Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > eupth2lemb | Structured version Visualization version GIF version |
Description: Lemma for eupth2 28020 (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 28020. (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 15718 | . . . . 5 ⊢ 2 ∥ 0 | |
2 | eupth2.v | . . . . . . . . . . . 12 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | 2 | fvexi 6686 | . . . . . . . . . . 11 ⊢ 𝑉 ∈ V |
4 | eupth2.i | . . . . . . . . . . . . 13 ⊢ 𝐼 = (iEdg‘𝐺) | |
5 | 4 | fvexi 6686 | . . . . . . . . . . . 12 ⊢ 𝐼 ∈ V |
6 | 5 | resex 5901 | . . . . . . . . . . 11 ⊢ (𝐼 ↾ (𝐹 “ (0..^0))) ∈ V |
7 | 3, 6 | pm3.2i 473 | . . . . . . . . . 10 ⊢ (𝑉 ∈ V ∧ (𝐼 ↾ (𝐹 “ (0..^0))) ∈ V) |
8 | opvtxfv 26791 | . . . . . . . . . 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 2829 | . . . . . . . 8 ⊢ (𝜑 → 𝑉 = (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉)) |
11 | 10 | eleq2d 2900 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑉 ↔ 𝑥 ∈ (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉))) |
12 | 11 | biimpa 479 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉)) |
13 | opiedgfv 26794 | . . . . . . . . 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 13064 | . . . . . . . . . . . 12 ⊢ (0..^0) = ∅ | |
16 | 15 | imaeq2i 5929 | . . . . . . . . . . 11 ⊢ (𝐹 “ (0..^0)) = (𝐹 “ ∅) |
17 | ima0 5947 | . . . . . . . . . . 11 ⊢ (𝐹 “ ∅) = ∅ | |
18 | 16, 17 | eqtri 2846 | . . . . . . . . . 10 ⊢ (𝐹 “ (0..^0)) = ∅ |
19 | 18 | reseq2i 5852 | . . . . . . . . 9 ⊢ (𝐼 ↾ (𝐹 “ (0..^0))) = (𝐼 ↾ ∅) |
20 | res0 5859 | . . . . . . . . 9 ⊢ (𝐼 ↾ ∅) = ∅ | |
21 | 19, 20 | eqtri 2846 | . . . . . . . 8 ⊢ (𝐼 ↾ (𝐹 “ (0..^0))) = ∅ |
22 | 14, 21 | syl6eq 2874 | . . . . . . 7 ⊢ (𝜑 → (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉) = ∅) |
23 | 22 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉) = ∅) |
24 | eqid 2823 | . . . . . . 7 ⊢ (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉) = (Vtx‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉) | |
25 | eqid 2823 | . . . . . . 7 ⊢ (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉) = (iEdg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉) | |
26 | 24, 25 | vtxdg0e 27258 | . . . . . 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 586 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((VtxDeg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉)‘𝑥) = 0) |
28 | 1, 27 | breqtrrid 5106 | . . . 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 3184 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 ¬ ¬ 2 ∥ ((VtxDeg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉)‘𝑥)) |
31 | rabeq0 4340 | . 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 236 | 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 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 {crab 3144 Vcvv 3496 ∅c0 4293 〈cop 4575 class class class wbr 5068 ↾ cres 5559 “ cima 5560 Fun wfun 6351 ‘cfv 6357 (class class class)co 7158 0cc0 10539 2c2 11695 ..^cfzo 13036 ∥ cdvds 15609 Vtxcvtx 26783 iEdgciedg 26784 UPGraphcupgr 26867 VtxDegcvtxdg 27249 EulerPathsceupth 27978 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-xadd 12511 df-fz 12896 df-fzo 13037 df-hash 13694 df-dvds 15610 df-vtx 26785 df-iedg 26786 df-vtxdg 27250 |
This theorem is referenced by: eupth2 28020 |
Copyright terms: Public domain | W3C validator |