wlkd - Metamath Proof Explorer

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

Theorem wlkd 28063

Description: Two words representing a walk in a graph. (Contributed by AV, 7-Feb-2021.)

**Hypotheses**
Ref	Expression
wlkd.p	⊢ (𝜑 → 𝑃 ∈ Word V)
wlkd.f	⊢ (𝜑 → 𝐹 ∈ Word V)
wlkd.l	⊢ (𝜑 → (♯‘𝑃) = ((♯‘𝐹) + 1))
wlkd.e	⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))
wlkd.n	⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)))
wlkd.g	⊢ (𝜑 → 𝐺 ∈ 𝑊)
wlkd.v	⊢ 𝑉 = (Vtx‘𝐺)
wlkd.i	⊢ 𝐼 = (iEdg‘𝐺)
wlkd.a	⊢ (𝜑 → ∀𝑘 ∈ (0...(♯‘𝐹))(𝑃‘𝑘) ∈ 𝑉)

**Assertion**
Ref	Expression
wlkd	⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃)

Distinct variable groups: 𝑘,𝐹 𝑃,𝑘 𝑘,𝐼 𝜑,𝑘 𝑘,𝐺 𝑘,𝑉 Allowed substitution hint: 𝑊(𝑘)

**Proof of Theorem wlkd**
Step	Hyp	Ref	Expression
1		wlkd.p	. . 3 ⊢ (𝜑 → 𝑃 ∈ Word V)
2		wlkd.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ Word V)
3		wlkd.l	. . 3 ⊢ (𝜑 → (♯‘𝑃) = ((♯‘𝐹) + 1))
4		wlkd.e	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))
5	1, 2, 3, 4	wlkdlem3 28061	. 2 ⊢ (𝜑 → 𝐹 ∈ Word dom 𝐼)
6		wlkd.a	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ (0...(♯‘𝐹))(𝑃‘𝑘) ∈ 𝑉)
7	1, 2, 3, 6	wlkdlem1 28059	. 2 ⊢ (𝜑 → 𝑃:(0...(♯‘𝐹))⟶𝑉)
8		wlkd.n	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)))
9	1, 2, 3, 4, 8	wlkdlem4 28062	. 2 ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))
10		wlkd.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑊)
11		wlkd.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
12		wlkd.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝐺)
13	11, 12	iswlk 27987	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹 ∈ Word V ∧ 𝑃 ∈ Word V) → (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, 2, 1, 13	syl3anc 1370	. 2 ⊢ (𝜑 → (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))))))
15	5, 7, 9, 14	mpbir3and 1341	1 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 if-wif 1060 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 Vcvv 3429 ⊆ wss 3886 {csn 4561 {cpr 4563 class class class wbr 5073 dom cdm 5584 ⟶wf 6422 ‘cfv 6426 (class class class)co 7267 0cc0 10881 1c1 10882 + caddc 10884 ...cfz 13249 ..^cfzo 13392 ♯chash 14054 Word cword 14227 Vtxcvtx 27376 iEdgciedg 27377 Walkscwlks 27973

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ifp 1061 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-om 7703 df-1st 7820 df-2nd 7821 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-1o 8284 df-er 8485 df-map 8604 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 df-card 9707 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-nn 11984 df-n0 12244 df-z 12330 df-uz 12593 df-fz 13250 df-fzo 13393 df-hash 14055 df-word 14228 df-wlks 27976

This theorem is referenced by: 2wlkd 28309 3wlkd 28542

W3C validator