wlknewwlksn - Metamath Proof Explorer

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

Theorem wlknewwlksn 30356

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 30089	. . . . . 6 ⊢ (𝑊 ∈ (Walks‘𝐺) ↔ (1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊))
2		wlkn0 30081	. . . . . 6 ⊢ ((1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊) → (2^nd ‘𝑊) ≠ ∅)
3	1, 2	sylbi 220	. . . . 5 ⊢ (𝑊 ∈ (Walks‘𝐺) → (2^nd ‘𝑊) ≠ ∅)
4	3	adantl 487	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) → (2^nd ‘𝑊) ≠ ∅)
5		eqid 2760	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
6		eqid 2760	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
7		eqid 2760	. . . . . . 7 ⊢ (1^st ‘𝑊) = (1^st ‘𝑊)
8		eqid 2760	. . . . . . 7 ⊢ (2^nd ‘𝑊) = (2^nd ‘𝑊)
9	5, 6, 7, 8	wlkelwrd 30093	. . . . . 6 ⊢ (𝑊 ∈ (Walks‘𝐺) → ((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺)))
10		ffz0iswrd 14607	. . . . . . 7 ⊢ ((2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺) → (2^nd ‘𝑊) ∈ Word (Vtx‘𝐺))
11	10	adantl 487	. . . . . 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 18	. . . . 5 ⊢ (𝑊 ∈ (Walks‘𝐺) → (2^nd ‘𝑊) ∈ Word (Vtx‘𝐺))
13	12	adantl 487	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) → (2^nd ‘𝑊) ∈ Word (Vtx‘𝐺))
14		eqid 2760	. . . . . . 7 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
15	14	upgrwlkvtxedg 30105	. . . . . 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 30082	. . . . . . . 8 ⊢ ((1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊) → (♯‘(1^st ‘𝑊)) = ((♯‘(2^nd ‘𝑊)) − 1))
17	16	adantl 487	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ (1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊)) → (♯‘(1^st ‘𝑊)) = ((♯‘(2^nd ‘𝑊)) − 1))
18	17	oveq2d 7430	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ (1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊)) → (0..^(♯‘(1^st ‘𝑊))) = (0..^((♯‘(2^nd ‘𝑊)) − 1)))
19	15, 18	raleqtrdv 3321	. . . . 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))
20	1, 19	sylan2b 606	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) → ∀𝑖 ∈ (0..^((♯‘(2^nd ‘𝑊)) − 1)){((2^nd ‘𝑊)‘𝑖), ((2^nd ‘𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺))
21	4, 13, 20	3jca 1146	. . 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)))
22	21	adantr 486	. 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)))
23		simpl 488	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ (♯‘(1^st ‘𝑊)) = 𝑁) → 𝑁 ∈ ℕ₀)
24		oveq2 7422	. . . . . . . . . . . . 13 ⊢ ((♯‘(1^st ‘𝑊)) = 𝑁 → (0...(♯‘(1^st ‘𝑊))) = (0...𝑁))
25	24	adantl 487	. . . . . . . . . . . 12 ⊢ (((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (♯‘(1^st ‘𝑊)) = 𝑁) → (0...(♯‘(1^st ‘𝑊))) = (0...𝑁))
26	25	feq2d 6687	. . . . . . . . . . 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)))
27	26	biimpd 232	. . . . . . . . . 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)))
28	27	impancom 457	. . . . . . . . 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)))
29	28	adantld 496	. . . . . . . 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)))
30	29	imp 412	. . . . . . 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))
31		ffz0hash 14513	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ (2^nd ‘𝑊):(0...𝑁)⟶(Vtx‘𝐺)) → (♯‘(2^nd ‘𝑊)) = (𝑁 + 1))
32	23, 30, 31	syl2an2 699	. . . . . 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))
33	32	ex 418	. . . . 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)))
34	9, 33	syl 18	. . . 4 ⊢ (𝑊 ∈ (Walks‘𝐺) → ((𝑁 ∈ ℕ₀ ∧ (♯‘(1^st ‘𝑊)) = 𝑁) → (♯‘(2^nd ‘𝑊)) = (𝑁 + 1)))
35	34	adantl 487	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) → ((𝑁 ∈ ℕ₀ ∧ (♯‘(1^st ‘𝑊)) = 𝑁) → (♯‘(2^nd ‘𝑊)) = (𝑁 + 1)))
36	35	imp 412	. 2 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) ∧ (𝑁 ∈ ℕ₀ ∧ (♯‘(1^st ‘𝑊)) = 𝑁)) → (♯‘(2^nd ‘𝑊)) = (𝑁 + 1))
37	23	adantl 487	. . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) ∧ (𝑁 ∈ ℕ₀ ∧ (♯‘(1^st ‘𝑊)) = 𝑁)) → 𝑁 ∈ ℕ₀)
38		iswwlksn 30307	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → ((2^nd ‘𝑊) ∈ (𝑁 WWalksN 𝐺) ↔ ((2^nd ‘𝑊) ∈ (WWalks‘𝐺) ∧ (♯‘(2^nd ‘𝑊)) = (𝑁 + 1))))
39	5, 14	iswwlks 30305	. . . . . 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)))
40	39	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))))
41	40	anbi1d 643	. . . 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))))
42	38, 41	bitrd 282	. . 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))))
43	37, 42	syl 18	. 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))))
44	22, 36, 43	mpbir2and 726	1 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) ∧ (𝑁 ∈ ℕ₀ ∧ (♯‘(1^st ‘𝑊)) = 𝑁)) → (2^nd ‘𝑊) ∈ (𝑁 WWalksN 𝐺))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ∀wral 3076 ∅c0 4279 {cpr 4586 class class class wbr 5103 dom cdm 5655 ⟶wf 6529 ‘cfv 6533 (class class class)co 7414 1^st c1st 7985 2^nd c2nd 7986 0cc0 11125 1c1 11126 + caddc 11128 − cmin 11466 ℕ₀cn0 12529 ...cfz 13562 ..^cfzo 13710 ♯chash 14395 Word cword 14579 Vtxcvtx 29454 iEdgciedg 29455 Edgcedg 29505 UPGraphcupgr 29538 Walkscwlks 30057 WWalkscwwlks 30294 WWalksN cwwlksn 30295

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-ifp 1079 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8456 df-2o 8457 df-oadd 8460 df-er 8697 df-map 8829 df-pm 8830 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-dju 9907 df-card 9945 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-nn 12259 df-2 12328 df-n0 12530 df-xnn0 12603 df-z 12617 df-uz 12889 df-fz 13563 df-fzo 13711 df-hash 14396 df-word 14580 df-edg 29506 df-uhgr 29516 df-upgr 29540 df-wlks 30060 df-wwlks 30299 df-wwlksn 30300

This theorem is used by: (None)

W3C validator