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Mirrors > Home > MPE Home > Th. List > eupthres | Structured version Visualization version GIF version |
Description: The restriction 〈𝐻, 𝑄〉 of an Eulerian path 〈𝐹, 𝑃〉 to an initial segment of the path (of length 𝑁) forms an Eulerian path on the subgraph 𝑆 consisting of the edges in the initial segment. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 6-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) |
Ref | Expression |
---|---|
eupth0.v | ⊢ 𝑉 = (Vtx‘𝐺) |
eupth0.i | ⊢ 𝐼 = (iEdg‘𝐺) |
eupthres.d | ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃) |
eupthres.n | ⊢ (𝜑 → 𝑁 ∈ (0..^(#‘𝐹))) |
eupthres.e | ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) |
eupthres.h | ⊢ 𝐻 = (𝐹 ↾ (0..^𝑁)) |
eupthres.q | ⊢ 𝑄 = (𝑃 ↾ (0...𝑁)) |
eupthres.s | ⊢ (Vtx‘𝑆) = 𝑉 |
Ref | Expression |
---|---|
eupthres | ⊢ (𝜑 → 𝐻(EulerPaths‘𝑆)𝑄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eupth0.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | eupth0.i | . . 3 ⊢ 𝐼 = (iEdg‘𝐺) | |
3 | eupthres.d | . . . 4 ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃) | |
4 | eupthistrl 27189 | . . . 4 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃) | |
5 | trliswlk 26650 | . . . 4 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | |
6 | 3, 4, 5 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) |
7 | eupthres.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ (0..^(#‘𝐹))) | |
8 | eupthres.s | . . . 4 ⊢ (Vtx‘𝑆) = 𝑉 | |
9 | 8 | a1i 11 | . . 3 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) |
10 | eupthres.e | . . 3 ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) | |
11 | eupthres.h | . . 3 ⊢ 𝐻 = (𝐹 ↾ (0..^𝑁)) | |
12 | eupthres.q | . . 3 ⊢ 𝑄 = (𝑃 ↾ (0...𝑁)) | |
13 | 1, 2, 6, 7, 9, 10, 11, 12 | wlkres 26623 | . 2 ⊢ (𝜑 → 𝐻(Walks‘𝑆)𝑄) |
14 | 3, 4 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) |
15 | 1, 2, 14, 7, 11 | trlreslem 26652 | . 2 ⊢ (𝜑 → 𝐻:(0..^(#‘𝐻))–1-1-onto→dom (𝐼 ↾ (𝐹 “ (0..^𝑁)))) |
16 | eqid 2651 | . . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆) | |
17 | 16 | iseupthf1o 27180 | . . 3 ⊢ (𝐻(EulerPaths‘𝑆)𝑄 ↔ (𝐻(Walks‘𝑆)𝑄 ∧ 𝐻:(0..^(#‘𝐻))–1-1-onto→dom (iEdg‘𝑆))) |
18 | 10 | dmeqd 5358 | . . . . 5 ⊢ (𝜑 → dom (iEdg‘𝑆) = dom (𝐼 ↾ (𝐹 “ (0..^𝑁)))) |
19 | 18 | f1oeq3d 6172 | . . . 4 ⊢ (𝜑 → (𝐻:(0..^(#‘𝐻))–1-1-onto→dom (iEdg‘𝑆) ↔ 𝐻:(0..^(#‘𝐻))–1-1-onto→dom (𝐼 ↾ (𝐹 “ (0..^𝑁))))) |
20 | 19 | anbi2d 740 | . . 3 ⊢ (𝜑 → ((𝐻(Walks‘𝑆)𝑄 ∧ 𝐻:(0..^(#‘𝐻))–1-1-onto→dom (iEdg‘𝑆)) ↔ (𝐻(Walks‘𝑆)𝑄 ∧ 𝐻:(0..^(#‘𝐻))–1-1-onto→dom (𝐼 ↾ (𝐹 “ (0..^𝑁)))))) |
21 | 17, 20 | syl5bb 272 | . 2 ⊢ (𝜑 → (𝐻(EulerPaths‘𝑆)𝑄 ↔ (𝐻(Walks‘𝑆)𝑄 ∧ 𝐻:(0..^(#‘𝐻))–1-1-onto→dom (𝐼 ↾ (𝐹 “ (0..^𝑁)))))) |
22 | 13, 15, 21 | mpbir2and 977 | 1 ⊢ (𝜑 → 𝐻(EulerPaths‘𝑆)𝑄) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 class class class wbr 4685 dom cdm 5143 ↾ cres 5145 “ cima 5146 –1-1-onto→wf1o 5925 ‘cfv 5926 (class class class)co 6690 0cc0 9974 ...cfz 12364 ..^cfzo 12504 #chash 13157 Vtxcvtx 25919 iEdgciedg 25920 Walkscwlks 26548 Trailsctrls 26643 EulerPathsceupth 27175 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-ifp 1033 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-map 7901 df-pm 7902 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-card 8803 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-n0 11331 df-z 11416 df-uz 11726 df-fz 12365 df-fzo 12505 df-hash 13158 df-word 13331 df-substr 13335 df-wlks 26551 df-trls 26645 df-eupth 27176 |
This theorem is referenced by: eucrct2eupth1 27222 |
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