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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 30049 | . . . 4 β’ (πΉ(EulerPathsβπΊ)π β πΉ(TrailsβπΊ)π) | |
5 | trliswlk 29539 | . . . 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 29512 | . 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 29541 | . 2 β’ (π β π»:(0..^(β―βπ»))β1-1-ontoβdom (πΌ βΎ (πΉ β (0..^π)))) |
16 | eqid 2728 | . . . 4 β’ (iEdgβπ) = (iEdgβπ) | |
17 | 16 | iseupthf1o 30040 | . . 3 β’ (π»(EulerPathsβπ)π β (π»(Walksβπ)π β§ π»:(0..^(β―βπ»))β1-1-ontoβdom (iEdgβπ))) |
18 | 10 | dmeqd 5912 | . . . . 5 β’ (π β dom (iEdgβπ) = dom (πΌ βΎ (πΉ β (0..^π)))) |
19 | 18 | f1oeq3d 6841 | . . . 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 282 | . 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 711 | 1 β’ (π β π»(EulerPathsβπ)π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 class class class wbr 5152 dom cdm 5682 βΎ cres 5684 β cima 5685 β1-1-ontoβwf1o 6552 βcfv 6553 (class class class)co 7426 0cc0 11148 ...cfz 13526 ..^cfzo 13669 β―chash 14331 prefix cpfx 14662 Vtxcvtx 28837 iEdgciedg 28838 Walkscwlks 29438 Trailsctrls 29532 EulerPathsceupth 30035 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ifp 1061 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-map 8855 df-pm 8856 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-card 9972 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-n0 12513 df-z 12599 df-uz 12863 df-fz 13527 df-fzo 13670 df-hash 14332 df-word 14507 df-substr 14633 df-pfx 14663 df-wlks 29441 df-trls 29534 df-eupth 30036 |
This theorem is referenced by: eucrct2eupth1 30082 |
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