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Mirrors > Home > MPE Home > Th. List > eupaseg | Structured version Visualization version GIF version |
Description: The 𝑁-th edge in an eulerian path is the edge from 𝑃(𝑁 − 1) to 𝑃(𝑁). (Contributed by Mario Carneiro, 12-Mar-2015.) |
Ref | Expression |
---|---|
eupaseg | ⊢ ((𝐹(𝑉 EulPaths 𝐸)𝑃 ∧ 𝑁 ∈ (1...(#‘𝐹))) → (𝐸‘(𝐹‘𝑁)) = {(𝑃‘(𝑁 − 1)), (𝑃‘𝑁)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eupagra 26493 | . . . . 5 ⊢ (𝐹(𝑉 EulPaths 𝐸)𝑃 → 𝑉 UMGrph 𝐸) | |
2 | umgraf2 25846 | . . . . 5 ⊢ (𝑉 UMGrph 𝐸 → 𝐸:dom 𝐸⟶{𝑘 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑘) ≤ 2}) | |
3 | ffn 5958 | . . . . 5 ⊢ (𝐸:dom 𝐸⟶{𝑘 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑘) ≤ 2} → 𝐸 Fn dom 𝐸) | |
4 | 1, 2, 3 | 3syl 18 | . . . 4 ⊢ (𝐹(𝑉 EulPaths 𝐸)𝑃 → 𝐸 Fn dom 𝐸) |
5 | eupai 26494 | . . . 4 ⊢ ((𝐹(𝑉 EulPaths 𝐸)𝑃 ∧ 𝐸 Fn dom 𝐸) → (((#‘𝐹) ∈ ℕ0 ∧ 𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)})) | |
6 | 4, 5 | mpdan 699 | . . 3 ⊢ (𝐹(𝑉 EulPaths 𝐸)𝑃 → (((#‘𝐹) ∈ ℕ0 ∧ 𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)})) |
7 | 6 | simprd 478 | . 2 ⊢ (𝐹(𝑉 EulPaths 𝐸)𝑃 → ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)}) |
8 | fveq2 6103 | . . . . 5 ⊢ (𝑘 = 𝑁 → (𝐹‘𝑘) = (𝐹‘𝑁)) | |
9 | 8 | fveq2d 6107 | . . . 4 ⊢ (𝑘 = 𝑁 → (𝐸‘(𝐹‘𝑘)) = (𝐸‘(𝐹‘𝑁))) |
10 | oveq1 6556 | . . . . . 6 ⊢ (𝑘 = 𝑁 → (𝑘 − 1) = (𝑁 − 1)) | |
11 | 10 | fveq2d 6107 | . . . . 5 ⊢ (𝑘 = 𝑁 → (𝑃‘(𝑘 − 1)) = (𝑃‘(𝑁 − 1))) |
12 | fveq2 6103 | . . . . 5 ⊢ (𝑘 = 𝑁 → (𝑃‘𝑘) = (𝑃‘𝑁)) | |
13 | 11, 12 | preq12d 4220 | . . . 4 ⊢ (𝑘 = 𝑁 → {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)} = {(𝑃‘(𝑁 − 1)), (𝑃‘𝑁)}) |
14 | 9, 13 | eqeq12d 2625 | . . 3 ⊢ (𝑘 = 𝑁 → ((𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)} ↔ (𝐸‘(𝐹‘𝑁)) = {(𝑃‘(𝑁 − 1)), (𝑃‘𝑁)})) |
15 | 14 | rspccva 3281 | . 2 ⊢ ((∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)} ∧ 𝑁 ∈ (1...(#‘𝐹))) → (𝐸‘(𝐹‘𝑁)) = {(𝑃‘(𝑁 − 1)), (𝑃‘𝑁)}) |
16 | 7, 15 | sylan 487 | 1 ⊢ ((𝐹(𝑉 EulPaths 𝐸)𝑃 ∧ 𝑁 ∈ (1...(#‘𝐹))) → (𝐸‘(𝐹‘𝑁)) = {(𝑃‘(𝑁 − 1)), (𝑃‘𝑁)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 ∖ cdif 3537 ∅c0 3874 𝒫 cpw 4108 {csn 4125 {cpr 4127 class class class wbr 4583 dom cdm 5038 Fn wfn 5799 ⟶wf 5800 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 ≤ cle 9954 − cmin 10145 2c2 10947 ℕ0cn0 11169 ...cfz 12197 #chash 12979 UMGrph cumg 25841 EulPaths ceup 26489 |
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 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 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-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-er 7629 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-hash 12980 df-umgra 25842 df-eupa 26490 |
This theorem is referenced by: eupares 26502 eupap1 26503 eupath2lem3 26506 |
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