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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 27071 | . . . 4 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃) | |
5 | trliswlk 26594 | . . . 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 26567 | . 2 ⊢ (𝜑 → 𝐻(Walks‘𝑆)𝑄) |
14 | 3, 4 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) |
15 | 1, 2, 14, 7, 11 | trlreslem 26596 | . 2 ⊢ (𝜑 → 𝐻:(0..^(#‘𝐻))–1-1-onto→dom (𝐼 ↾ (𝐹 “ (0..^𝑁)))) |
16 | eqid 2622 | . . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆) | |
17 | 16 | iseupthf1o 27062 | . . 3 ⊢ (𝐻(EulerPaths‘𝑆)𝑄 ↔ (𝐻(Walks‘𝑆)𝑄 ∧ 𝐻:(0..^(#‘𝐻))–1-1-onto→dom (iEdg‘𝑆))) |
18 | 10 | dmeqd 5326 | . . . . 5 ⊢ (𝜑 → dom (iEdg‘𝑆) = dom (𝐼 ↾ (𝐹 “ (0..^𝑁)))) |
19 | 18 | f1oeq3d 6134 | . . . 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 957 | 1 ⊢ (𝜑 → 𝐻(EulerPaths‘𝑆)𝑄) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 dom cdm 5114 ↾ cres 5116 “ cima 5117 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 0cc0 9936 ...cfz 12326 ..^cfzo 12465 #chash 13117 Vtxcvtx 25874 iEdgciedg 25875 Walkscwlks 26492 Trailsctrls 26587 EulerPathsceupth 27057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ifp 1013 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-substr 13303 df-wlks 26495 df-trls 26589 df-eupth 27058 |
This theorem is referenced by: eucrct2eupth1 27104 |
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