![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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.) Hypothesis revised using the prefix operation. (Revised by AV, 30-Nov-2022.) |
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 prefix 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 27996 | . . . 4 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃) | |
5 | trliswlk 27487 | . . . 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 prefix 𝑁) | |
12 | eupthres.q | . . 3 ⊢ 𝑄 = (𝑃 ↾ (0...𝑁)) | |
13 | 1, 2, 6, 7, 9, 10, 11, 12 | wlkres 27460 | . 2 ⊢ (𝜑 → 𝐻(Walks‘𝑆)𝑄) |
14 | 3, 4 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) |
15 | 1, 2, 14, 7, 11 | trlreslem 27489 | . 2 ⊢ (𝜑 → 𝐻:(0..^(♯‘𝐻))–1-1-onto→dom (𝐼 ↾ (𝐹 “ (0..^𝑁)))) |
16 | eqid 2798 | . . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆) | |
17 | 16 | iseupthf1o 27987 | . . 3 ⊢ (𝐻(EulerPaths‘𝑆)𝑄 ↔ (𝐻(Walks‘𝑆)𝑄 ∧ 𝐻:(0..^(♯‘𝐻))–1-1-onto→dom (iEdg‘𝑆))) |
18 | 10 | dmeqd 5738 | . . . . 5 ⊢ (𝜑 → dom (iEdg‘𝑆) = dom (𝐼 ↾ (𝐹 “ (0..^𝑁)))) |
19 | 18 | f1oeq3d 6587 | . . . 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 631 | . . 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 286 | . 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 712 | 1 ⊢ (𝜑 → 𝐻(EulerPaths‘𝑆)𝑄) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 dom cdm 5519 ↾ cres 5521 “ cima 5522 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 0cc0 10526 ...cfz 12885 ..^cfzo 13028 ♯chash 13686 prefix cpfx 14023 Vtxcvtx 26789 iEdgciedg 26790 Walkscwlks 27386 Trailsctrls 27480 EulerPathsceupth 27982 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ifp 1059 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-hash 13687 df-word 13858 df-substr 13994 df-pfx 14024 df-wlks 27389 df-trls 27482 df-eupth 27983 |
This theorem is referenced by: eucrct2eupth1 28029 |
Copyright terms: Public domain | W3C validator |