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Mirrors > Home > MPE Home > Th. List > eucrct2eupth1 | Structured version Visualization version GIF version |
Description: Removing one edge (πΌβ(πΉβπ)) from a nonempty graph πΊ with an Eulerian circuit β¨πΉ, πβ© results in a graph π with an Eulerian path β¨π», πβ©. This is the special case of eucrct2eupth 29487 (with π½ = (π β 1)) where the last segment/edge of the circuit is removed. (Contributed by AV, 11-Mar-2021.) Hypothesis revised using the prefix operation. (Revised by AV, 30-Nov-2022.) |
Ref | Expression |
---|---|
eucrct2eupth1.v | β’ π = (VtxβπΊ) |
eucrct2eupth1.i | β’ πΌ = (iEdgβπΊ) |
eucrct2eupth1.d | β’ (π β πΉ(EulerPathsβπΊ)π) |
eucrct2eupth1.c | β’ (π β πΉ(CircuitsβπΊ)π) |
eucrct2eupth1.s | β’ (Vtxβπ) = π |
eucrct2eupth1.g | β’ (π β 0 < (β―βπΉ)) |
eucrct2eupth1.n | β’ (π β π = ((β―βπΉ) β 1)) |
eucrct2eupth1.e | β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) |
eucrct2eupth1.h | β’ π» = (πΉ prefix π) |
eucrct2eupth1.q | β’ π = (π βΎ (0...π)) |
Ref | Expression |
---|---|
eucrct2eupth1 | β’ (π β π»(EulerPathsβπ)π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eucrct2eupth1.v | . 2 β’ π = (VtxβπΊ) | |
2 | eucrct2eupth1.i | . 2 β’ πΌ = (iEdgβπΊ) | |
3 | eucrct2eupth1.d | . 2 β’ (π β πΉ(EulerPathsβπΊ)π) | |
4 | eucrct2eupth1.n | . . 3 β’ (π β π = ((β―βπΉ) β 1)) | |
5 | eucrct2eupth1.g | . . . . 5 β’ (π β 0 < (β―βπΉ)) | |
6 | eupthiswlk 29454 | . . . . . 6 β’ (πΉ(EulerPathsβπΊ)π β πΉ(WalksβπΊ)π) | |
7 | wlkcl 28861 | . . . . . . 7 β’ (πΉ(WalksβπΊ)π β (β―βπΉ) β β0) | |
8 | nn0z 12579 | . . . . . . . . . 10 β’ ((β―βπΉ) β β0 β (β―βπΉ) β β€) | |
9 | 8 | anim1i 615 | . . . . . . . . 9 β’ (((β―βπΉ) β β0 β§ 0 < (β―βπΉ)) β ((β―βπΉ) β β€ β§ 0 < (β―βπΉ))) |
10 | elnnz 12564 | . . . . . . . . 9 β’ ((β―βπΉ) β β β ((β―βπΉ) β β€ β§ 0 < (β―βπΉ))) | |
11 | 9, 10 | sylibr 233 | . . . . . . . 8 β’ (((β―βπΉ) β β0 β§ 0 < (β―βπΉ)) β (β―βπΉ) β β) |
12 | 11 | ex 413 | . . . . . . 7 β’ ((β―βπΉ) β β0 β (0 < (β―βπΉ) β (β―βπΉ) β β)) |
13 | 7, 12 | syl 17 | . . . . . 6 β’ (πΉ(WalksβπΊ)π β (0 < (β―βπΉ) β (β―βπΉ) β β)) |
14 | 3, 6, 13 | 3syl 18 | . . . . 5 β’ (π β (0 < (β―βπΉ) β (β―βπΉ) β β)) |
15 | 5, 14 | mpd 15 | . . . 4 β’ (π β (β―βπΉ) β β) |
16 | fzo0end 13720 | . . . 4 β’ ((β―βπΉ) β β β ((β―βπΉ) β 1) β (0..^(β―βπΉ))) | |
17 | 15, 16 | syl 17 | . . 3 β’ (π β ((β―βπΉ) β 1) β (0..^(β―βπΉ))) |
18 | 4, 17 | eqeltrd 2833 | . 2 β’ (π β π β (0..^(β―βπΉ))) |
19 | eucrct2eupth1.e | . 2 β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) | |
20 | eucrct2eupth1.h | . 2 β’ π» = (πΉ prefix π) | |
21 | eucrct2eupth1.q | . 2 β’ π = (π βΎ (0...π)) | |
22 | eucrct2eupth1.s | . 2 β’ (Vtxβπ) = π | |
23 | 1, 2, 3, 18, 19, 20, 21, 22 | eupthres 29457 | 1 β’ (π β π»(EulerPathsβπ)π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 class class class wbr 5147 βΎ cres 5677 β cima 5678 βcfv 6540 (class class class)co 7405 0cc0 11106 1c1 11107 < clt 11244 β cmin 11440 βcn 12208 β0cn0 12468 β€cz 12554 ...cfz 13480 ..^cfzo 13623 β―chash 14286 prefix cpfx 14616 Vtxcvtx 28245 iEdgciedg 28246 Walkscwlks 28842 Circuitsccrcts 29030 EulerPathsceupth 29439 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ifp 1062 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-fzo 13624 df-hash 14287 df-word 14461 df-substr 14587 df-pfx 14617 df-wlks 28845 df-trls 28938 df-eupth 29440 |
This theorem is referenced by: eucrct2eupth 29487 |
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