wlkcomp - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > wlkcomp		Structured version Visualization version GIF version

Theorem wlkcomp 29611

Description: A walk expressed by properties of its components. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 1-Jan-2021.)

**Hypotheses**
Ref	Expression
wlkcomp.v	⊢ 𝑉 = (Vtx‘𝐺)
wlkcomp.i	⊢ 𝐼 = (iEdg‘𝐺)
wlkcomp.1	⊢ 𝐹 = (1^st ‘𝑊)
wlkcomp.2	⊢ 𝑃 = (2^nd ‘𝑊)

**Assertion**
Ref	Expression
wlkcomp	⊢ ((𝐺 ∈ 𝑈 ∧ 𝑊 ∈ (𝑆 × 𝑇)) → (𝑊 ∈ (Walks‘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))))))

Distinct variable groups: 𝑘,𝐹 𝑘,𝐺 𝑃,𝑘 Allowed substitution hints: 𝑆(𝑘) 𝑇(𝑘) 𝑈(𝑘) 𝐼(𝑘) 𝑉(𝑘) 𝑊(𝑘)

**Proof of Theorem wlkcomp**
Step	Hyp	Ref	Expression
1		wlkcomp.1	. . . . . . 7 ⊢ 𝐹 = (1^st ‘𝑊)
2	1	eqcomi 2744	. . . . . 6 ⊢ (1^st ‘𝑊) = 𝐹
3		wlkcomp.2	. . . . . . 7 ⊢ 𝑃 = (2^nd ‘𝑊)
4	3	eqcomi 2744	. . . . . 6 ⊢ (2^nd ‘𝑊) = 𝑃
5	2, 4	pm3.2i 470	. . . . 5 ⊢ ((1^st ‘𝑊) = 𝐹 ∧ (2^nd ‘𝑊) = 𝑃)
6		eqop 8030	. . . . 5 ⊢ (𝑊 ∈ (𝑆 × 𝑇) → (𝑊 = ⟨𝐹, 𝑃⟩ ↔ ((1^st ‘𝑊) = 𝐹 ∧ (2^nd ‘𝑊) = 𝑃)))
7	5, 6	mpbiri 258	. . . 4 ⊢ (𝑊 ∈ (𝑆 × 𝑇) → 𝑊 = ⟨𝐹, 𝑃⟩)
8	7	eleq1d 2819	. . 3 ⊢ (𝑊 ∈ (𝑆 × 𝑇) → (𝑊 ∈ (Walks‘𝐺) ↔ ⟨𝐹, 𝑃⟩ ∈ (Walks‘𝐺)))
9		df-br 5120	. . 3 ⊢ (𝐹(Walks‘𝐺)𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ (Walks‘𝐺))
10	8, 9	bitr4di 289	. 2 ⊢ (𝑊 ∈ (𝑆 × 𝑇) → (𝑊 ∈ (Walks‘𝐺) ↔ 𝐹(Walks‘𝐺)𝑃))
11		wlkcomp.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
12		wlkcomp.i	. . 3 ⊢ 𝐼 = (iEdg‘𝐺)
13	11, 12	iswlkg 29593	. 2 ⊢ (𝐺 ∈ 𝑈 → (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, 13	sylan9bbr 510	1 ⊢ ((𝐺 ∈ 𝑈 ∧ 𝑊 ∈ (𝑆 × 𝑇)) → (𝑊 ∈ (Walks‘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))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 if-wif 1062 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ∀wral 3051 ⊆ wss 3926 {csn 4601 {cpr 4603 ⟨cop 4607 class class class wbr 5119 × cxp 5652 dom cdm 5654 ⟶wf 6527 ‘cfv 6531 (class class class)co 7405 1^st c1st 7986 2^nd c2nd 7987 0cc0 11129 1c1 11130 + caddc 11132 ...cfz 13524 ..^cfzo 13671 ♯chash 14348 Word cword 14531 Vtxcvtx 28975 iEdgciedg 28976 Walkscwlks 29576

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8719 df-map 8842 df-pm 8843 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-nn 12241 df-n0 12502 df-z 12589 df-uz 12853 df-fz 13525 df-fzo 13672 df-hash 14349 df-word 14532 df-wlks 29579

This theorem is referenced by: wlkcompim 29612

W3C validator