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Mirrors > Home > MPE Home > Th. List > eupth2lem3 | Structured version Visualization version GIF version |
Description: Lemma for eupth2 27694. (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 27666 | . . . 4 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | |
7 | wlkcl 27068 | . . . 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 12918 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (♯‘𝐹) ∈ ℕ0 ∧ (𝑁 + 1) ≤ (♯‘𝐹)) → 𝑁 ∈ (0..^(♯‘𝐹))) | |
11 | 4, 8, 9, 10 | syl3anc 1362 | . 2 ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) |
12 | eupth2.u | . 2 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
13 | eupthistrl 27665 | . . 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 6533 | . . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))〉) |
17 | 1 | fvexi 6544 | . . . . 5 ⊢ 𝑉 ∈ V |
18 | 2 | fvexi 6544 | . . . . . 6 ⊢ 𝐼 ∈ V |
19 | 18 | resex 5772 | . . . . 5 ⊢ (𝐼 ↾ (𝐹 “ (0..^𝑁))) ∈ V |
20 | 17, 19 | opvtxfvi 26465 | . . . 4 ⊢ (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))〉) = 𝑉 |
21 | 16, 20 | eqtri 2817 | . . 3 ⊢ (Vtx‘𝐻) = 𝑉 |
22 | 21 | a1i 11 | . 2 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) |
23 | snex 5216 | . . . 4 ⊢ {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉} ∈ V | |
24 | 17, 23 | opvtxfvi 26465 | . . 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 6533 | . . . 4 ⊢ (Vtx‘𝑋) = (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))〉) |
28 | 18 | resex 5772 | . . . . 5 ⊢ (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1)))) ∈ V |
29 | 17, 28 | opvtxfvi 26465 | . . . 4 ⊢ (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))〉) = 𝑉 |
30 | 27, 29 | eqtri 2817 | . . 3 ⊢ (Vtx‘𝑋) = 𝑉 |
31 | 30 | a1i 11 | . 2 ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉) |
32 | 15 | fveq2i 6533 | . . . 4 ⊢ (iEdg‘𝐻) = (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))〉) |
33 | 17, 19 | opiedgfvi 26466 | . . . 4 ⊢ (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))〉) = (𝐼 ↾ (𝐹 “ (0..^𝑁))) |
34 | 32, 33 | eqtri 2817 | . . 3 ⊢ (iEdg‘𝐻) = (𝐼 ↾ (𝐹 “ (0..^𝑁))) |
35 | 34 | a1i 11 | . 2 ⊢ (𝜑 → (iEdg‘𝐻) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) |
36 | 17, 23 | opiedgfvi 26466 | . . 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 6533 | . . . 4 ⊢ (iEdg‘𝑋) = (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))〉) |
39 | 17, 28 | opiedgfvi 26466 | . . . 4 ⊢ (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))〉) = (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1)))) |
40 | 38, 39 | eqtri 2817 | . . 3 ⊢ (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1)))) |
41 | 4 | nn0zd 11923 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
42 | fzval3 12944 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (0...𝑁) = (0..^(𝑁 + 1))) | |
43 | 42 | eqcomd 2799 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (0..^(𝑁 + 1)) = (0...𝑁)) |
44 | 41, 43 | syl 17 | . . . . 5 ⊢ (𝜑 → (0..^(𝑁 + 1)) = (0...𝑁)) |
45 | 44 | imaeq2d 5798 | . . . 4 ⊢ (𝜑 → (𝐹 “ (0..^(𝑁 + 1))) = (𝐹 “ (0...𝑁))) |
46 | 45 | reseq2d 5726 | . . 3 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1)))) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) |
47 | 40, 46 | syl5eq 2841 | . 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 6535 | . . . 4 ⊢ (𝑘 = 𝑁 → (𝐼‘(𝐹‘𝑘)) = (𝐼‘(𝐹‘𝑁))) | |
50 | fveq2 6530 | . . . . 5 ⊢ (𝑘 = 𝑁 → (𝑃‘𝑘) = (𝑃‘𝑁)) | |
51 | fvoveq1 7030 | . . . . 5 ⊢ (𝑘 = 𝑁 → (𝑃‘(𝑘 + 1)) = (𝑃‘(𝑁 + 1))) | |
52 | 50, 51 | preq12d 4578 | . . . 4 ⊢ (𝑘 = 𝑁 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))}) |
53 | 49, 52 | eqeq12d 2808 | . . 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 27094 | . . . 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 3560 | . 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 27689 | 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 207 = wceq 1520 ∈ wcel 2079 ∀wral 3103 {crab 3107 ∅c0 4206 ifcif 4375 {csn 4466 {cpr 4468 〈cop 4472 class class class wbr 4956 ↾ cres 5437 “ cima 5438 Fun wfun 6211 ‘cfv 6217 (class class class)co 7007 0cc0 10372 1c1 10373 + caddc 10375 ≤ cle 10511 2c2 11529 ℕ0cn0 11734 ℤcz 11818 ...cfz 12731 ..^cfzo 12872 ♯chash 13528 ∥ cdvds 15428 Vtxcvtx 26452 iEdgciedg 26453 UPGraphcupgr 26536 VtxDegcvtxdg 26918 Walkscwlks 27049 Trailsctrls 27145 EulerPathsceupth 27651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 ax-pre-sup 10450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-ifp 1054 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-int 4777 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-om 7428 df-1st 7536 df-2nd 7537 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-1o 7944 df-2o 7945 df-oadd 7948 df-er 8130 df-map 8249 df-pm 8250 df-en 8348 df-dom 8349 df-sdom 8350 df-fin 8351 df-sup 8742 df-inf 8743 df-dju 9165 df-card 9203 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-div 11135 df-nn 11476 df-2 11537 df-3 11538 df-n0 11735 df-xnn0 11805 df-z 11819 df-uz 12083 df-rp 12229 df-xadd 12347 df-fz 12732 df-fzo 12873 df-seq 13208 df-exp 13268 df-hash 13529 df-word 13696 df-cj 14280 df-re 14281 df-im 14282 df-sqrt 14416 df-abs 14417 df-dvds 15429 df-vtx 26454 df-iedg 26455 df-edg 26504 df-uhgr 26514 df-ushgr 26515 df-upgr 26538 df-uspgr 26606 df-vtxdg 26919 df-wlks 27052 df-trls 27147 df-eupth 27652 |
This theorem is referenced by: eupth2lems 27693 |
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