grtriclwlk3 - Mathbox for Alexander van der Vekens

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Theorem grtriclwlk3 49137

Description: A triangle induces a closed walk of length 3 . (Contributed by AV, 26-Jul-2025.)

**Hypotheses**
Ref	Expression
grtriclwlk3.t	⊢ (𝜑 → 𝑇 ∈ (GrTriangles‘𝐺))
grtriclwlk3.p	⊢ (𝜑 → 𝑃:(0..^3)–1-1-onto→𝑇)

**Assertion**
Ref	Expression
grtriclwlk3	⊢ (𝜑 → 𝑃 ∈ (3 ClWWalksN 𝐺))

Proof of Theorem grtriclwlk3 Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grtriclwlk3.p	. . . . 5 ⊢ (𝜑 → 𝑃:(0..^3)–1-1-onto→𝑇)
2		f1ofn 6826	. . . . 5 ⊢ (𝑃:(0..^3)–1-1-onto→𝑇 → 𝑃 Fn (0..^3))
3		hashfn 14523	. . . . 5 ⊢ (𝑃 Fn (0..^3) → (♯‘𝑃) = (♯‘(0..^3)))
4	1, 2, 3	3syl 19	. . . 4 ⊢ (𝜑 → (♯‘𝑃) = (♯‘(0..^3)))
5		3nn0 12628	. . . . 5 ⊢ 3 ∈ ℕ₀
6		hashfzo0 14579	. . . . 5 ⊢ (3 ∈ ℕ₀ → (♯‘(0..^3)) = 3)
7	5, 6	mp1i 14	. . . 4 ⊢ (𝜑 → (♯‘(0..^3)) = 3)
8	4, 7	eqtrd 2795	. . 3 ⊢ (𝜑 → (♯‘𝑃) = 3)
9		f1of 6825	. . . . . . . . 9 ⊢ (𝑃:(0..^3)–1-1-onto→𝑇 → 𝑃:(0..^3)⟶𝑇)
10	1, 9	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑃:(0..^3)⟶𝑇)
11		grtriclwlk3.t	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ (GrTriangles‘𝐺))
12		eqid 2760	. . . . . . . . . 10 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
13	12	grtrissvtx 49136	. . . . . . . . 9 ⊢ (𝑇 ∈ (GrTriangles‘𝐺) → 𝑇 ⊆ (Vtx‘𝐺))
14	11, 13	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ⊆ (Vtx‘𝐺))
15	10, 14	jca 521	. . . . . . 7 ⊢ (𝜑 → (𝑃:(0..^3)⟶𝑇 ∧ 𝑇 ⊆ (Vtx‘𝐺)))
16	15	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → (𝑃:(0..^3)⟶𝑇 ∧ 𝑇 ⊆ (Vtx‘𝐺)))
17		fss 6727	. . . . . 6 ⊢ ((𝑃:(0..^3)⟶𝑇 ∧ 𝑇 ⊆ (Vtx‘𝐺)) → 𝑃:(0..^3)⟶(Vtx‘𝐺))
18		iswrdi 14666	. . . . . 6 ⊢ (𝑃:(0..^3)⟶(Vtx‘𝐺) → 𝑃 ∈ Word (Vtx‘𝐺))
19	16, 17, 18	3syl 19	. . . . 5 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → 𝑃 ∈ Word (Vtx‘𝐺))
20		oveq1 7428	. . . . . . . . . . . 12 ⊢ ((♯‘𝑃) = 3 → ((♯‘𝑃) − 1) = (3 − 1))
21		3m1e2 12474	. . . . . . . . . . . 12 ⊢ (3 − 1) = 2
22	20, 21	eqtrdi 2811	. . . . . . . . . . 11 ⊢ ((♯‘𝑃) = 3 → ((♯‘𝑃) − 1) = 2)
23	22	oveq2d 7437	. . . . . . . . . 10 ⊢ ((♯‘𝑃) = 3 → (0..^((♯‘𝑃) − 1)) = (0..^2))
24		fzo0to2pr 13889	. . . . . . . . . 10 ⊢ (0..^2) = {0, 1}
25	23, 24	eqtrdi 2811	. . . . . . . . 9 ⊢ ((♯‘𝑃) = 3 → (0..^((♯‘𝑃) − 1)) = {0, 1})
26	25	eleq2d 2846	. . . . . . . 8 ⊢ ((♯‘𝑃) = 3 → (𝑖 ∈ (0..^((♯‘𝑃) − 1)) ↔ 𝑖 ∈ {0, 1}))
27	26	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → (𝑖 ∈ (0..^((♯‘𝑃) − 1)) ↔ 𝑖 ∈ {0, 1}))
28	11, 1	jca 521	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑇 ∈ (GrTriangles‘𝐺) ∧ 𝑃:(0..^3)–1-1-onto→𝑇))
29		eqid 2760	. . . . . . . . . . . . 13 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
30	12, 29	grtrif1o 49134	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ (GrTriangles‘𝐺) ∧ 𝑃:(0..^3)–1-1-onto→𝑇) → ({(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺) ∧ {(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺) ∧ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺)))
31		simp1 1154	. . . . . . . . . . . 12 ⊢ (({(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺) ∧ {(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺) ∧ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺)) → {(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺))
32	28, 30, 31	3syl 19	. . . . . . . . . . 11 ⊢ (𝜑 → {(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺))
33	32	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → {(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺))
34		fveq2 6886	. . . . . . . . . . . 12 ⊢ (𝑖 = 0 → (𝑃‘𝑖) = (𝑃‘0))
35		fv0p1e1 12468	. . . . . . . . . . . 12 ⊢ (𝑖 = 0 → (𝑃‘(𝑖 + 1)) = (𝑃‘1))
36	34, 35	preq12d 4698	. . . . . . . . . . 11 ⊢ (𝑖 = 0 → {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} = {(𝑃‘0), (𝑃‘1)})
37	36	eleq1d 2845	. . . . . . . . . 10 ⊢ (𝑖 = 0 → ({(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺)))
38	33, 37	imbitrrid 249	. . . . . . . . 9 ⊢ (𝑖 = 0 → ((𝜑 ∧ (♯‘𝑃) = 3) → {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
39		simp3 1156	. . . . . . . . . . . 12 ⊢ (({(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺) ∧ {(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺) ∧ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺)) → {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺))
40	28, 30, 39	3syl 19	. . . . . . . . . . 11 ⊢ (𝜑 → {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺))
41	40	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺))
42		fveq2 6886	. . . . . . . . . . . 12 ⊢ (𝑖 = 1 → (𝑃‘𝑖) = (𝑃‘1))
