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Mirrors > Home > MPE Home > Th. List > upgreupthseg | Structured version Visualization version GIF version |
Description: The 𝑁-th edge in an eulerian path is the edge from 𝑃(𝑁) to 𝑃(𝑁 + 1). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) |
Ref | Expression |
---|---|
upgreupthseg.i | ⊢ 𝐼 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
upgreupthseg | ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃 ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | upgreupthseg.i | . . . 4 ⊢ 𝐼 = (iEdg‘𝐺) | |
2 | eqid 2771 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
3 | 1, 2 | upgreupthi 27388 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) → (𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑛 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑛)) = {(𝑃‘𝑛), (𝑃‘(𝑛 + 1))})) |
4 | fveq2 6332 | . . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝐹‘𝑛) = (𝐹‘𝑁)) | |
5 | 4 | fveq2d 6336 | . . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝐼‘(𝐹‘𝑛)) = (𝐼‘(𝐹‘𝑁))) |
6 | fveq2 6332 | . . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑃‘𝑛) = (𝑃‘𝑁)) | |
7 | fvoveq1 6816 | . . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑃‘(𝑛 + 1)) = (𝑃‘(𝑁 + 1))) | |
8 | 6, 7 | preq12d 4412 | . . . . . . 7 ⊢ (𝑛 = 𝑁 → {(𝑃‘𝑛), (𝑃‘(𝑛 + 1))} = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))}) |
9 | 5, 8 | eqeq12d 2786 | . . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝐼‘(𝐹‘𝑛)) = {(𝑃‘𝑛), (𝑃‘(𝑛 + 1))} ↔ (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))})) |
10 | 9 | rspcva 3458 | . . . . 5 ⊢ ((𝑁 ∈ (0..^(♯‘𝐹)) ∧ ∀𝑛 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑛)) = {(𝑃‘𝑛), (𝑃‘(𝑛 + 1))}) → (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))}) |
11 | 10 | expcom 398 | . . . 4 ⊢ (∀𝑛 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑛)) = {(𝑃‘𝑛), (𝑃‘(𝑛 + 1))} → (𝑁 ∈ (0..^(♯‘𝐹)) → (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))})) |
12 | 11 | 3ad2ant3 1129 | . . 3 ⊢ ((𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑛 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑛)) = {(𝑃‘𝑛), (𝑃‘(𝑛 + 1))}) → (𝑁 ∈ (0..^(♯‘𝐹)) → (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))})) |
13 | 3, 12 | syl 17 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) → (𝑁 ∈ (0..^(♯‘𝐹)) → (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))})) |
14 | 13 | 3impia 1109 | 1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃 ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ∀wral 3061 {cpr 4318 class class class wbr 4786 dom cdm 5249 ⟶wf 6027 –1-1-onto→wf1o 6030 ‘cfv 6031 (class class class)co 6793 0cc0 10138 1c1 10139 + caddc 10141 ...cfz 12533 ..^cfzo 12673 ♯chash 13321 Vtxcvtx 26095 iEdgciedg 26096 UPGraphcupgr 26196 EulerPathsceupth 27377 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-ifp 1050 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-2o 7714 df-oadd 7717 df-er 7896 df-map 8011 df-pm 8012 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-card 8965 df-cda 9192 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-nn 11223 df-2 11281 df-n0 11495 df-xnn0 11566 df-z 11580 df-uz 11889 df-fz 12534 df-fzo 12674 df-hash 13322 df-word 13495 df-edg 26161 df-uhgr 26174 df-upgr 26198 df-wlks 26730 df-trls 26824 df-eupth 27378 |
This theorem is referenced by: (None) |
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