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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 2830 | . . . . . 6 ⊢ (1st ‘𝑊) = 𝐹 |
3 | clwlkcomp.2 | . . . . . . 7 ⊢ 𝑃 = (2nd ‘𝑊) | |
4 | 3 | eqcomi 2830 | . . . . . 6 ⊢ (2nd ‘𝑊) = 𝑃 |
5 | 2, 4 | pm3.2i 473 | . . . . 5 ⊢ ((1st ‘𝑊) = 𝐹 ∧ (2nd ‘𝑊) = 𝑃) |
6 | eqop 7725 | . . . . 5 ⊢ (𝑊 ∈ (𝑆 × 𝑇) → (𝑊 = 〈𝐹, 𝑃〉 ↔ ((1st ‘𝑊) = 𝐹 ∧ (2nd ‘𝑊) = 𝑃))) | |
7 | 5, 6 | mpbiri 260 | . . . 4 ⊢ (𝑊 ∈ (𝑆 × 𝑇) → 𝑊 = 〈𝐹, 𝑃〉) |
8 | 7 | eleq1d 2897 | . . 3 ⊢ (𝑊 ∈ (𝑆 × 𝑇) → (𝑊 ∈ (ClWalks‘𝐺) ↔ 〈𝐹, 𝑃〉 ∈ (ClWalks‘𝐺))) |
9 | df-br 5060 | . . 3 ⊢ (𝐹(ClWalks‘𝐺)𝑃 ↔ 〈𝐹, 𝑃〉 ∈ (ClWalks‘𝐺)) | |
10 | 8, 9 | syl6bbr 291 | . 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 27552 | . 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 513 | 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 208 ∧ wa 398 if-wif 1057 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ⊆ wss 3936 {csn 4561 {cpr 4563 〈cop 4567 class class class wbr 5059 × cxp 5548 dom cdm 5550 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 1st c1st 7681 2nd c2nd 7682 0cc0 10531 1c1 10532 + caddc 10534 ...cfz 12886 ..^cfzo 13027 ♯chash 13684 Word cword 13855 Vtxcvtx 26775 iEdgciedg 26776 ClWalkscclwlks 27545 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
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 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-hash 13685 df-word 13856 df-wlks 27375 df-clwlks 27546 |
This theorem is referenced by: clwlkcompim 27555 |
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