43		oveq1 7428	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 1 → (𝑖 + 1) = (1 + 1))
44		1p1e2 12470	. . . . . . . . . . . . . 14 ⊢ (1 + 1) = 2
45	43, 44	eqtrdi 2811	. . . . . . . . . . . . 13 ⊢ (𝑖 = 1 → (𝑖 + 1) = 2)
46	45	fveq2d 6890	. . . . . . . . . . . 12 ⊢ (𝑖 = 1 → (𝑃‘(𝑖 + 1)) = (𝑃‘2))
47	42, 46	preq12d 4698	. . . . . . . . . . 11 ⊢ (𝑖 = 1 → {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} = {(𝑃‘1), (𝑃‘2)})
48	47	eleq1d 2845	. . . . . . . . . 10 ⊢ (𝑖 = 1 → ({(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺)))
49	41, 48	imbitrrid 249	. . . . . . . . 9 ⊢ (𝑖 = 1 → ((𝜑 ∧ (♯‘𝑃) = 3) → {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
50	38, 49	jaoi 871	. . . . . . . 8 ⊢ ((𝑖 = 0 ∨ 𝑖 = 1) → ((𝜑 ∧ (♯‘𝑃) = 3) → {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
51		elpri 4604	. . . . . . . 8 ⊢ (𝑖 ∈ {0, 1} → (𝑖 = 0 ∨ 𝑖 = 1))
52	50, 51	syl11 34	. . . . . . 7 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → (𝑖 ∈ {0, 1} → {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
53	27, 52	sylbid 243	. . . . . 6 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → (𝑖 ∈ (0..^((♯‘𝑃) − 1)) → {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
54	53	ralrimiv 3153	. . . . 5 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))
55		ovexd 7456	. . . . . . . . . . . 12 ⊢ (𝑃:(0..^3)–1-1-onto→𝑇 → (0..^3) ∈ V)
56	9, 55	jca 521	. . . . . . . . . . 11 ⊢ (𝑃:(0..^3)–1-1-onto→𝑇 → (𝑃:(0..^3)⟶𝑇 ∧ (0..^3) ∈ V))
57		fex 7233	. . . . . . . . . . 11 ⊢ ((𝑃:(0..^3)⟶𝑇 ∧ (0..^3) ∈ V) → 𝑃 ∈ V)
58	1, 56, 57	3syl 19	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ V)
59	58	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → 𝑃 ∈ V)
60		lsw 14713	. . . . . . . . 9 ⊢ (𝑃 ∈ V → (lastS‘𝑃) = (𝑃‘((♯‘𝑃) − 1)))
61	59, 60	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → (lastS‘𝑃) = (𝑃‘((♯‘𝑃) − 1)))
62	22	fveq2d 6890	. . . . . . . . 9 ⊢ ((♯‘𝑃) = 3 → (𝑃‘((♯‘𝑃) − 1)) = (𝑃‘2))
63	62	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → (𝑃‘((♯‘𝑃) − 1)) = (𝑃‘2))
64	61, 63	eqtrd 2795	. . . . . . 7 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → (lastS‘𝑃) = (𝑃‘2))
65	64	preq1d 4696	. . . . . 6 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → {(lastS‘𝑃), (𝑃‘0)} = {(𝑃‘2), (𝑃‘0)})
66		prcom 4689	. . . . . . . . . . 11 ⊢ {(𝑃‘0), (𝑃‘2)} = {(𝑃‘2), (𝑃‘0)}
67	66	eleq1i 2851	. . . . . . . . . 10 ⊢ ({(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺) ↔ {(𝑃‘2), (𝑃‘0)} ∈ (Edg‘𝐺))
68	67	biimpi 219	. . . . . . . . 9 ⊢ ({(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺) → {(𝑃‘2), (𝑃‘0)} ∈ (Edg‘𝐺))
69	68	3ad2ant2 1152	. . . . . . . 8 ⊢ (({(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺) ∧ {(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺) ∧ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺)) → {(𝑃‘2), (𝑃‘0)} ∈ (Edg‘𝐺))
70	28, 30, 69	3syl 19	. . . . . . 7 ⊢ (𝜑 → {(𝑃‘2), (𝑃‘0)} ∈ (Edg‘𝐺))
71	70	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → {(𝑃‘2), (𝑃‘0)} ∈ (Edg‘𝐺))
72	65, 71	eqeltrd 2860	. . . . 5 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → {(lastS‘𝑃), (𝑃‘0)} ∈ (Edg‘𝐺))
73	19, 54, 72	3jca 1146	. . . 4 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → (𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑃), (𝑃‘0)} ∈ (Edg‘𝐺)))
74		simpr 490	. . . 4 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → (♯‘𝑃) = 3)
75	73, 74	jca 521	. . 3 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → ((𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑃), (𝑃‘0)} ∈ (Edg‘𝐺)) ∧ (♯‘𝑃) = 3))
76	8, 75	mpdan 700	. 2 ⊢ (𝜑 → ((𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑃), (𝑃‘0)} ∈ (Edg‘𝐺)) ∧ (♯‘𝑃) = 3))
77		3nn 12426	. . 3 ⊢ 3 ∈ ℕ
78	12, 29	isclwwlknx 30633	. . 3 ⊢ (3 ∈ ℕ → (𝑃 ∈ (3 ClWWalksN 𝐺) ↔ ((𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑃), (𝑃‘0)} ∈ (Edg‘𝐺)) ∧ (♯‘𝑃) = 3)))
79	77, 78	mp1i 14	. 2 ⊢ (𝜑 → (𝑃 ∈ (3 ClWWalksN 𝐺) ↔ ((𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑃), (𝑃‘0)} ∈ (Edg‘𝐺)) ∧ (♯‘𝑃) = 3)))
80	76, 79	mpbird 260	1 ⊢ (𝜑 → 𝑃 ∈ (3 ClWWalksN 𝐺))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ∀wral 3076 Vcvv 3450 ⊆ wss 3895 {cpr 4586 Fn wfn 6534 ⟶wf 6535 –1-1-onto→wf1o 6538 ‘cfv 6539 (class class class)co 7421 0cc0 11204 1c1 11205 + caddc 11207 − cmin 11545 ℕcn 12339 2c2 12401 3c3 12402 ℕ₀cn0 12610 ..^cfzo 13792 ♯chash 14478 Word cword 14662 lastSclsw 14711 Vtxcvtx 29580 Edgcedg 29631 ClWWalksN cclwwlkn 30621 GrTrianglescgrtri 49129

