![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > wlkf | Structured version Visualization version GIF version |
Description: The mapping enumerating the (indices of the) edges of a walk is a word over the indices of the edges of the graph. (Contributed by AV, 5-Apr-2021.) |
Ref | Expression |
---|---|
wlkf.i | ⊢ 𝐼 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
wlkf | ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2725 | . . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | wlkf.i | . . 3 ⊢ 𝐼 = (iEdg‘𝐺) | |
3 | 1, 2 | wlkprop 29497 | . 2 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))) |
4 | 3 | simp1d 1139 | 1 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 if-wif 1060 = wceq 1533 ∈ wcel 2098 ∀wral 3050 ⊆ wss 3944 {csn 4630 {cpr 4632 class class class wbr 5149 dom cdm 5678 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 0cc0 11140 1c1 11141 + caddc 11143 ...cfz 13519 ..^cfzo 13662 ♯chash 14325 Word cword 14500 Vtxcvtx 28881 iEdgciedg 28882 Walkscwlks 29482 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ifp 1061 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-n0 12506 df-z 12592 df-uz 12856 df-fz 13520 df-fzo 13663 df-hash 14326 df-word 14501 df-wlks 29485 |
This theorem is referenced by: wlkcl 29501 wksvOLD 29506 iedginwlk 29523 wlk1ewlk 29526 wlkv0 29537 wlkonwlk1l 29549 wlkres 29556 redwlk 29558 wlkp1lem2 29560 wlkp1lem3 29561 wlkp1lem4 29562 wlkp1lem6 29564 wlkp1lem8 29566 wlkp1 29567 lfgriswlk 29574 trlf1 29584 trlreslem 29585 upgr2pthnlp 29618 uhgrwkspthlem1 29639 usgr2wlkspthlem1 29643 crctcshlem2 29701 crctcshlem4 29703 crctcshwlkn0 29704 eupth2eucrct 30099 eucrctshift 30125 eucrct2eupth 30127 pfxwlk 34864 revwlk 34865 swrdwlk 34867 |
Copyright terms: Public domain | W3C validator |