grtriclwlk3 - Mathbox for Alexander van der Vekens

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Theorem grtriclwlk3 47850

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 6850	. . . . 5 ⊢ (𝑃:(0..^3)–1-1-onto→𝑇 → 𝑃 Fn (0..^3))
3		hashfn 14411	. . . . 5 ⊢ (𝑃 Fn (0..^3) → (♯‘𝑃) = (♯‘(0..^3)))
4	1, 2, 3	3syl 18	. . . 4 ⊢ (𝜑 → (♯‘𝑃) = (♯‘(0..^3)))
5		3nn0 12542	. . . . 5 ⊢ 3 ∈ ℕ₀
6		hashfzo0 14466	. . . . 5 ⊢ (3 ∈ ℕ₀ → (♯‘(0..^3)) = 3)
7	5, 6	mp1i 13	. . . 4 ⊢ (𝜑 → (♯‘(0..^3)) = 3)
8	4, 7	eqtrd 2775	. . 3 ⊢ (𝜑 → (♯‘𝑃) = 3)
9		f1of 6849	. . . . . . . . 9 ⊢ (𝑃:(0..^3)–1-1-onto→𝑇 → 𝑃:(0..^3)⟶𝑇)
10	1, 9	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑃:(0..^3)⟶𝑇)
11		grtriclwlk3.t	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ (GrTriangles‘𝐺))
12		eqid 2735	. . . . . . . . . 10 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
13	12	grtrissvtx 47849	. . . . . . . . 9 ⊢ (𝑇 ∈ (GrTriangles‘𝐺) → 𝑇 ⊆ (Vtx‘𝐺))
14	11, 13	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ⊆ (Vtx‘𝐺))
15	10, 14	jca 511	. . . . . . 7 ⊢ (𝜑 → (𝑃:(0..^3)⟶𝑇 ∧ 𝑇 ⊆ (Vtx‘𝐺)))
16	15	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → (𝑃:(0..^3)⟶𝑇 ∧ 𝑇 ⊆ (Vtx‘𝐺)))
17		fss 6753	. . . . . 6 ⊢ ((𝑃:(0..^3)⟶𝑇 ∧ 𝑇 ⊆ (Vtx‘𝐺)) → 𝑃:(0..^3)⟶(Vtx‘𝐺))
18		iswrdi 14553	. . . . . 6 ⊢ (𝑃:(0..^3)⟶(Vtx‘𝐺) → 𝑃 ∈ Word (Vtx‘𝐺))
19	16, 17, 18	3syl 18	. . . . 5 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → 𝑃 ∈ Word (Vtx‘𝐺))
20		oveq1 7438	. . . . . . . . . . . 12 ⊢ ((♯‘𝑃) = 3 → ((♯‘𝑃) − 1) = (3 − 1))
21		3m1e2 12392	. . . . . . . . . . . 12 ⊢ (3 − 1) = 2
22	20, 21	eqtrdi 2791	. . . . . . . . . . 11 ⊢ ((♯‘𝑃) = 3 → ((♯‘𝑃) − 1) = 2)
23	22	oveq2d 7447	. . . . . . . . . 10 ⊢ ((♯‘𝑃) = 3 → (0..^((♯‘𝑃) − 1)) = (0..^2))
24		fzo0to2pr 13786	. . . . . . . . . 10 ⊢ (0..^2) = {0, 1}
25	23, 24	eqtrdi 2791	. . . . . . . . 9 ⊢ ((♯‘𝑃) = 3 → (0..^((♯‘𝑃) − 1)) = {0, 1})
26	25	eleq2d 2825	. . . . . . . 8 ⊢ ((♯‘𝑃) = 3 → (𝑖 ∈ (0..^((♯‘𝑃) − 1)) ↔ 𝑖 ∈ {0, 1}))
27	26	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → (𝑖 ∈ (0..^((♯‘𝑃) − 1)) ↔ 𝑖 ∈ {0, 1}))
28	11, 1	jca 511	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑇 ∈ (GrTriangles‘𝐺) ∧ 𝑃:(0..^3)–1-1-onto→𝑇))
29		eqid 2735	. . . . . . . . . . . . 13 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
30	12, 29	grtrif1o 47847	. . . . . . . . . . . 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 1135	. . . . . . . . . . . 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 18	. . . . . . . . . . 11 ⊢ (𝜑 → {(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺))
33	32	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → {(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺))
34		fveq2 6907	. . . . . . . . . . . 12 ⊢ (𝑖 = 0 → (𝑃‘𝑖) = (𝑃‘0))
35		fv0p1e1 12387	. . . . . . . . . . . 12 ⊢ (𝑖 = 0 → (𝑃‘(𝑖 + 1)) = (𝑃‘1))
36	34, 35	preq12d 4746	. . . . . . . . . . 11 ⊢ (𝑖 = 0 → {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} = {(𝑃‘0), (𝑃‘1)})
37	36	eleq1d 2824	. . . . . . . . . 10 ⊢ (𝑖 = 0 → ({(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑃‘0), (𝑃‘1)} ∈ (Edg‘𝐺)))
38	33, 37	imbitrrid 246	. . . . . . . . 9 ⊢ (𝑖 = 0 → ((𝜑 ∧ (♯‘𝑃) = 3) → {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
39		simp3 1137	. . . . . . . . . . . 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 18	. . . . . . . . . . 11 ⊢ (𝜑 → {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺))
41	40	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺))
42		fveq2 6907	. . . . . . . . . . . 12 ⊢ (𝑖 = 1 → (𝑃‘𝑖) = (𝑃‘1))
43		oveq1 7438	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 1 → (𝑖 + 1) = (1 + 1))
44		1p1e2 12389	. . . . . . . . . . . . . 14 ⊢ (1 + 1) = 2
45	43, 44	eqtrdi 2791	. . . . . . . . . . . . 13 ⊢ (𝑖 = 1 → (𝑖 + 1) = 2)
46	45	fveq2d 6911	. . . . . . . . . . . 12 ⊢ (𝑖 = 1 → (𝑃‘(𝑖 + 1)) = (𝑃‘2))
