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Mirrors > Home > MPE Home > Th. List > clwlkcomp | Structured version Visualization version GIF version |
Description: A closed walk expressed by properties of its components. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 17-Feb-2021.) |
Ref | Expression |
---|---|
isclwlke.v | ⊢ 𝑉 = (Vtx‘𝐺) |
isclwlke.i | ⊢ 𝐼 = (iEdg‘𝐺) |
clwlkcomp.1 | ⊢ 𝐹 = (1st ‘𝑊) |
clwlkcomp.2 | ⊢ 𝑃 = (2nd ‘𝑊) |
Ref | Expression |
---|---|
clwlkcomp | ⊢ ((𝐺 ∈ 𝑋 ∧ 𝑊 ∈ (𝑆 × 𝑇)) → (𝑊 ∈ (ClWalks‘𝐺) ↔ ((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))) ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clwlkcomp.1 | . . . . . . 7 ⊢ 𝐹 = (1st ‘𝑊) | |
2 | 1 | eqcomi 2834 | . . . . . 6 ⊢ (1st ‘𝑊) = 𝐹 |
3 | clwlkcomp.2 | . . . . . . 7 ⊢ 𝑃 = (2nd ‘𝑊) | |
4 | 3 | eqcomi 2834 | . . . . . 6 ⊢ (2nd ‘𝑊) = 𝑃 |
5 | 2, 4 | pm3.2i 464 | . . . . 5 ⊢ ((1st ‘𝑊) = 𝐹 ∧ (2nd ‘𝑊) = 𝑃) |
6 | eqop 7475 | . . . . 5 ⊢ (𝑊 ∈ (𝑆 × 𝑇) → (𝑊 = 〈𝐹, 𝑃〉 ↔ ((1st ‘𝑊) = 𝐹 ∧ (2nd ‘𝑊) = 𝑃))) | |
7 | 5, 6 | mpbiri 250 | . . . 4 ⊢ (𝑊 ∈ (𝑆 × 𝑇) → 𝑊 = 〈𝐹, 𝑃〉) |
8 | 7 | eleq1d 2891 | . . 3 ⊢ (𝑊 ∈ (𝑆 × 𝑇) → (𝑊 ∈ (ClWalks‘𝐺) ↔ 〈𝐹, 𝑃〉 ∈ (ClWalks‘𝐺))) |
9 | df-br 4876 | . . 3 ⊢ (𝐹(ClWalks‘𝐺)𝑃 ↔ 〈𝐹, 𝑃〉 ∈ (ClWalks‘𝐺)) | |
10 | 8, 9 | syl6bbr 281 | . 2 ⊢ (𝑊 ∈ (𝑆 × 𝑇) → (𝑊 ∈ (ClWalks‘𝐺) ↔ 𝐹(ClWalks‘𝐺)𝑃)) |
11 | isclwlke.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
12 | isclwlke.i | . . 3 ⊢ 𝐼 = (iEdg‘𝐺) | |
13 | 11, 12 | isclwlke 27086 | . 2 ⊢ (𝐺 ∈ 𝑋 → (𝐹(ClWalks‘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))) ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹)))))) |
14 | 10, 13 | sylan9bbr 506 | 1 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝑊 ∈ (𝑆 × 𝑇)) → (𝑊 ∈ (ClWalks‘𝐺) ↔ ((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))) ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹)))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 if-wif 1089 = wceq 1656 ∈ wcel 2164 ∀wral 3117 ⊆ wss 3798 {csn 4399 {cpr 4401 〈cop 4405 class class class wbr 4875 × cxp 5344 dom cdm 5346 ⟶wf 6123 ‘cfv 6127 (class class class)co 6910 1st c1st 7431 2nd c2nd 7432 0cc0 10259 1c1 10260 + caddc 10262 ...cfz 12626 ..^cfzo 12767 ♯chash 13417 Word cword 13581 Vtxcvtx 26301 iEdgciedg 26302 ClWalkscclwlks 27079 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-ifp 1090 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-er 8014 df-map 8129 df-pm 8130 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-card 9085 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-n0 11626 df-z 11712 df-uz 11976 df-fz 12627 df-fzo 12768 df-hash 13418 df-word 13582 df-wlks 26904 df-clwlks 27080 |
This theorem is referenced by: clwlkcompim 27089 |
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