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Theorem wlknewwlksn 29408

Description: If a walk in a pseudograph has length 𝑁, then the sequence of the vertices of the walk is a word representing the walk as word of length 𝑁. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 11-Apr-2021.)

**Assertion**
Ref	Expression
wlknewwlksn	⊢ (((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) ∧ (𝑁 ∈ ℕ₀ ∧ (♯‘(1^st ‘𝑊)) = 𝑁)) → (2^nd ‘𝑊) ∈ (𝑁 WWalksN 𝐺))

Proof of Theorem wlknewwlksn Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wlkcpr 29153	. . . . . 6 ⊢ (𝑊 ∈ (Walks‘𝐺) ↔ (1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊))
2		wlkn0 29145	. . . . . 6 ⊢ ((1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊) → (2^nd ‘𝑊) ≠ ∅)
3	1, 2	sylbi 216	. . . . 5 ⊢ (𝑊 ∈ (Walks‘𝐺) → (2^nd ‘𝑊) ≠ ∅)
4	3	adantl 480	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) → (2^nd ‘𝑊) ≠ ∅)
5		eqid 2730	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
6		eqid 2730	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
7		eqid 2730	. . . . . . 7 ⊢ (1^st ‘𝑊) = (1^st ‘𝑊)
8		eqid 2730	. . . . . . 7 ⊢ (2^nd ‘𝑊) = (2^nd ‘𝑊)
9	5, 6, 7, 8	wlkelwrd 29157	. . . . . 6 ⊢ (𝑊 ∈ (Walks‘𝐺) → ((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺)))
10		ffz0iswrd 14495	. . . . . . 7 ⊢ ((2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺) → (2^nd ‘𝑊) ∈ Word (Vtx‘𝐺))
11	10	adantl 480	. . . . . 6 ⊢ (((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺)) → (2^nd ‘𝑊) ∈ Word (Vtx‘𝐺))
12	9, 11	syl 17	. . . . 5 ⊢ (𝑊 ∈ (Walks‘𝐺) → (2^nd ‘𝑊) ∈ Word (Vtx‘𝐺))
13	12	adantl 480	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) → (2^nd ‘𝑊) ∈ Word (Vtx‘𝐺))
14		eqid 2730	. . . . . . 7 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
15	14	upgrwlkvtxedg 29169	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ (1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊)) → ∀𝑖 ∈ (0..^(♯‘(1^st ‘𝑊))){((2^nd ‘𝑊)‘𝑖), ((2^nd ‘𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺))
16		wlklenvm1 29146	. . . . . . . . 9 ⊢ ((1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊) → (♯‘(1^st ‘𝑊)) = ((♯‘(2^nd ‘𝑊)) − 1))
17	16	adantl 480	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ (1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊)) → (♯‘(1^st ‘𝑊)) = ((♯‘(2^nd ‘𝑊)) − 1))
18	17	oveq2d 7427	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ (1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊)) → (0..^(♯‘(1^st ‘𝑊))) = (0..^((♯‘(2^nd ‘𝑊)) − 1)))
19	18	raleqdv 3323	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ (1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊)) → (∀𝑖 ∈ (0..^(♯‘(1^st ‘𝑊))){((2^nd ‘𝑊)‘𝑖), ((2^nd ‘𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^((♯‘(2^nd ‘𝑊)) − 1)){((2^nd ‘𝑊)‘𝑖), ((2^nd ‘𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
