wlknewwlksn - Mathbox for Alexander van der Vekens

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Theorem wlknewwlksn 41084

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 ∈ (1Walks‘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		1wlkcpr 40833	. . . . . 6 ⊢ (𝑊 ∈ (1Walks‘𝐺) ↔ (1^st ‘𝑊)(1Walks‘𝐺)(2^nd ‘𝑊))
2		1wlkn0 40825	. . . . . 6 ⊢ ((1^st ‘𝑊)(1Walks‘𝐺)(2^nd ‘𝑊) → (2^nd ‘𝑊) ≠ ∅)
3	1, 2	sylbi 206	. . . . 5 ⊢ (𝑊 ∈ (1Walks‘𝐺) → (2^nd ‘𝑊) ≠ ∅)
4	3	adantl 481	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (1Walks‘𝐺)) → (2^nd ‘𝑊) ≠ ∅)
5		eqid 2610	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
6		eqid 2610	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
7		eqid 2610	. . . . . . 7 ⊢ (1^st ‘𝑊) = (1^st ‘𝑊)
8		eqid 2610	. . . . . . 7 ⊢ (2^nd ‘𝑊) = (2^nd ‘𝑊)
9	5, 6, 7, 8	1wlkelwrd 40837	. . . . . 6 ⊢ (𝑊 ∈ (1Walks‘𝐺) → ((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝑊):(0...(#‘(1^st ‘𝑊)))⟶(Vtx‘𝐺)))
10		ffz0iswrd 13187	. . . . . . 7 ⊢ ((2^nd ‘𝑊):(0...(#‘(1^st ‘𝑊)))⟶(Vtx‘𝐺) → (2^nd ‘𝑊) ∈ Word (Vtx‘𝐺))
11	10	adantl 481	. . . . . 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 ∈ (1Walks‘𝐺) → (2^nd ‘𝑊) ∈ Word (Vtx‘𝐺))
13	12	adantl 481	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (1Walks‘𝐺)) → (2^nd ‘𝑊) ∈ Word (Vtx‘𝐺))
14		eqid 2610	. . . . . . 7 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
15	14	upgr1wlkvtxedg 40853	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ (1^st ‘𝑊)(1Walks‘𝐺)(2^nd ‘𝑊)) → ∀𝑖 ∈ (0..^(#‘(1^st ‘𝑊))){((2^nd ‘𝑊)‘𝑖), ((2^nd ‘𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺))
16		1wlklenvm1 40826	. . . . . . . . 9 ⊢ ((1^st ‘𝑊)(1Walks‘𝐺)(2^nd ‘𝑊) → (#‘(1^st ‘𝑊)) = ((#‘(2^nd ‘𝑊)) − 1))
17	16	adantl 481	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ (1^st ‘𝑊)(1Walks‘𝐺)(2^nd ‘𝑊)) → (#‘(1^st ‘𝑊)) = ((#‘(2^nd ‘𝑊)) − 1))
18	17	oveq2d 6565	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ (1^st ‘𝑊)(1Walks‘𝐺)(2^nd ‘𝑊)) → (0..^(#‘(1^st ‘𝑊))) = (0..^((#‘(2^nd ‘𝑊)) − 1)))
19	18	raleqdv 3121	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ (1^st ‘𝑊)(1Walks‘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 221	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ (1^st ‘𝑊)(1Walks‘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 491	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (1Walks‘𝐺)) → ∀𝑖 ∈ (0..^((#‘(2^nd ‘𝑊)) − 1)){((2^nd ‘𝑊)‘𝑖), ((2^nd ‘𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺))
22	4, 13, 21	3jca 1235	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (1Walks‘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 480	. 2 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (1Walks‘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 472	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ (#‘(1^st ‘𝑊)) = 𝑁) → 𝑁 ∈ ℕ₀)
25		oveq2 6557	. . . . . . . . . . . . 13 ⊢ ((#‘(1^st ‘𝑊)) = 𝑁 → (0...(#‘(1^st ‘𝑊))) = (0...𝑁))
26	25	adantl 481	. . . . . . . . . . . 12 ⊢ (((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (#‘(1^st ‘𝑊)) = 𝑁) → (0...(#‘(1^st ‘𝑊))) = (0...𝑁))
27	26	feq2d 5944	. . . . . . . . . . 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 218	. . . . . . . . . 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 455	. . . . . . . . 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 482	. . . . . . . 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 444	. . . . . . 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 13088	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ (2^nd ‘𝑊):(0...𝑁)⟶(Vtx‘𝐺)) → (#‘(2^nd ‘𝑊)) = (𝑁 + 1))
33	24, 31, 32	syl2an2 871	. . . . . 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 449	. . . . 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 ∈ (1Walks‘𝐺) → ((𝑁 ∈ ℕ₀ ∧ (#‘(1^st ‘𝑊)) = 𝑁) → (#‘(2^nd ‘𝑊)) = (𝑁 + 1)))
36	35	adantl 481	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (1Walks‘𝐺)) → ((𝑁 ∈ ℕ₀ ∧ (#‘(1^st ‘𝑊)) = 𝑁) → (#‘(2^nd ‘𝑊)) = (𝑁 + 1)))
37	36	imp 444	. 2 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (1Walks‘𝐺)) ∧ (𝑁 ∈ ℕ₀ ∧ (#‘(1^st ‘𝑊)) = 𝑁)) → (#‘(2^nd ‘𝑊)) = (𝑁 + 1))
38	24	adantl 481	. . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (1Walks‘𝐺)) ∧ (𝑁 ∈ ℕ₀ ∧ (#‘(1^st ‘𝑊)) = 𝑁)) → 𝑁 ∈ ℕ₀)
39		iswwlksn 41041	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → ((2^nd ‘𝑊) ∈ (𝑁 WWalkSN 𝐺) ↔ ((2^nd ‘𝑊) ∈ (WWalkS‘𝐺) ∧ (#‘(2^nd ‘𝑊)) = (𝑁 + 1))))
40	5, 14	iswwlks 41039	. . . . . 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 737	. . . 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 267	. . 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 ∈ (1Walks‘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 959	1 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (1Walks‘𝐺)) ∧ (𝑁 ∈ ℕ₀ ∧ (#‘(1^st ‘𝑊)) = 𝑁)) → (2^nd ‘𝑊) ∈ (𝑁 WWalkSN 𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∅c0 3874 {cpr 4127 class class class wbr 4583 dom cdm 5038 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 1^st c1st 7057 2^nd c2nd 7058 0cc0 9815 1c1 9816 + caddc 9818 − cmin 10145 ℕ₀cn0 11169 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 Vtxcvtx 25673 iEdgciedg 25674 UPGraph cupgr 25747 Edgcedga 25792 1Walksc1wlks 40796 WWalkScwwlks 41028 WWalkSN cwwlksn 41029

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ifp 1007 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-uhgr 25724 df-upgr 25749 df-edga 25793 df-1wlks 40800 df-wlks 40801 df-wwlks 41033 df-wwlksn 41034

This theorem is referenced by: wlknwwlksnfun 41085 wlkwwlkfun 41092

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