 7895 df-map 8010 df-pm 8011 df-en 8109 df-dom 8110 df-sdom 8111 df-fin 8112 df-sup 8503 df-inf 8504 df-card 8964 df-cda 9191 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-sub 10469 df-neg 10470 df-div 10886 df-nn 11222 df-2 11280 df-3 11281 df-n0 11494 df-xnn0 11565 df-z 11579 df-uz 11888 df-rp 12035 df-xadd 12151 df-fz 12533 df-fzo 12673 df-seq 13008 df-exp 13067 df-hash 13321 df-word 13494 df-cj 14046 df-re 14047 df-im 14048 df-sqrt 14182 df-abs 14183 df-dvds 15189 df-vtx 26096 df-iedg 26097 df-edg 26160 df-uhgr 26173 df-ushgr 26174 df-upgr 26197 df-uspgr 26266 df-vtxdg 26596 df-wlks 26729 df-trls 26823 df-eupth 27375 |
This theorem is referenced by: eupth2lems 27415 |
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