20	15, 19	mpbid 231	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ (1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊)) → ∀𝑖 ∈ (0..^((♯‘(2^nd ‘𝑊)) − 1)){((2^nd ‘𝑊)‘𝑖), ((2^nd ‘𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺))
21	1, 20	sylan2b 592	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) → ∀𝑖 ∈ (0..^((♯‘(2^nd ‘𝑊)) − 1)){((2^nd ‘𝑊)‘𝑖), ((2^nd ‘𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺))
22	4, 13, 21	3jca 1126	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) → ((2^nd ‘𝑊) ≠ ∅ ∧ (2^nd ‘𝑊) ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘(2^nd ‘𝑊)) − 1)){((2^nd ‘𝑊)‘𝑖), ((2^nd ‘𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
23	22	adantr 479	. 2 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) ∧ (𝑁 ∈ ℕ₀ ∧ (♯‘(1^st ‘𝑊)) = 𝑁)) → ((2^nd ‘𝑊) ≠ ∅ ∧ (2^nd ‘𝑊) ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘(2^nd ‘𝑊)) − 1)){((2^nd ‘𝑊)‘𝑖), ((2^nd ‘𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
24		simpl 481	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ (♯‘(1^st ‘𝑊)) = 𝑁) → 𝑁 ∈ ℕ₀)
25		oveq2 7419	. . . . . . . . . . . . 13 ⊢ ((♯‘(1^st ‘𝑊)) = 𝑁 → (0...(♯‘(1^st ‘𝑊))) = (0...𝑁))
26	25	adantl 480	. . . . . . . . . . . 12 ⊢ (((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (♯‘(1^st ‘𝑊)) = 𝑁) → (0...(♯‘(1^st ‘𝑊))) = (0...𝑁))
27	26	feq2d 6702	. . . . . . . . . . 11 ⊢ (((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (♯‘(1^st ‘𝑊)) = 𝑁) → ((2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺) ↔ (2^nd ‘𝑊):(0...𝑁)⟶(Vtx‘𝐺)))
28	27	biimpd 228	. . . . . . . . . 10 ⊢ (((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (♯‘(1^st ‘𝑊)) = 𝑁) → ((2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺) → (2^nd ‘𝑊):(0...𝑁)⟶(Vtx‘𝐺)))
29	28	impancom 450	. . . . . . . . 9 ⊢ (((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺)) → ((♯‘(1^st ‘𝑊)) = 𝑁 → (2^nd ‘𝑊):(0...𝑁)⟶(Vtx‘𝐺)))
30	29	adantld 489	. . . . . . . 8 ⊢ (((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺)) → ((𝑁 ∈ ℕ₀ ∧ (♯‘(1^st ‘𝑊)) = 𝑁) → (2^nd ‘𝑊):(0...𝑁)⟶(Vtx‘𝐺)))
31	30	imp 405	. . . . . . 7 ⊢ ((((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ₀ ∧ (♯‘(1^st ‘𝑊)) = 𝑁)) → (2^nd ‘𝑊):(0...𝑁)⟶(Vtx‘𝐺))
32		ffz0hash 14410	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ (2^nd ‘𝑊):(0...𝑁)⟶(Vtx‘𝐺)) → (♯‘(2^nd ‘𝑊)) = (𝑁 + 1))
33	24, 31, 32	syl2an2 682	. . . . . 6 ⊢ ((((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ₀ ∧ (♯‘(1^st ‘𝑊)) = 𝑁)) → (♯‘(2^nd ‘𝑊)) = (𝑁 + 1))
34	33	ex 411	. . . . 5 ⊢ (((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺)) → ((𝑁 ∈ ℕ₀ ∧ (♯‘(1^st ‘𝑊)) = 𝑁) → (♯‘(2^nd ‘𝑊)) = (𝑁 + 1)))
35	9, 34	syl 17	. . . 4 ⊢ (𝑊 ∈ (Walks‘𝐺) → ((𝑁 ∈ ℕ₀ ∧ (♯‘(1^st ‘𝑊)) = 𝑁) → (♯‘(2^nd ‘𝑊)) = (𝑁 + 1)))
36	35	adantl 480	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) → ((𝑁 ∈ ℕ₀ ∧ (♯‘(1^st ‘𝑊)) = 𝑁) → (♯‘(2^nd ‘𝑊)) = (𝑁 + 1)))
37	36	imp 405	. 2 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) ∧ (𝑁 ∈ ℕ₀ ∧ (♯‘(1^st ‘𝑊)) = 𝑁)) → (♯‘(2^nd ‘𝑊)) = (𝑁 + 1))
