grtriclwlk3 - Mathbox for Alexander van der Vekens

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Theorem grtriclwlk3 48565

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 6811	. . . . 5 ⊢ (𝑃:(0..^3)–1-1-onto→𝑇 → 𝑃 Fn (0..^3))
3		hashfn 14402	. . . . 5 ⊢ (𝑃 Fn (0..^3) → (♯‘𝑃) = (♯‘(0..^3)))
4	1, 2, 3	3syl 19	. . . 4 ⊢ (𝜑 → (♯‘𝑃) = (♯‘(0..^3)))
5		3nn0 12513	. . . . 5 ⊢ 3 ∈ ℕ₀
6		hashfzo0 14457	. . . . 5 ⊢ (3 ∈ ℕ₀ → (♯‘(0..^3)) = 3)
7	5, 6	mp1i 14	. . . 4 ⊢ (𝜑 → (♯‘(0..^3)) = 3)
8	4, 7	eqtrd 2800	. . 3 ⊢ (𝜑 → (♯‘𝑃) = 3)
9		f1of 6810	. . . . . . . . 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 2765	. . . . . . . . . 10 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
13	12	grtrissvtx 48564	. . . . . . . . 9 ⊢ (𝑇 ∈ (GrTriangles‘𝐺) → 𝑇 ⊆ (Vtx‘𝐺))
14	11, 13	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ⊆ (Vtx‘𝐺))
15	10, 14	jca 520	. . . . . . 7 ⊢ (𝜑 → (𝑃:(0..^3)⟶𝑇 ∧ 𝑇 ⊆ (Vtx‘𝐺)))
16	15	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → (𝑃:(0..^3)⟶𝑇 ∧ 𝑇 ⊆ (Vtx‘𝐺)))
17		fss 6712	. . . . . 6 ⊢ ((𝑃:(0..^3)⟶𝑇 ∧ 𝑇 ⊆ (Vtx‘𝐺)) → 𝑃:(0..^3)⟶(Vtx‘𝐺))
18		iswrdi 14544	. . . . . 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 7407	. . . . . . . . . . . 12 ⊢ ((♯‘𝑃) = 3 → ((♯‘𝑃) − 1) = (3 − 1))
21		3m1e2 12359	. . . . . . . . . . . 12 ⊢ (3 − 1) = 2
22	20, 21	eqtrdi 2816	. . . . . . . . . . 11 ⊢ ((♯‘𝑃) = 3 → ((♯‘𝑃) − 1) = 2)
23	22	oveq2d 7416	. . . . . . . . . 10 ⊢ ((♯‘𝑃) = 3 → (0..^((♯‘𝑃) − 1)) = (0..^2))
24		fzo0to2pr 13770	. . . . . . . . . 10 ⊢ (0..^2) = {0, 1}
25	23, 24	eqtrdi 2816	. . . . . . . . 9 ⊢ ((♯‘𝑃) = 3 → (0..^((♯‘𝑃) − 1)) = {0, 1})
26	25	eleq2d 2851	. . . . . . . 8 ⊢ ((♯‘𝑃) = 3 → (𝑖 ∈ (0..^((♯‘𝑃) − 1)) ↔ 𝑖 ∈ {0, 1}))
27	26	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → (𝑖 ∈ (0..^((♯‘𝑃) − 1)) ↔ 𝑖 ∈ {0, 1}))
28	11, 1	jca 520	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑇 ∈ (GrTriangles‘𝐺) ∧ 𝑃:(0..^3)–1-1-onto→𝑇))
29		eqid 2765	. . . . . . . . . . . . 13 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
30	12, 29	grtrif1o 48562	. . . . . . . . . . . 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 1152	. . . . . . . . . . . 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 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → {(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺))
34		fveq2 6871	. . . . . . . . . . . 12 ⊢ (𝑖 = 0 → (𝑃‘𝑖) = (𝑃‘0))
35		fv0p1e1 12353	. . . . . . . . . . . 12 ⊢ (𝑖 = 0 → (𝑃‘(𝑖 + 1)) = (𝑃‘1))
36	34, 35	preq12d 4703	. . . . . . . . . . 11 ⊢ (𝑖 = 0 → {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} = {(𝑃‘0), (𝑃‘1)})
37	36	eleq1d 2850	. . . . . . . . . 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 1154	. . . . . . . . . . . 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 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺))
42		fveq2 6871	. . . . . . . . . . . 12 ⊢ (𝑖 = 1 → (𝑃‘𝑖) = (𝑃‘1))
43		oveq1 7407	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 1 → (𝑖 + 1) = (1 + 1))
44		1p1e2 12355	. . . . . . . . . . . . . 14 ⊢ (1 + 1) = 2
45	43, 44	eqtrdi 2816	. . . . . . . . . . . . 13 ⊢ (𝑖 = 1 → (𝑖 + 1) = 2)
46	45	fveq2d 6875	. . . . . . . . . . . 12 ⊢ (𝑖 = 1 → (𝑃‘(𝑖 + 1)) = (𝑃‘2))
47	42, 46	preq12d 4703	. . . . . . . . . . 11 ⊢ (𝑖 = 1 → {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} = {(𝑃‘1), (𝑃‘2)})
