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Mirrors > Home > MPE Home > Th. List > eupth2lem3 | Structured version Visualization version GIF version |
Description: Lemma for eupth2 27945. (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 27918 | . . . 4 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | |
7 | wlkcl 27324 | . . . 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 13068 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (♯‘𝐹) ∈ ℕ0 ∧ (𝑁 + 1) ≤ (♯‘𝐹)) → 𝑁 ∈ (0..^(♯‘𝐹))) | |
11 | 4, 8, 9, 10 | syl3anc 1363 | . 2 ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) |
12 | eupth2.u | . 2 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
13 | eupthistrl 27917 | . . 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 6666 | . . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))〉) |
17 | 1 | fvexi 6677 | . . . . 5 ⊢ 𝑉 ∈ V |
18 | 2 | fvexi 6677 | . . . . . 6 ⊢ 𝐼 ∈ V |
19 | 18 | resex 5892 | . . . . 5 ⊢ (𝐼 ↾ (𝐹 “ (0..^𝑁))) ∈ V |
20 | 17, 19 | opvtxfvi 26721 | . . . 4 ⊢ (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))〉) = 𝑉 |
21 | 16, 20 | eqtri 2841 | . . 3 ⊢ (Vtx‘𝐻) = 𝑉 |
22 | 21 | a1i 11 | . 2 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) |
23 | snex 5322 | . . . 4 ⊢ {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉} ∈ V | |
24 | 17, 23 | opvtxfvi 26721 | . . 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 6666 | . . . 4 ⊢ (Vtx‘𝑋) = (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))〉) |
28 | 18 | resex 5892 | . . . . 5 ⊢ (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1)))) ∈ V |
29 | 17, 28 | opvtxfvi 26721 | . . . 4 ⊢ (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))〉) = 𝑉 |
30 | 27, 29 | eqtri 2841 | . . 3 ⊢ (Vtx‘𝑋) = 𝑉 |
31 | 30 | a1i 11 | . 2 ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉) |
32 | 15 | fveq2i 6666 | . . . 4 ⊢ (iEdg‘𝐻) = (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))〉) |
33 | 17, 19 | opiedgfvi 26722 | . . . 4 ⊢ (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))〉) = (𝐼 ↾ (𝐹 “ (0..^𝑁))) |
34 | 32, 33 | eqtri 2841 | . . 3 ⊢ (iEdg‘𝐻) = (𝐼 ↾ (𝐹 “ (0..^𝑁))) |
35 | 34 | a1i 11 | . 2 ⊢ (𝜑 → (iEdg‘𝐻) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) |
36 | 17, 23 | opiedgfvi 26722 | . . 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 6666 | . . . 4 ⊢ (iEdg‘𝑋) = (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))〉) |
39 | 17, 28 | opiedgfvi 26722 | . . . 4 ⊢ (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))〉) = (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1)))) |
40 | 38, 39 | eqtri 2841 | . . 3 ⊢ (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1)))) |
41 | 4 | nn0zd 12073 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
42 | fzval3 13094 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (0...𝑁) = (0..^(𝑁 + 1))) | |
43 | 42 | eqcomd 2824 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (0..^(𝑁 + 1)) = (0...𝑁)) |
44 | 41, 43 | syl 17 | . . . . 5 ⊢ (𝜑 → (0..^(𝑁 + 1)) = (0...𝑁)) |
45 | 44 | imaeq2d 5922 | . . . 4 ⊢ (𝜑 → (𝐹 “ (0..^(𝑁 + 1))) = (𝐹 “ (0...𝑁))) |
46 | 45 | reseq2d 5846 | . . 3 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1)))) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) |
47 | 40, 46 | syl5eq 2865 | . 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 6668 | . . . 4 ⊢ (𝑘 = 𝑁 → (𝐼‘(𝐹‘𝑘)) = (𝐼‘(𝐹‘𝑁))) | |
50 | fveq2 6663 | . . . . 5 ⊢ (𝑘 = 𝑁 → (𝑃‘𝑘) = (𝑃‘𝑁)) | |
51 | fvoveq1 7168 | . . . . 5 ⊢ (𝑘 = 𝑁 → (𝑃‘(𝑘 + 1)) = (𝑃‘(𝑁 + 1))) | |
52 | 50, 51 | preq12d 4669 | . . . 4 ⊢ (𝑘 = 𝑁 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))}) |
53 | 49, 52 | eqeq12d 2834 | . . 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 27350 | . . . 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 3622 | . 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 27940 | 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 1528 ∈ wcel 2105 ∀wral 3135 {crab 3139 ∅c0 4288 ifcif 4463 {csn 4557 {cpr 4559 〈cop 4563 class class class wbr 5057 ↾ cres 5550 “ cima 5551 Fun wfun 6342 ‘cfv 6348 (class class class)co 7145 0cc0 10525 1c1 10526 + caddc 10528 ≤ cle 10664 2c2 11680 ℕ0cn0 11885 ℤcz 11969 ...cfz 12880 ..^cfzo 13021 ♯chash 13678 ∥ cdvds 15595 Vtxcvtx 26708 iEdgciedg 26709 UPGraphcupgr 26792 VtxDegcvtxdg 27174 Walkscwlks 27305 Trailsctrls 27399 EulerPathsceupth 27903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ifp 1055 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-dju 9318 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-rp 12378 df-xadd 12496 df-fz 12881 df-fzo 13022 df-seq 13358 df-exp 13418 df-hash 13679 df-word 13850 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-dvds 15596 df-vtx 26710 df-iedg 26711 df-edg 26760 df-uhgr 26770 df-ushgr 26771 df-upgr 26794 df-uspgr 26862 df-vtxdg 27175 df-wlks 27308 df-trls 27401 df-eupth 27904 |
This theorem is referenced by: eupth2lems 27944 |
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