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Theorem wlknewwlksn 28252

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 27996	. . . . . 6 ⊢ (𝑊 ∈ (Walks‘𝐺) ↔ (1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊))
2		wlkn0 27988	. . . . . 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 482	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) → (2^nd ‘𝑊) ≠ ∅)
5		eqid 2738	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
6		eqid 2738	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
7		eqid 2738	. . . . . . 7 ⊢ (1^st ‘𝑊) = (1^st ‘𝑊)
8		eqid 2738	. . . . . . 7 ⊢ (2^nd ‘𝑊) = (2^nd ‘𝑊)
9	5, 6, 7, 8	wlkelwrd 28000	. . . . . 6 ⊢ (𝑊 ∈ (Walks‘𝐺) → ((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺)))
10		ffz0iswrd 14244	. . . . . . 7 ⊢ ((2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺) → (2^nd ‘𝑊) ∈ Word (Vtx‘𝐺))
11	10	adantl 482	. . . . . 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 482	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) → (2^nd ‘𝑊) ∈ Word (Vtx‘𝐺))
14		eqid 2738	. . . . . . 7 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
15	14	upgrwlkvtxedg 28012	. . . . . 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 27989	. . . . . . . . 9 ⊢ ((1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊) → (♯‘(1^st ‘𝑊)) = ((♯‘(2^nd ‘𝑊)) − 1))
17	16	adantl 482	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ (1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊)) → (♯‘(1^st ‘𝑊)) = ((♯‘(2^nd ‘𝑊)) − 1))
18	17	oveq2d 7291	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ (1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊)) → (0..^(♯‘(1^st ‘𝑊))) = (0..^((♯‘(2^nd ‘𝑊)) − 1)))
19	18	raleqdv 3348	. . . . . 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 594	. . . 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 1127	. . 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 481	. 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 483	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ (♯‘(1^st ‘𝑊)) = 𝑁) → 𝑁 ∈ ℕ₀)
25		oveq2 7283	. . . . . . . . . . . . 13 ⊢ ((♯‘(1^st ‘𝑊)) = 𝑁 → (0...(♯‘(1^st ‘𝑊))) = (0...𝑁))
26	25	adantl 482	. . . . . . . . . . . 12 ⊢ (((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (♯‘(1^st ‘𝑊)) = 𝑁) → (0...(♯‘(1^st ‘𝑊))) = (0...𝑁))
27	26	feq2d 6586	. . . . . . . . . . 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 452	. . . . . . . . 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 491	. . . . . . . 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 407	. . . . . . 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 14159	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ (2^nd ‘𝑊):(0...𝑁)⟶(Vtx‘𝐺)) → (♯‘(2^nd ‘𝑊)) = (𝑁 + 1))
33	24, 31, 32	syl2an2 683	. . . . . 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 413	. . . . 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 482	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) → ((𝑁 ∈ ℕ₀ ∧ (♯‘(1^st ‘𝑊)) = 𝑁) → (♯‘(2^nd ‘𝑊)) = (𝑁 + 1)))
37	36	imp 407	. 2 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) ∧ (𝑁 ∈ ℕ₀ ∧ (♯‘(1^st ‘𝑊)) = 𝑁)) → (♯‘(2^nd ‘𝑊)) = (𝑁 + 1))
38	24	adantl 482	. . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) ∧ (𝑁 ∈ ℕ₀ ∧ (♯‘(1^st ‘𝑊)) = 𝑁)) → 𝑁 ∈ ℕ₀)
39		iswwlksn 28203	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → ((2^nd ‘𝑊) ∈ (𝑁 WWalksN 𝐺) ↔ ((2^nd ‘𝑊) ∈ (WWalks‘𝐺) ∧ (♯‘(2^nd ‘𝑊)) = (𝑁 + 1))))
40	5, 14	iswwlks 28201	. . . . . 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 630	. . . 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 710	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 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∅c0 4256 {cpr 4563 class class class wbr 5074 dom cdm 5589 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 1^st c1st 7829 2^nd c2nd 7830 0cc0 10871 1c1 10872 + caddc 10874 − cmin 11205 ℕ₀cn0 12233 ...cfz 13239 ..^cfzo 13382 ♯chash 14044 Word cword 14217 Vtxcvtx 27366 iEdgciedg 27367 Edgcedg 27417 UPGraphcupgr 27450 Walkscwlks 27963 WWalkscwwlks 28190 WWalksN cwwlksn 28191

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 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

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 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 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 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-er 8498 df-map 8617 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-dju 9659 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-n0 12234 df-xnn0 12306 df-z 12320 df-uz 12583 df-fz 13240 df-fzo 13383 df-hash 14045 df-word 14218 df-edg 27418 df-uhgr 27428 df-upgr 27452 df-wlks 27966 df-wwlks 28195 df-wwlksn 28196

This theorem is referenced by: (None)

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