48	47	eleq1d 2850	. . . . . . . . . 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 870	. . . . . . . 8 ⊢ ((𝑖 = 0 ∨ 𝑖 = 1) → ((𝜑 ∧ (♯‘𝑃) = 3) → {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
51		elpri 4609	. . . . . . . 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 3156	. . . . 5 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))
55		ovexd 7435	. . . . . . . . . . . 12 ⊢ (𝑃:(0..^3)–1-1-onto→𝑇 → (0..^3) ∈ V)
56	9, 55	jca 520	. . . . . . . . . . 11 ⊢ (𝑃:(0..^3)–1-1-onto→𝑇 → (𝑃:(0..^3)⟶𝑇 ∧ (0..^3) ∈ V))
57		fex 7214	. . . . . . . . . . 11 ⊢ ((𝑃:(0..^3)⟶𝑇 ∧ (0..^3) ∈ V) → 𝑃 ∈ V)
58	1, 56, 57	3syl 19	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ V)
59	58	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → 𝑃 ∈ V)
60		lsw 14591	. . . . . . . . 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 6875	. . . . . . . . 9 ⊢ ((♯‘𝑃) = 3 → (𝑃‘((♯‘𝑃) − 1)) = (𝑃‘2))
63	62	adantl 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → (𝑃‘((♯‘𝑃) − 1)) = (𝑃‘2))
64	61, 63	eqtrd 2800	. . . . . . 7 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → (lastS‘𝑃) = (𝑃‘2))
65	64	preq1d 4701	. . . . . 6 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → {(lastS‘𝑃), (𝑃‘0)} = {(𝑃‘2), (𝑃‘0)})
66		prcom 4694	. . . . . . . . . . 11 ⊢ {(𝑃‘0), (𝑃‘2)} = {(𝑃‘2), (𝑃‘0)}
67	66	eleq1i 2856	. . . . . . . . . 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 1150	. . . . . . . 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 485	. . . . . 6 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → {(𝑃‘2), (𝑃‘0)} ∈ (Edg‘𝐺))
72	65, 71	eqeltrd 2865	. . . . 5 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → {(lastS‘𝑃), (𝑃‘0)} ∈ (Edg‘𝐺))
73	19, 54, 72	3jca 1144	. . . 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 489	. . . 4 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → (♯‘𝑃) = 3)
75	73, 74	jca 520	. . 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 699	. 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 12311	. . 3 ⊢ 3 ∈ ℕ
78	12, 29	isclwwlknx 30296	. . 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

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 ∧ w3a 1101 = wceq 1563 ∈ wcel 2145 ∀wral 3079 Vcvv 3457 ⊆ wss 3907 {cpr 4587 Fn wfn 6520 ⟶wf 6521 –1-1-onto→wf1o 6524 ‘cfv 6525 (class class class)co 7400 0cc0 11088 1c1 11089 + caddc 11091 − cmin 11429 ℕcn 12224 2c2 12286 3c3 12287 ℕ₀cn0 12495 ..^cfzo 13673 ♯chash 14357 Word cword 14540 lastSclsw 14589 Vtxcvtx 29255 Edgcedg 29306 ClWWalksN cclwwlkn 30284 GrTrianglescgrtri 48557

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4869 df-int 4909 df-iun 4954 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-1st 7974 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-2o 8442 df-3o 8443 df-oadd 8445 df-er 8682 df-map 8814 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-dju 9875 df-card 9913 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-nn 12225 df-2 12294 df-3 12295 df-n0 12496 df-xnn0 12569 df-z 12583 df-uz 12854 df-fz 13527 df-fzo 13674 df-hash 14358 df-word 14541 df-lsw 14590 df-clwwlk 30242 df-clwwlkn 30285 df-grtri 48558

This theorem is referenced by: (None)

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