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752 ax-cnex 11260 ax-resscn 11261 ax-1cn 11262 ax-icn 11263 ax-addcl 11264 ax-addrcl 11265 ax-mulcl 11266 ax-mulrcl 11267 ax-mulcom 11268 ax-addass 11269 ax-mulass 11270 ax-distr 11271 ax-i2m1 11272 ax-1ne0 11273 ax-1rid 11274 ax-rnegex 11275 ax-rrecex 11276 ax-cnre 11277 ax-pre-lttri 11278 ax-pre-lttrn 11279 ax-pre-ltadd 11280 ax-pre-mulgt0 11281

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 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 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-om 7879 df-1st 8002 df-2nd 8003 df-frecs 8303 df-wrecs 8334 df-recs 8383 df-rdg 8422 df-1o 8480 df-2o 8481 df-3o 8482 df-oadd 8484 df-er 8721 df-map 8853 df-en 8978 df-dom 8979 df-sdom 8980 df-fin 8981 df-dju 9986 df-card 10024 df-pnf 11349 df-mnf 11350 df-xr 11351 df-ltxr 11352 df-le 11353 df-sub 11547 df-neg 11548 df-nn 12340 df-2 12409 df-3 12410 df-n0 12611 df-xnn0 12684 df-z 12698 df-uz 12970 df-fz 13644 df-fzo 13793 df-hash 14479 df-word 14663 df-lsw 14712 df-clwwlk 30579 df-clwwlkn 30622 df-grtri 49130

This theorem is used by: (None)

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