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Mirrors > Home > MPE Home > Th. List > wlkd | Structured version Visualization version GIF version |
Description: Two words representing a walk in a graph. (Contributed by AV, 7-Feb-2021.) |
Ref | Expression |
---|---|
wlkd.p | ⊢ (𝜑 → 𝑃 ∈ Word V) |
wlkd.f | ⊢ (𝜑 → 𝐹 ∈ Word V) |
wlkd.l | ⊢ (𝜑 → (♯‘𝑃) = ((♯‘𝐹) + 1)) |
wlkd.e | ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) |
wlkd.n | ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))) |
wlkd.g | ⊢ (𝜑 → 𝐺 ∈ 𝑊) |
wlkd.v | ⊢ 𝑉 = (Vtx‘𝐺) |
wlkd.i | ⊢ 𝐼 = (iEdg‘𝐺) |
wlkd.a | ⊢ (𝜑 → ∀𝑘 ∈ (0...(♯‘𝐹))(𝑃‘𝑘) ∈ 𝑉) |
Ref | Expression |
---|---|
wlkd | ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wlkd.p | . . 3 ⊢ (𝜑 → 𝑃 ∈ Word V) | |
2 | wlkd.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ Word V) | |
3 | wlkd.l | . . 3 ⊢ (𝜑 → (♯‘𝑃) = ((♯‘𝐹) + 1)) | |
4 | wlkd.e | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) | |
5 | 1, 2, 3, 4 | wlkdlem3 27447 | . 2 ⊢ (𝜑 → 𝐹 ∈ Word dom 𝐼) |
6 | wlkd.a | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ (0...(♯‘𝐹))(𝑃‘𝑘) ∈ 𝑉) | |
7 | 1, 2, 3, 6 | wlkdlem1 27445 | . 2 ⊢ (𝜑 → 𝑃:(0...(♯‘𝐹))⟶𝑉) |
8 | wlkd.n | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))) | |
9 | 1, 2, 3, 4, 8 | wlkdlem4 27448 | . 2 ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))) |
10 | wlkd.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑊) | |
11 | wlkd.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
12 | wlkd.i | . . . 4 ⊢ 𝐼 = (iEdg‘𝐺) | |
13 | 11, 12 | iswlk 27373 | . . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹 ∈ Word V ∧ 𝑃 ∈ Word V) → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))))) |
14 | 10, 2, 1, 13 | syl3anc 1367 | . 2 ⊢ (𝜑 → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))))) |
15 | 5, 7, 9, 14 | mpbir3and 1338 | 1 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 if-wif 1057 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3011 ∀wral 3133 Vcvv 3481 ⊆ wss 3919 {csn 4548 {cpr 4550 class class class wbr 5047 dom cdm 5536 ⟶wf 6332 ‘cfv 6336 (class class class)co 7137 0cc0 10518 1c1 10519 + caddc 10521 ...cfz 12877 ..^cfzo 13018 ♯chash 13675 Word cword 13846 Vtxcvtx 26762 iEdgciedg 26763 Walkscwlks 27359 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 ax-rep 5171 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 ax-un 7442 ax-cnex 10574 ax-resscn 10575 ax-1cn 10576 ax-icn 10577 ax-addcl 10578 ax-addrcl 10579 ax-mulcl 10580 ax-mulrcl 10581 ax-mulcom 10582 ax-addass 10583 ax-mulass 10584 ax-distr 10585 ax-i2m1 10586 ax-1ne0 10587 ax-1rid 10588 ax-rnegex 10589 ax-rrecex 10590 ax-cnre 10591 ax-pre-lttri 10592 ax-pre-lttrn 10593 ax-pre-ltadd 10594 ax-pre-mulgt0 10595 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3012 df-nel 3119 df-ral 3138 df-rex 3139 df-reu 3140 df-rab 3142 df-v 3483 df-sbc 3759 df-csb 3867 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-pss 3937 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-tp 4553 df-op 4555 df-uni 4820 df-int 4858 df-iun 4902 df-br 5048 df-opab 5110 df-mpt 5128 df-tr 5154 df-id 5441 df-eprel 5446 df-po 5455 df-so 5456 df-fr 5495 df-we 5497 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-riota 7095 df-ov 7140 df-oprab 7141 df-mpo 7142 df-om 7562 df-1st 7670 df-2nd 7671 df-wrecs 7928 df-recs 7989 df-rdg 8027 df-1o 8083 df-oadd 8087 df-er 8270 df-map 8389 df-en 8491 df-dom 8492 df-sdom 8493 df-fin 8494 df-card 9349 df-pnf 10658 df-mnf 10659 df-xr 10660 df-ltxr 10661 df-le 10662 df-sub 10853 df-neg 10854 df-nn 11620 df-n0 11880 df-z 11964 df-uz 12226 df-fz 12878 df-fzo 13019 df-hash 13676 df-word 13847 df-wlks 27362 |
This theorem is referenced by: 2wlkd 27695 3wlkd 27928 |
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