Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2631 | . . . . . 6 ⊢ (1st ‘𝑊) = 𝐹 |
3 | clwlkcomp.2 | . . . . . . 7 ⊢ 𝑃 = (2nd ‘𝑊) | |
4 | 3 | eqcomi 2631 | . . . . . 6 ⊢ (2nd ‘𝑊) = 𝑃 |
5 | 2, 4 | pm3.2i 471 | . . . . 5 ⊢ ((1st ‘𝑊) = 𝐹 ∧ (2nd ‘𝑊) = 𝑃) |
6 | eqop 7208 | . . . . 5 ⊢ (𝑊 ∈ (𝑆 × 𝑇) → (𝑊 = 〈𝐹, 𝑃〉 ↔ ((1st ‘𝑊) = 𝐹 ∧ (2nd ‘𝑊) = 𝑃))) | |
7 | 5, 6 | mpbiri 248 | . . . 4 ⊢ (𝑊 ∈ (𝑆 × 𝑇) → 𝑊 = 〈𝐹, 𝑃〉) |
8 | 7 | eleq1d 2686 | . . 3 ⊢ (𝑊 ∈ (𝑆 × 𝑇) → (𝑊 ∈ (ClWalks‘𝐺) ↔ 〈𝐹, 𝑃〉 ∈ (ClWalks‘𝐺))) |
9 | df-br 4654 | . . 3 ⊢ (𝐹(ClWalks‘𝐺)𝑃 ↔ 〈𝐹, 𝑃〉 ∈ (ClWalks‘𝐺)) | |
10 | 8, 9 | syl6bbr 278 | . 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 26673 | . 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 737 | 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 196 ∧ wa 384 if-wif 1012 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ⊆ wss 3574 {csn 4177 {cpr 4179 〈cop 4183 class class class wbr 4653 × cxp 5112 dom cdm 5114 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 1st c1st 7166 2nd c2nd 7167 0cc0 9936 1c1 9937 + caddc 9939 ...cfz 12326 ..^cfzo 12465 #chash 13117 Word cword 13291 Vtxcvtx 25874 iEdgciedg 25875 ClWalkscclwlks 26666 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ifp 1013 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-wlks 26495 df-clwlks 26667 |
This theorem is referenced by: clwlkcompim 26676 |
Copyright terms: Public domain | W3C validator |