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Mirrors > Home > MPE Home > Th. List > eupth2lemb | Structured version Visualization version GIF version |
Description: Lemma for eupth2 27945 (induction basis): There are no vertices of odd degree in an Eulerian path of length 0, having no edge and identical endpoints (the single vertex of the Eulerian path). Formerly part of proof 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) |
Ref | Expression |
---|---|
eupth2lemb | ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉)‘𝑥)} = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | z0even 15704 | . . . . 5 ⊢ 2 ∥ 0 | |
2 | eupth2.v | . . . . . . . . . . . 12 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | 2 | fvexi 6677 | . . . . . . . . . . 11 ⊢ 𝑉 ∈ V |
4 | eupth2.i | . . . . . . . . . . . . 13 ⊢ 𝐼 = (iEdg‘𝐺) | |
5 | 4 | fvexi 6677 | . . . . . . . . . . . 12 ⊢ 𝐼 ∈ V |
6 | 5 | resex 5892 | . . . . . . . . . . 11 ⊢ (𝐼 ↾ (𝐹 “ (0..^0))) ∈ V |
7 | 3, 6 | pm3.2i 471 | . . . . . . . . . 10 ⊢ (𝑉 ∈ V ∧ (𝐼 ↾ (𝐹 “ (0..^0))) ∈ V) |
8 | opvtxfv 26716 | . . . . . . . . . 10 ⊢ ((𝑉 ∈ V ∧ (𝐼 ↾ (𝐹 “ (0..^0))) ∈ V) → (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉) = 𝑉) | |
9 | 7, 8 | mp1i 13 | . . . . . . . . 9 ⊢ (𝜑 → (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉) = 𝑉) |
10 | 9 | eqcomd 2824 | . . . . . . . 8 ⊢ (𝜑 → 𝑉 = (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉)) |
11 | 10 | eleq2d 2895 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑉 ↔ 𝑥 ∈ (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉))) |
12 | 11 | biimpa 477 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉)) |
13 | opiedgfv 26719 | . . . . . . . . 9 ⊢ ((𝑉 ∈ V ∧ (𝐼 ↾ (𝐹 “ (0..^0))) ∈ V) → (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉) = (𝐼 ↾ (𝐹 “ (0..^0)))) | |
14 | 7, 13 | mp1i 13 | . . . . . . . 8 ⊢ (𝜑 → (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉) = (𝐼 ↾ (𝐹 “ (0..^0)))) |
15 | fzo0 13049 | . . . . . . . . . . . 12 ⊢ (0..^0) = ∅ | |
16 | 15 | imaeq2i 5920 | . . . . . . . . . . 11 ⊢ (𝐹 “ (0..^0)) = (𝐹 “ ∅) |
17 | ima0 5938 | . . . . . . . . . . 11 ⊢ (𝐹 “ ∅) = ∅ | |
18 | 16, 17 | eqtri 2841 | . . . . . . . . . 10 ⊢ (𝐹 “ (0..^0)) = ∅ |
19 | 18 | reseq2i 5843 | . . . . . . . . 9 ⊢ (𝐼 ↾ (𝐹 “ (0..^0))) = (𝐼 ↾ ∅) |
20 | res0 5850 | . . . . . . . . 9 ⊢ (𝐼 ↾ ∅) = ∅ | |
21 | 19, 20 | eqtri 2841 | . . . . . . . 8 ⊢ (𝐼 ↾ (𝐹 “ (0..^0))) = ∅ |
22 | 14, 21 | syl6eq 2869 | . . . . . . 7 ⊢ (𝜑 → (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉) = ∅) |
23 | 22 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉) = ∅) |
24 | eqid 2818 | . . . . . . 7 ⊢ (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉) = (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉) | |
25 | eqid 2818 | . . . . . . 7 ⊢ (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉) = (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉) | |
26 | 24, 25 | vtxdg0e 27183 | . . . . . 6 ⊢ ((𝑥 ∈ (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉) ∧ (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉) = ∅) → ((VtxDeg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉)‘𝑥) = 0) |
27 | 12, 23, 26 | syl2anc 584 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((VtxDeg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉)‘𝑥) = 0) |
28 | 1, 27 | breqtrrid 5095 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 2 ∥ ((VtxDeg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉)‘𝑥)) |
29 | 28 | notnotd 146 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ¬ ¬ 2 ∥ ((VtxDeg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉)‘𝑥)) |
30 | 29 | ralrimiva 3179 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 ¬ ¬ 2 ∥ ((VtxDeg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉)‘𝑥)) |
31 | rabeq0 4335 | . 2 ⊢ ({𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉)‘𝑥)} = ∅ ↔ ∀𝑥 ∈ 𝑉 ¬ ¬ 2 ∥ ((VtxDeg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉)‘𝑥)) | |
32 | 30, 31 | sylibr 235 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉)‘𝑥)} = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 {crab 3139 Vcvv 3492 ∅c0 4288 〈cop 4563 class class class wbr 5057 ↾ cres 5550 “ cima 5551 Fun wfun 6342 ‘cfv 6348 (class class class)co 7145 0cc0 10525 2c2 11680 ..^cfzo 13021 ∥ cdvds 15595 Vtxcvtx 26708 iEdgciedg 26709 UPGraphcupgr 26792 VtxDegcvtxdg 27174 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 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 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-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-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 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-nn 11627 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-xadd 12496 df-fz 12881 df-fzo 13022 df-hash 13679 df-dvds 15596 df-vtx 26710 df-iedg 26711 df-vtxdg 27175 |
This theorem is referenced by: eupth2 27945 |
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