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Mirrors > Home > MPE Home > Th. List > wlk1ewlk | Structured version Visualization version GIF version |
Description: A walk is an s-walk "on the edge level" (with s=1) according to Aksoy et al. (Contributed by AV, 5-Jan-2021.) |
Ref | Expression |
---|---|
wlk1ewlk | ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ (𝐺 EdgWalks 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
2 | 1 | wlkf 27969 | . 2 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom (iEdg‘𝐺)) |
3 | 1 | wlk1walk 27993 | . 2 ⊢ (𝐹(Walks‘𝐺)𝑃 → ∀𝑘 ∈ (1..^(♯‘𝐹))1 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘))))) |
4 | wlkv 27967 | . . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V)) | |
5 | 4 | simp1d 1141 | . . 3 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐺 ∈ V) |
6 | 1nn0 12237 | . . . 4 ⊢ 1 ∈ ℕ0 | |
7 | nn0xnn0 12297 | . . . 4 ⊢ (1 ∈ ℕ0 → 1 ∈ ℕ0*) | |
8 | 6, 7 | mp1i 13 | . . 3 ⊢ (𝐹(Walks‘𝐺)𝑃 → 1 ∈ ℕ0*) |
9 | 1 | isewlk 27957 | . . 3 ⊢ ((𝐺 ∈ V ∧ 1 ∈ ℕ0* ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (𝐹 ∈ (𝐺 EdgWalks 1) ↔ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(♯‘𝐹))1 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘))))))) |
10 | 5, 8, 2, 9 | syl3anc 1370 | . 2 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐹 ∈ (𝐺 EdgWalks 1) ↔ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(♯‘𝐹))1 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘))))))) |
11 | 2, 3, 10 | mpbir2and 710 | 1 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ (𝐺 EdgWalks 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 ∀wral 3064 Vcvv 3430 ∩ cin 3886 class class class wbr 5074 dom cdm 5585 ‘cfv 6427 (class class class)co 7268 1c1 10860 ≤ cle 10998 − cmin 11193 ℕ0cn0 12221 ℕ0*cxnn0 12293 ..^cfzo 13370 ♯chash 14032 Word cword 14205 iEdgciedg 27355 EdgWalks cewlks 27950 Walkscwlks 27951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-cnex 10915 ax-resscn 10916 ax-1cn 10917 ax-icn 10918 ax-addcl 10919 ax-addrcl 10920 ax-mulcl 10921 ax-mulrcl 10922 ax-mulcom 10923 ax-addass 10924 ax-mulass 10925 ax-distr 10926 ax-i2m1 10927 ax-1ne0 10928 ax-1rid 10929 ax-rnegex 10930 ax-rrecex 10931 ax-cnre 10932 ax-pre-lttri 10933 ax-pre-lttrn 10934 ax-pre-ltadd 10935 ax-pre-mulgt0 10936 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ifp 1061 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-int 4881 df-iun 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6263 df-on 6264 df-lim 6265 df-suc 6266 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7704 df-1st 7821 df-2nd 7822 df-frecs 8085 df-wrecs 8116 df-recs 8190 df-rdg 8229 df-1o 8285 df-er 8486 df-map 8605 df-en 8722 df-dom 8723 df-sdom 8724 df-fin 8725 df-card 9685 df-pnf 10999 df-mnf 11000 df-xr 11001 df-ltxr 11002 df-le 11003 df-sub 11195 df-neg 11196 df-nn 11962 df-n0 12222 df-xnn0 12294 df-z 12308 df-uz 12571 df-fz 13228 df-fzo 13371 df-hash 14033 df-word 14206 df-ewlks 27953 df-wlks 27954 |
This theorem is referenced by: (None) |
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