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Theorem wlkd 29778

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 29776	. 2 ⊢ (𝜑 → 𝐹 ∈ Word dom 𝐼)
6		wlkd.a	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ (0...(♯‘𝐹))(𝑃‘𝑘) ∈ 𝑉)
7	1, 2, 3, 6	wlkdlem1 29774	. 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 29777	. 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 29704	. . 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 1379	. 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 1349	1 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 if-wif 1068 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ≠ wne 2935 ∀wral 3054 Vcvv 3432 ⊆ wss 3890 {csn 4562 {cpr 4564 class class class wbr 5079 dom cdm 5625 ⟶wf 6488 ‘cfv 6492 (class class class)co 7363 0cc0 11036 1c1 11037 + caddc 11039 ...cfz 13459 ..^cfzo 13606 ♯chash 14290 Word cword 14473 Vtxcvtx 29090 iEdgciedg 29091 Walkscwlks 29690

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-ifp 1069 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-om 7814 df-1st 7938 df-2nd 7939 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-er 8640 df-map 8772 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-card 9861 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-nn 12173 df-n0 12436 df-z 12523 df-uz 12787 df-fz 13460 df-fzo 13607 df-hash 14291 df-word 14474 df-wlks 29693

This theorem is referenced by: 2wlkd 30029 3wlkd 30265

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