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Theorem wlkcompim 29398

Description: Implications for the properties of the components of a walk. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 2-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
wlkcompim	⊢ (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: 𝐼(𝑘) 𝑉(𝑘) 𝑊(𝑘)

**Proof of Theorem wlkcompim**
Step	Hyp	Ref	Expression
1		elfvex 6923	. 2 ⊢ (𝑊 ∈ (Walks‘𝐺) → 𝐺 ∈ V)
2		wlkcpr 29395	. . 3 ⊢ (𝑊 ∈ (Walks‘𝐺) ↔ (1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊))
3		wlkvv 29393	. . 3 ⊢ ((1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊) → 𝑊 ∈ (V × V))
4	2, 3	sylbi 216	. 2 ⊢ (𝑊 ∈ (Walks‘𝐺) → 𝑊 ∈ (V × V))
5		wlkcomp.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
6		wlkcomp.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝐺)
7		wlkcomp.1	. . . 4 ⊢ 𝐹 = (1^st ‘𝑊)
8		wlkcomp.2	. . . 4 ⊢ 𝑃 = (2^nd ‘𝑊)
9	5, 6, 7, 8	wlkcomp 29397	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝑊 ∈ (V × V)) → (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))))))
10	9	biimpcd 248	. 2 ⊢ (𝑊 ∈ (Walks‘𝐺) → ((𝐺 ∈ V ∧ 𝑊 ∈ (V × V)) → (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))))))
11	1, 4, 10	mp2and 696	1 ⊢ (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 ∧ wa 395 if-wif 1059 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3055 Vcvv 3468 ⊆ wss 3943 {csn 4623 {cpr 4625 class class class wbr 5141 × cxp 5667 dom cdm 5669 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 1^st c1st 7972 2^nd c2nd 7973 0cc0 11112 1c1 11113 + caddc 11115 ...cfz 13490 ..^cfzo 13633 ♯chash 14295 Word cword 14470 Vtxcvtx 28764 iEdgciedg 28765 Walkscwlks 29362

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-ifp 1060 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-fzo 13634 df-hash 14296 df-word 14471 df-wlks 29365

This theorem is referenced by: wlkelwrd 29399

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