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Mirrors > Home > MPE Home > Th. List > eupth0 | Structured version Visualization version GIF version |
Description: There is an Eulerian path on an empty graph, i.e. a graph with at least one vertex, but without an edge. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 5-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) |
Ref | Expression |
---|---|
eupth0.v | β’ π = (VtxβπΊ) |
eupth0.i | β’ πΌ = (iEdgβπΊ) |
Ref | Expression |
---|---|
eupth0 | β’ ((π΄ β π β§ πΌ = β ) β β (EulerPathsβπΊ){β¨0, π΄β©}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2738 | . . . 4 β’ (π΄ β π β {β¨0, π΄β©} = {β¨0, π΄β©}) | |
2 | eupth0.v | . . . . 5 β’ π = (VtxβπΊ) | |
3 | 2 | is0wlk 29064 | . . . 4 β’ (({β¨0, π΄β©} = {β¨0, π΄β©} β§ π΄ β π) β β (WalksβπΊ){β¨0, π΄β©}) |
4 | 1, 3 | mpancom 687 | . . 3 β’ (π΄ β π β β (WalksβπΊ){β¨0, π΄β©}) |
5 | f1o0 6822 | . . . 4 β’ β :β β1-1-ontoββ | |
6 | eqidd 2738 | . . . . 5 β’ (πΌ = β β β = β ) | |
7 | hash0 14268 | . . . . . . . 8 β’ (β―ββ ) = 0 | |
8 | 7 | oveq2i 7369 | . . . . . . 7 β’ (0..^(β―ββ )) = (0..^0) |
9 | fzo0 13597 | . . . . . . 7 β’ (0..^0) = β | |
10 | 8, 9 | eqtri 2765 | . . . . . 6 β’ (0..^(β―ββ )) = β |
11 | 10 | a1i 11 | . . . . 5 β’ (πΌ = β β (0..^(β―ββ )) = β ) |
12 | dmeq 5860 | . . . . . 6 β’ (πΌ = β β dom πΌ = dom β ) | |
13 | dm0 5877 | . . . . . 6 β’ dom β = β | |
14 | 12, 13 | eqtrdi 2793 | . . . . 5 β’ (πΌ = β β dom πΌ = β ) |
15 | 6, 11, 14 | f1oeq123d 6779 | . . . 4 β’ (πΌ = β β (β :(0..^(β―ββ ))β1-1-ontoβdom πΌ β β :β β1-1-ontoββ )) |
16 | 5, 15 | mpbiri 258 | . . 3 β’ (πΌ = β β β :(0..^(β―ββ ))β1-1-ontoβdom πΌ) |
17 | 4, 16 | anim12i 614 | . 2 β’ ((π΄ β π β§ πΌ = β ) β (β (WalksβπΊ){β¨0, π΄β©} β§ β :(0..^(β―ββ ))β1-1-ontoβdom πΌ)) |
18 | eupth0.i | . . 3 β’ πΌ = (iEdgβπΊ) | |
19 | 18 | iseupthf1o 29149 | . 2 β’ (β (EulerPathsβπΊ){β¨0, π΄β©} β (β (WalksβπΊ){β¨0, π΄β©} β§ β :(0..^(β―ββ ))β1-1-ontoβdom πΌ)) |
20 | 17, 19 | sylibr 233 | 1 β’ ((π΄ β π β§ πΌ = β ) β β (EulerPathsβπΊ){β¨0, π΄β©}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β c0 4283 {csn 4587 β¨cop 4593 class class class wbr 5106 dom cdm 5634 β1-1-ontoβwf1o 6496 βcfv 6497 (class class class)co 7358 0cc0 11052 ..^cfzo 13568 β―chash 14231 Vtxcvtx 27950 iEdgciedg 27951 Walkscwlks 28547 EulerPathsceupth 29144 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ifp 1063 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8649 df-map 8768 df-pm 8769 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-card 9876 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-nn 12155 df-n0 12415 df-z 12501 df-uz 12765 df-fz 13426 df-fzo 13569 df-hash 14232 df-word 14404 df-wlks 28550 df-trls 28643 df-eupth 29145 |
This theorem is referenced by: (None) |
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