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Theorem wlkd 29770

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 29768	. 2 ⊢ (𝜑 → 𝐹 ∈ Word dom 𝐼)
6		wlkd.a	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ (0...(♯‘𝐹))(𝑃‘𝑘) ∈ 𝑉)
7	1, 2, 3, 6	wlkdlem1 29766	. 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 29769	. 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 29696	. . 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 1374	. 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 1344	1 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 if-wif 1063 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 Vcvv 3442 ⊆ wss 3903 {csn 4582 {cpr 4584 class class class wbr 5100 dom cdm 5632 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 0cc0 11038 1c1 11039 + caddc 11041 ...cfz 13435 ..^cfzo 13582 ♯chash 14265 Word cword 14448 Vtxcvtx 29081 iEdgciedg 29082 Walkscwlks 29682

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ifp 1064 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-n0 12414 df-z 12501 df-uz 12764 df-fz 13436 df-fzo 13583 df-hash 14266 df-word 14449 df-wlks 29685

This theorem is referenced by: 2wlkd 30021 3wlkd 30257

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