Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > eulerpathpr | Structured version Visualization version GIF version |
Description: A graph with an Eulerian path has either zero or two vertices of odd degree. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 26-Feb-2021.) |
Ref | Expression |
---|---|
eulerpathpr.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
eulerpathpr | ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) → (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eulerpathpr.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | eqid 2610 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
3 | simpl 472 | . . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) → 𝐺 ∈ UPGraph ) | |
4 | upgruhgr 25768 | . . . . . 6 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph ) | |
5 | 2 | uhgrfun 25732 | . . . . . 6 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺)) |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝐺 ∈ UPGraph → Fun (iEdg‘𝐺)) |
7 | 6 | adantr 480 | . . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) → Fun (iEdg‘𝐺)) |
8 | simpr 476 | . . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) → 𝐹(EulerPaths‘𝐺)𝑃) | |
9 | 1, 2, 3, 7, 8 | eupth2 41407 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} = if((𝑃‘0) = (𝑃‘(#‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(#‘𝐹))})) |
10 | 9 | fveq2d 6107 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) → (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = (#‘if((𝑃‘0) = (𝑃‘(#‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(#‘𝐹))}))) |
11 | fveq2 6103 | . . . 4 ⊢ (∅ = if((𝑃‘0) = (𝑃‘(#‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(#‘𝐹))}) → (#‘∅) = (#‘if((𝑃‘0) = (𝑃‘(#‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(#‘𝐹))}))) | |
12 | 11 | eleq1d 2672 | . . 3 ⊢ (∅ = if((𝑃‘0) = (𝑃‘(#‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(#‘𝐹))}) → ((#‘∅) ∈ {0, 2} ↔ (#‘if((𝑃‘0) = (𝑃‘(#‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(#‘𝐹))})) ∈ {0, 2})) |
13 | fveq2 6103 | . . . 4 ⊢ ({(𝑃‘0), (𝑃‘(#‘𝐹))} = if((𝑃‘0) = (𝑃‘(#‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(#‘𝐹))}) → (#‘{(𝑃‘0), (𝑃‘(#‘𝐹))}) = (#‘if((𝑃‘0) = (𝑃‘(#‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(#‘𝐹))}))) | |
14 | 13 | eleq1d 2672 | . . 3 ⊢ ({(𝑃‘0), (𝑃‘(#‘𝐹))} = if((𝑃‘0) = (𝑃‘(#‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(#‘𝐹))}) → ((#‘{(𝑃‘0), (𝑃‘(#‘𝐹))}) ∈ {0, 2} ↔ (#‘if((𝑃‘0) = (𝑃‘(#‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(#‘𝐹))})) ∈ {0, 2})) |
15 | hash0 13019 | . . . . 5 ⊢ (#‘∅) = 0 | |
16 | c0ex 9913 | . . . . . 6 ⊢ 0 ∈ V | |
17 | 16 | prid1 4241 | . . . . 5 ⊢ 0 ∈ {0, 2} |
18 | 15, 17 | eqeltri 2684 | . . . 4 ⊢ (#‘∅) ∈ {0, 2} |
19 | 18 | a1i 11 | . . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) → (#‘∅) ∈ {0, 2}) |
20 | simpr 476 | . . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) ∧ ¬ (𝑃‘0) = (𝑃‘(#‘𝐹))) → ¬ (𝑃‘0) = (𝑃‘(#‘𝐹))) | |
21 | 20 | neqned 2789 | . . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) ∧ ¬ (𝑃‘0) = (𝑃‘(#‘𝐹))) → (𝑃‘0) ≠ (𝑃‘(#‘𝐹))) |
22 | fvex 6113 | . . . . . 6 ⊢ (𝑃‘0) ∈ V | |
23 | fvex 6113 | . . . . . 6 ⊢ (𝑃‘(#‘𝐹)) ∈ V | |
24 | hashprg 13043 | . . . . . 6 ⊢ (((𝑃‘0) ∈ V ∧ (𝑃‘(#‘𝐹)) ∈ V) → ((𝑃‘0) ≠ (𝑃‘(#‘𝐹)) ↔ (#‘{(𝑃‘0), (𝑃‘(#‘𝐹))}) = 2)) | |
25 | 22, 23, 24 | mp2an 704 | . . . . 5 ⊢ ((𝑃‘0) ≠ (𝑃‘(#‘𝐹)) ↔ (#‘{(𝑃‘0), (𝑃‘(#‘𝐹))}) = 2) |
26 | 21, 25 | sylib 207 | . . . 4 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) ∧ ¬ (𝑃‘0) = (𝑃‘(#‘𝐹))) → (#‘{(𝑃‘0), (𝑃‘(#‘𝐹))}) = 2) |
27 | 2ex 10969 | . . . . 5 ⊢ 2 ∈ V | |
28 | 27 | prid2 4242 | . . . 4 ⊢ 2 ∈ {0, 2} |
29 | 26, 28 | syl6eqel 2696 | . . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) ∧ ¬ (𝑃‘0) = (𝑃‘(#‘𝐹))) → (#‘{(𝑃‘0), (𝑃‘(#‘𝐹))}) ∈ {0, 2}) |
30 | 12, 14, 19, 29 | ifbothda 4073 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) → (#‘if((𝑃‘0) = (𝑃‘(#‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(#‘𝐹))})) ∈ {0, 2}) |
31 | 10, 30 | eqeltrd 2688 | 1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) → (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 {crab 2900 Vcvv 3173 ∅c0 3874 ifcif 4036 {cpr 4127 class class class wbr 4583 Fun wfun 5798 ‘cfv 5804 0cc0 9815 2c2 10947 #chash 12979 ∥ cdvds 14821 Vtxcvtx 25673 iEdgciedg 25674 UHGraph cuhgr 25722 UPGraph cupgr 25747 VtxDegcvtxdg 40681 EulerPathsceupth 41364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ifp 1007 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-rp 11709 df-xadd 11823 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-word 13154 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 df-vtx 25675 df-iedg 25676 df-uhgr 25724 df-ushgr 25725 df-upgr 25749 df-edga 25793 df-uspgr 40380 df-vtxdg 40682 df-1wlks 40800 df-wlks 40801 df-trls 40901 df-eupth 41365 |
This theorem is referenced by: eulerpath 41409 |
Copyright terms: Public domain | W3C validator |