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 2818 | . . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | wlkf.i | . . 3 ⊢ 𝐼 = (iEdg‘𝐺) | |
3 | 1, 2 | wlkprop 27320 | . 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 1134 | 1 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 if-wif 1054 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ⊆ wss 3933 {csn 4557 {cpr 4559 class class class wbr 5057 dom cdm 5548 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 0cc0 10525 1c1 10526 + caddc 10528 ...cfz 12880 ..^cfzo 13021 ♯chash 13678 Word cword 13849 Vtxcvtx 26708 iEdgciedg 26709 Walkscwlks 27305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ifp 1055 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-fzo 13022 df-hash 13679 df-word 13850 df-wlks 27308 |
This theorem is referenced by: wlkcl 27324 wksv 27328 iedginwlk 27345 wlk1ewlk 27348 wlkv0 27359 wlkonwlk1l 27372 wlkres 27379 redwlk 27381 wlkp1lem2 27383 wlkp1lem3 27384 wlkp1lem4 27385 wlkp1lem6 27387 wlkp1lem8 27389 wlkp1 27390 lfgriswlk 27397 trlf1 27407 trlreslem 27408 upgr2pthnlp 27440 uhgrwkspthlem1 27461 usgr2wlkspthlem1 27465 crctcshlem2 27523 crctcshlem4 27525 crctcshwlkn0 27526 eupth2eucrct 27923 eucrctshift 27949 eucrct2eupth 27951 pfxwlk 32267 revwlk 32268 swrdwlk 32270 |
Copyright terms: Public domain | W3C validator |