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Theorem wlknewwlksn 29972

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 29714	. . . . . 6 ⊢ (𝑊 ∈ (Walks‘𝐺) ↔ (1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊))
2		wlkn0 29706	. . . . . 6 ⊢ ((1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊) → (2^nd ‘𝑊) ≠ ∅)
3	1, 2	sylbi 217	. . . . 5 ⊢ (𝑊 ∈ (Walks‘𝐺) → (2^nd ‘𝑊) ≠ ∅)
4	3	adantl 481	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) → (2^nd ‘𝑊) ≠ ∅)
5		eqid 2737	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
6		eqid 2737	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
7		eqid 2737	. . . . . . 7 ⊢ (1^st ‘𝑊) = (1^st ‘𝑊)
8		eqid 2737	. . . . . . 7 ⊢ (2^nd ‘𝑊) = (2^nd ‘𝑊)
9	5, 6, 7, 8	wlkelwrd 29718	. . . . . 6 ⊢ (𝑊 ∈ (Walks‘𝐺) → ((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺)))
10		ffz0iswrd 14476	. . . . . . 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 ∈ (Walks‘𝐺) → (2^nd ‘𝑊) ∈ Word (Vtx‘𝐺))
13	12	adantl 481	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) → (2^nd ‘𝑊) ∈ Word (Vtx‘𝐺))
14		eqid 2737	. . . . . . 7 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
15	14	upgrwlkvtxedg 29730	. . . . . 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 29707	. . . . . . . 8 ⊢ ((1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊) → (♯‘(1^st ‘𝑊)) = ((♯‘(2^nd ‘𝑊)) − 1))
17	16	adantl 481	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ (1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊)) → (♯‘(1^st ‘𝑊)) = ((♯‘(2^nd ‘𝑊)) − 1))
18	17	oveq2d 7384	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ (1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊)) → (0..^(♯‘(1^st ‘𝑊))) = (0..^((♯‘(2^nd ‘𝑊)) − 1)))
19	15, 18	raleqtrdv 3300	. . . . 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 595	. . . 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 1129	. . 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 480	. 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 482	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ (♯‘(1^st ‘𝑊)) = 𝑁) → 𝑁 ∈ ℕ₀)
24		oveq2 7376	. . . . . . . . . . . . 13 ⊢ ((♯‘(1^st ‘𝑊)) = 𝑁 → (0...(♯‘(1^st ‘𝑊))) = (0...𝑁))
25	24	adantl 481	. . . . . . . . . . . 12 ⊢ (((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (♯‘(1^st ‘𝑊)) = 𝑁) → (0...(♯‘(1^st ‘𝑊))) = (0...𝑁))
26	25	feq2d 6654	. . . . . . . . . . 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 229	. . . . . . . . . 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 451	. . . . . . . . 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 490	. . . . . . . 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 406	. . . . . . 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 14382	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ (2^nd ‘𝑊):(0...𝑁)⟶(Vtx‘𝐺)) → (♯‘(2^nd ‘𝑊)) = (𝑁 + 1))
32	23, 30, 31	syl2an2 687	. . . . . 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 412	. . . . 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 17	. . . 4 ⊢ (𝑊 ∈ (Walks‘𝐺) → ((𝑁 ∈ ℕ₀ ∧ (♯‘(1^st ‘𝑊)) = 𝑁) → (♯‘(2^nd ‘𝑊)) = (𝑁 + 1)))
35	34	adantl 481	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) → ((𝑁 ∈ ℕ₀ ∧ (♯‘(1^st ‘𝑊)) = 𝑁) → (♯‘(2^nd ‘𝑊)) = (𝑁 + 1)))
36	35	imp 406	. 2 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) ∧ (𝑁 ∈ ℕ₀ ∧ (♯‘(1^st ‘𝑊)) = 𝑁)) → (♯‘(2^nd ‘𝑊)) = (𝑁 + 1))
37	23	adantl 481	. . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) ∧ (𝑁 ∈ ℕ₀ ∧ (♯‘(1^st ‘𝑊)) = 𝑁)) → 𝑁 ∈ ℕ₀)
38		iswwlksn 29923	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → ((2^nd ‘𝑊) ∈ (𝑁 WWalksN 𝐺) ↔ ((2^nd ‘𝑊) ∈ (WWalks‘𝐺) ∧ (♯‘(2^nd ‘𝑊)) = (𝑁 + 1))))
39	5, 14	iswwlks 29921	. . . . . 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 632	. . . 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 279	. . 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 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))))
44	22, 36, 43	mpbir2and 714	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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∅c0 4287 {cpr 4584 class class class wbr 5100 dom cdm 5632 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 1^st c1st 7941 2^nd c2nd 7942 0cc0 11038 1c1 11039 + caddc 11041 − cmin 11376 ℕ₀cn0 12413 ...cfz 13435 ..^cfzo 13582 ♯chash 14265 Word cword 14448 Vtxcvtx 29081 iEdgciedg 29082 Edgcedg 29132 UPGraphcupgr 29165 Walkscwlks 29682 WWalkscwwlks 29910 WWalksN cwwlksn 29911

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-2o 8408 df-oadd 8411 df-er 8645 df-map 8777 df-pm 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-dju 9825 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-2 12220 df-n0 12414 df-xnn0 12487 df-z 12501 df-uz 12764 df-fz 13436 df-fzo 13583 df-hash 14266 df-word 14449 df-edg 29133 df-uhgr 29143 df-upgr 29167 df-wlks 29685 df-wwlks 29915 df-wwlksn 29916

This theorem is referenced by: (None)

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