![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2798 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
3 | 1, 2 | upgreupthi 27545 | . . 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 | 2fveq3 6415 | . . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝐼‘(𝐹‘𝑛)) = (𝐼‘(𝐹‘𝑁))) | |
5 | fveq2 6410 | . . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑃‘𝑛) = (𝑃‘𝑁)) | |
6 | fvoveq1 6900 | . . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑃‘(𝑛 + 1)) = (𝑃‘(𝑁 + 1))) | |
7 | 5, 6 | preq12d 4464 | . . . . . . 7 ⊢ (𝑛 = 𝑁 → {(𝑃‘𝑛), (𝑃‘(𝑛 + 1))} = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))}) |
8 | 4, 7 | eqeq12d 2813 | . . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝐼‘(𝐹‘𝑛)) = {(𝑃‘𝑛), (𝑃‘(𝑛 + 1))} ↔ (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))})) |
9 | 8 | rspcva 3494 | . . . . 5 ⊢ ((𝑁 ∈ (0..^(♯‘𝐹)) ∧ ∀𝑛 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑛)) = {(𝑃‘𝑛), (𝑃‘(𝑛 + 1))}) → (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))}) |
10 | 9 | expcom 403 | . . . 4 ⊢ (∀𝑛 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑛)) = {(𝑃‘𝑛), (𝑃‘(𝑛 + 1))} → (𝑁 ∈ (0..^(♯‘𝐹)) → (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))})) |
11 | 10 | 3ad2ant3 1166 | . . 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))})) |
12 | 3, 11 | syl 17 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) → (𝑁 ∈ (0..^(♯‘𝐹)) → (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))})) |
13 | 12 | 3impia 1146 | 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 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ∀wral 3088 {cpr 4369 class class class wbr 4842 dom cdm 5311 ⟶wf 6096 –1-1-onto→wf1o 6099 ‘cfv 6100 (class class class)co 6877 0cc0 10223 1c1 10224 + caddc 10226 ...cfz 12577 ..^cfzo 12717 ♯chash 13367 Vtxcvtx 26224 iEdgciedg 26225 UPGraphcupgr 26308 EulerPathsceupth 27534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2776 ax-rep 4963 ax-sep 4974 ax-nul 4982 ax-pow 5034 ax-pr 5096 ax-un 7182 ax-cnex 10279 ax-resscn 10280 ax-1cn 10281 ax-icn 10282 ax-addcl 10283 ax-addrcl 10284 ax-mulcl 10285 ax-mulrcl 10286 ax-mulcom 10287 ax-addass 10288 ax-mulass 10289 ax-distr 10290 ax-i2m1 10291 ax-1ne0 10292 ax-1rid 10293 ax-rnegex 10294 ax-rrecex 10295 ax-cnre 10296 ax-pre-lttri 10297 ax-pre-lttrn 10298 ax-pre-ltadd 10299 ax-pre-mulgt0 10300 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-ifp 1087 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2785 df-cleq 2791 df-clel 2794 df-nfc 2929 df-ne 2971 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3386 df-sbc 3633 df-csb 3728 df-dif 3771 df-un 3773 df-in 3775 df-ss 3782 df-pss 3784 df-nul 4115 df-if 4277 df-pw 4350 df-sn 4368 df-pr 4370 df-tp 4372 df-op 4374 df-uni 4628 df-int 4667 df-iun 4711 df-br 4843 df-opab 4905 df-mpt 4922 df-tr 4945 df-id 5219 df-eprel 5224 df-po 5232 df-so 5233 df-fr 5270 df-we 5272 df-xp 5317 df-rel 5318 df-cnv 5319 df-co 5320 df-dm 5321 df-rn 5322 df-res 5323 df-ima 5324 df-pred 5897 df-ord 5943 df-on 5944 df-lim 5945 df-suc 5946 df-iota 6063 df-fun 6102 df-fn 6103 df-f 6104 df-f1 6105 df-fo 6106 df-f1o 6107 df-fv 6108 df-riota 6838 df-ov 6880 df-oprab 6881 df-mpt2 6882 df-om 7299 df-1st 7400 df-2nd 7401 df-wrecs 7644 df-recs 7706 df-rdg 7744 df-1o 7798 df-2o 7799 df-oadd 7802 df-er 7981 df-map 8096 df-pm 8097 df-en 8195 df-dom 8196 df-sdom 8197 df-fin 8198 df-card 9050 df-cda 9277 df-pnf 10364 df-mnf 10365 df-xr 10366 df-ltxr 10367 df-le 10368 df-sub 10557 df-neg 10558 df-nn 11312 df-2 11373 df-n0 11578 df-xnn0 11650 df-z 11664 df-uz 11928 df-fz 12578 df-fzo 12718 df-hash 13368 df-word 13532 df-edg 26276 df-uhgr 26286 df-upgr 26310 df-wlks 26842 df-trls 26938 df-eupth 27535 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |