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Mirrors > Home > MPE Home > Th. List > eupthres | Structured version Visualization version GIF version |
Description: The restriction β¨π», πβ© of an Eulerian path β¨πΉ, πβ© to an initial segment of the path (of length π) forms an Eulerian path on the subgraph π 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 | β’ π = (VtxβπΊ) |
eupth0.i | β’ πΌ = (iEdgβπΊ) |
eupthres.d | β’ (π β πΉ(EulerPathsβπΊ)π) |
eupthres.n | β’ (π β π β (0..^(β―βπΉ))) |
eupthres.e | β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) |
eupthres.h | β’ π» = (πΉ prefix π) |
eupthres.q | β’ π = (π βΎ (0...π)) |
eupthres.s | β’ (Vtxβπ) = π |
Ref | Expression |
---|---|
eupthres | β’ (π β π»(EulerPathsβπ)π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eupth0.v | . . 3 β’ π = (VtxβπΊ) | |
2 | eupth0.i | . . 3 β’ πΌ = (iEdgβπΊ) | |
3 | eupthres.d | . . . 4 β’ (π β πΉ(EulerPathsβπΊ)π) | |
4 | eupthistrl 29973 | . . . 4 β’ (πΉ(EulerPathsβπΊ)π β πΉ(TrailsβπΊ)π) | |
5 | trliswlk 29463 | . . . 4 β’ (πΉ(TrailsβπΊ)π β πΉ(WalksβπΊ)π) | |
6 | 3, 4, 5 | 3syl 18 | . . 3 β’ (π β πΉ(WalksβπΊ)π) |
7 | eupthres.n | . . 3 β’ (π β π β (0..^(β―βπΉ))) | |
8 | eupthres.s | . . . 4 β’ (Vtxβπ) = π | |
9 | 8 | a1i 11 | . . 3 β’ (π β (Vtxβπ) = π) |
10 | eupthres.e | . . 3 β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) | |
11 | eupthres.h | . . 3 β’ π» = (πΉ prefix π) | |
12 | eupthres.q | . . 3 β’ π = (π βΎ (0...π)) | |
13 | 1, 2, 6, 7, 9, 10, 11, 12 | wlkres 29436 | . 2 β’ (π β π»(Walksβπ)π) |
14 | 3, 4 | syl 17 | . . 3 β’ (π β πΉ(TrailsβπΊ)π) |
15 | 1, 2, 14, 7, 11 | trlreslem 29465 | . 2 β’ (π β π»:(0..^(β―βπ»))β1-1-ontoβdom (πΌ βΎ (πΉ β (0..^π)))) |
16 | eqid 2726 | . . . 4 β’ (iEdgβπ) = (iEdgβπ) | |
17 | 16 | iseupthf1o 29964 | . . 3 β’ (π»(EulerPathsβπ)π β (π»(Walksβπ)π β§ π»:(0..^(β―βπ»))β1-1-ontoβdom (iEdgβπ))) |
18 | 10 | dmeqd 5899 | . . . . 5 β’ (π β dom (iEdgβπ) = dom (πΌ βΎ (πΉ β (0..^π)))) |
19 | 18 | f1oeq3d 6824 | . . . 4 β’ (π β (π»:(0..^(β―βπ»))β1-1-ontoβdom (iEdgβπ) β π»:(0..^(β―βπ»))β1-1-ontoβdom (πΌ βΎ (πΉ β (0..^π))))) |
20 | 19 | anbi2d 628 | . . 3 β’ (π β ((π»(Walksβπ)π β§ π»:(0..^(β―βπ»))β1-1-ontoβdom (iEdgβπ)) β (π»(Walksβπ)π β§ π»:(0..^(β―βπ»))β1-1-ontoβdom (πΌ βΎ (πΉ β (0..^π)))))) |
21 | 17, 20 | bitrid 283 | . 2 β’ (π β (π»(EulerPathsβπ)π β (π»(Walksβπ)π β§ π»:(0..^(β―βπ»))β1-1-ontoβdom (πΌ βΎ (πΉ β (0..^π)))))) |
22 | 13, 15, 21 | mpbir2and 710 | 1 β’ (π β π»(EulerPathsβπ)π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 class class class wbr 5141 dom cdm 5669 βΎ cres 5671 β cima 5672 β1-1-ontoβwf1o 6536 βcfv 6537 (class class class)co 7405 0cc0 11112 ...cfz 13490 ..^cfzo 13633 β―chash 14295 prefix cpfx 14626 Vtxcvtx 28764 iEdgciedg 28765 Walkscwlks 29362 Trailsctrls 29456 EulerPathsceupth 29959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-ifp 1060 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-fzo 13634 df-hash 14296 df-word 14471 df-substr 14597 df-pfx 14627 df-wlks 29365 df-trls 29458 df-eupth 29960 |
This theorem is referenced by: eucrct2eupth1 30006 |
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