47	42, 46	preq12d 4746	. . . . . . . . . . 11 ⊢ (𝑖 = 1 → {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} = {(𝑃‘1), (𝑃‘2)})
48	47	eleq1d 2824	. . . . . . . . . 10 ⊢ (𝑖 = 1 → ({(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑃‘1), (𝑃‘2)} ∈ (Edg‘𝐺)))
49	41, 48	imbitrrid 246	. . . . . . . . 9 ⊢ (𝑖 = 1 → ((𝜑 ∧ (♯‘𝑃) = 3) → {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
50	38, 49	jaoi 857	. . . . . . . 8 ⊢ ((𝑖 = 0 ∨ 𝑖 = 1) → ((𝜑 ∧ (♯‘𝑃) = 3) → {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
51		elpri 4654	. . . . . . . 8 ⊢ (𝑖 ∈ {0, 1} → (𝑖 = 0 ∨ 𝑖 = 1))
52	50, 51	syl11 33	. . . . . . 7 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → (𝑖 ∈ {0, 1} → {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
53	27, 52	sylbid 240	. . . . . 6 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → (𝑖 ∈ (0..^((♯‘𝑃) − 1)) → {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
54	53	ralrimiv 3143	. . . . 5 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))
55		ovexd 7466	. . . . . . . . . . . 12 ⊢ (𝑃:(0..^3)–1-1-onto→𝑇 → (0..^3) ∈ V)
56	9, 55	jca 511	. . . . . . . . . . 11 ⊢ (𝑃:(0..^3)–1-1-onto→𝑇 → (𝑃:(0..^3)⟶𝑇 ∧ (0..^3) ∈ V))
57		fex 7246	. . . . . . . . . . 11 ⊢ ((𝑃:(0..^3)⟶𝑇 ∧ (0..^3) ∈ V) → 𝑃 ∈ V)
58	1, 56, 57	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ V)
59	58	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → 𝑃 ∈ V)
60		lsw 14599	. . . . . . . . 9 ⊢ (𝑃 ∈ V → (lastS‘𝑃) = (𝑃‘((♯‘𝑃) − 1)))
61	59, 60	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → (lastS‘𝑃) = (𝑃‘((♯‘𝑃) − 1)))
62	22	fveq2d 6911	. . . . . . . . 9 ⊢ ((♯‘𝑃) = 3 → (𝑃‘((♯‘𝑃) − 1)) = (𝑃‘2))
63	62	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → (𝑃‘((♯‘𝑃) − 1)) = (𝑃‘2))
64	61, 63	eqtrd 2775	. . . . . . 7 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → (lastS‘𝑃) = (𝑃‘2))
65	64	preq1d 4744	. . . . . 6 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → {(lastS‘𝑃), (𝑃‘0)} = {(𝑃‘2), (𝑃‘0)})
66		prcom 4737	. . . . . . . . . . 11 ⊢ {(𝑃‘0), (𝑃‘2)} = {(𝑃‘2), (𝑃‘0)}
67	66	eleq1i 2830	. . . . . . . . . 10 ⊢ ({(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺) ↔ {(𝑃‘2), (𝑃‘0)} ∈ (Edg‘𝐺))
68	67	biimpi 216	. . . . . . . . 9 ⊢ ({(𝑃‘0), (𝑃‘2)} ∈ (Edg‘𝐺) → {(𝑃‘2), (𝑃‘0)} ∈ (Edg‘𝐺))
69	68	3ad2ant2 1133	. . . . . . . 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 18	. . . . . . 7 ⊢ (𝜑 → {(𝑃‘2), (𝑃‘0)} ∈ (Edg‘𝐺))
71	70	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → {(𝑃‘2), (𝑃‘0)} ∈ (Edg‘𝐺))
72	65, 71	eqeltrd 2839	. . . . 5 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → {(lastS‘𝑃), (𝑃‘0)} ∈ (Edg‘𝐺))
73	19, 54, 72	3jca 1127	. . . 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 484	. . . 4 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → (♯‘𝑃) = 3)
75	73, 74	jca 511	. . 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 687	. 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 12343	. . 3 ⊢ 3 ∈ ℕ
78	12, 29	isclwwlknx 30065	. . 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 13	. 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 257	1 ⊢ (𝜑 → 𝑃 ∈ (3 ClWWalksN 𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 ⊆ wss 3963 {cpr 4633 Fn wfn 6558 ⟶wf 6559 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 0cc0 11153 1c1 11154 + caddc 11156 − cmin 11490 ℕcn 12264 2c2 12319 3c3 12320 ℕ₀cn0 12524 ..^cfzo 13691 ♯chash 14366 Word cword 14549 lastSclsw 14597 Vtxcvtx 29028 Edgcedg 29079 ClWWalksN cclwwlkn 30053 GrTrianglescgrtri 47842

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-3o 8507 df-oadd 8509 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-hash 14367 df-word 14550 df-lsw 14598 df-clwwlk 30011 df-clwwlkn 30054 df-grtri 47843

This theorem is referenced by: (None)

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