38	24	adantl 480	. . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) ∧ (𝑁 ∈ ℕ₀ ∧ (♯‘(1^st ‘𝑊)) = 𝑁)) → 𝑁 ∈ ℕ₀)
39		iswwlksn 29359	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → ((2^nd ‘𝑊) ∈ (𝑁 WWalksN 𝐺) ↔ ((2^nd ‘𝑊) ∈ (WWalks‘𝐺) ∧ (♯‘(2^nd ‘𝑊)) = (𝑁 + 1))))
40	5, 14	iswwlks 29357	. . . . . 6 ⊢ ((2^nd ‘𝑊) ∈ (WWalks‘𝐺) ↔ ((2^nd ‘𝑊) ≠ ∅ ∧ (2^nd ‘𝑊) ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘(2^nd ‘𝑊)) − 1)){((2^nd ‘𝑊)‘𝑖), ((2^nd ‘𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
41	40	a1i 11	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → ((2^nd ‘𝑊) ∈ (WWalks‘𝐺) ↔ ((2^nd ‘𝑊) ≠ ∅ ∧ (2^nd ‘𝑊) ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘(2^nd ‘𝑊)) − 1)){((2^nd ‘𝑊)‘𝑖), ((2^nd ‘𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺))))
42	41	anbi1d 628	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → (((2^nd ‘𝑊) ∈ (WWalks‘𝐺) ∧ (♯‘(2^nd ‘𝑊)) = (𝑁 + 1)) ↔ (((2^nd ‘𝑊) ≠ ∅ ∧ (2^nd ‘𝑊) ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘(2^nd ‘𝑊)) − 1)){((2^nd ‘𝑊)‘𝑖), ((2^nd ‘𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘(2^nd ‘𝑊)) = (𝑁 + 1))))
43	39, 42	bitrd 278	. . 3 ⊢ (𝑁 ∈ ℕ₀ → ((2^nd ‘𝑊) ∈ (𝑁 WWalksN 𝐺) ↔ (((2^nd ‘𝑊) ≠ ∅ ∧ (2^nd ‘𝑊) ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘(2^nd ‘𝑊)) − 1)){((2^nd ‘𝑊)‘𝑖), ((2^nd ‘𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘(2^nd ‘𝑊)) = (𝑁 + 1))))
44	38, 43	syl 17	. 2 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) ∧ (𝑁 ∈ ℕ₀ ∧ (♯‘(1^st ‘𝑊)) = 𝑁)) → ((2^nd ‘𝑊) ∈ (𝑁 WWalksN 𝐺) ↔ (((2^nd ‘𝑊) ≠ ∅ ∧ (2^nd ‘𝑊) ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘(2^nd ‘𝑊)) − 1)){((2^nd ‘𝑊)‘𝑖), ((2^nd ‘𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘(2^nd ‘𝑊)) = (𝑁 + 1))))
45	23, 37, 44	mpbir2and 709	1 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) ∧ (𝑁 ∈ ℕ₀ ∧ (♯‘(1^st ‘𝑊)) = 𝑁)) → (2^nd ‘𝑊) ∈ (𝑁 WWalksN 𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ∀wral 3059 ∅c0 4321 {cpr 4629 class class class wbr 5147 dom cdm 5675 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 1^st c1st 7975 2^nd c2nd 7976 0cc0 11112 1c1 11113 + caddc 11115 − cmin 11448 ℕ₀cn0 12476 ...cfz 13488 ..^cfzo 13631 ♯chash 14294 Word cword 14468 Vtxcvtx 28523 iEdgciedg 28524 Edgcedg 28574 UPGraphcupgr 28607 Walkscwlks 29120 WWalkscwwlks 29346 WWalksN cwwlksn 29347

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 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 395 df-or 844 df-ifp 1060 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-oadd 8472 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-dju 9898 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-2 12279 df-n0 12477 df-xnn0 12549 df-z 12563 df-uz 12827 df-fz 13489 df-fzo 13632 df-hash 14295 df-word 14469 df-edg 28575 df-uhgr 28585 df-upgr 28609 df-wlks 29123 df-wwlks 29351 df-wwlksn 29352

This theorem is referenced by: (None)

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