grtriclwlk3 - Mathbox for Alexander van der Vekens

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Theorem grtriclwlk3 48227

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 6776	. . . . 5 ⊢ (𝑃:(0..^3)–1-1-onto→𝑇 → 𝑃 Fn (0..^3))
3		hashfn 14302	. . . . 5 ⊢ (𝑃 Fn (0..^3) → (♯‘𝑃) = (♯‘(0..^3)))
4	1, 2, 3	3syl 18	. . . 4 ⊢ (𝜑 → (♯‘𝑃) = (♯‘(0..^3)))
5		3nn0 12423	. . . . 5 ⊢ 3 ∈ ℕ₀
6		hashfzo0 14357	. . . . 5 ⊢ (3 ∈ ℕ₀ → (♯‘(0..^3)) = 3)
7	5, 6	mp1i 13	. . . 4 ⊢ (𝜑 → (♯‘(0..^3)) = 3)
8	4, 7	eqtrd 2772	. . 3 ⊢ (𝜑 → (♯‘𝑃) = 3)
9		f1of 6775	. . . . . . . . 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 2737	. . . . . . . . . 10 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
13	12	grtrissvtx 48226	. . . . . . . . 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 6679	. . . . . 6 ⊢ ((𝑃:(0..^3)⟶𝑇 ∧ 𝑇 ⊆ (Vtx‘𝐺)) → 𝑃:(0..^3)⟶(Vtx‘𝐺))
18		iswrdi 14444	. . . . . 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 7367	. . . . . . . . . . . 12 ⊢ ((♯‘𝑃) = 3 → ((♯‘𝑃) − 1) = (3 − 1))
21		3m1e2 12272	. . . . . . . . . . . 12 ⊢ (3 − 1) = 2
22	20, 21	eqtrdi 2788	. . . . . . . . . . 11 ⊢ ((♯‘𝑃) = 3 → ((♯‘𝑃) − 1) = 2)
23	22	oveq2d 7376	. . . . . . . . . 10 ⊢ ((♯‘𝑃) = 3 → (0..^((♯‘𝑃) − 1)) = (0..^2))
24		fzo0to2pr 13670	. . . . . . . . . 10 ⊢ (0..^2) = {0, 1}
25	23, 24	eqtrdi 2788	. . . . . . . . 9 ⊢ ((♯‘𝑃) = 3 → (0..^((♯‘𝑃) − 1)) = {0, 1})
26	25	eleq2d 2823	. . . . . . . 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 2737	. . . . . . . . . . . . 13 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
30	12, 29	grtrif1o 48224	. . . . . . . . . . . 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 1137	. . . . . . . . . . . 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 6835	. . . . . . . . . . . 12 ⊢ (𝑖 = 0 → (𝑃‘𝑖) = (𝑃‘0))
35		fv0p1e1 12267	. . . . . . . . . . . 12 ⊢ (𝑖 = 0 → (𝑃‘(𝑖 + 1)) = (𝑃‘1))
36	34, 35	preq12d 4699	. . . . . . . . . . 11 ⊢ (𝑖 = 0 → {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} = {(𝑃‘0), (𝑃‘1)})
37	36	eleq1d 2822	. . . . . . . . . 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 1139	. . . . . . . . . . . 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 6835	. . . . . . . . . . . 12 ⊢ (𝑖 = 1 → (𝑃‘𝑖) = (𝑃‘1))
43		oveq1 7367	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 1 → (𝑖 + 1) = (1 + 1))
44		1p1e2 12269	. . . . . . . . . . . . . 14 ⊢ (1 + 1) = 2
45	43, 44	eqtrdi 2788	. . . . . . . . . . . . 13 ⊢ (𝑖 = 1 → (𝑖 + 1) = 2)
46	45	fveq2d 6839	. . . . . . . . . . . 12 ⊢ (𝑖 = 1 → (𝑃‘(𝑖 + 1)) = (𝑃‘2))
47	42, 46	preq12d 4699	. . . . . . . . . . 11 ⊢ (𝑖 = 1 → {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} = {(𝑃‘1), (𝑃‘2)})
48	47	eleq1d 2822	. . . . . . . . . 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 858	. . . . . . . 8 ⊢ ((𝑖 = 0 ∨ 𝑖 = 1) → ((𝜑 ∧ (♯‘𝑃) = 3) → {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
51		elpri 4605	. . . . . . . 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 3128	. . . . 5 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))
55		ovexd 7395	. . . . . . . . . . . 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 7174	. . . . . . . . . . 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 14491	. . . . . . . . 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 6839	. . . . . . . . 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 2772	. . . . . . 7 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → (lastS‘𝑃) = (𝑃‘2))
65	64	preq1d 4697	. . . . . 6 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → {(lastS‘𝑃), (𝑃‘0)} = {(𝑃‘2), (𝑃‘0)})
66		prcom 4690	. . . . . . . . . . 11 ⊢ {(𝑃‘0), (𝑃‘2)} = {(𝑃‘2), (𝑃‘0)}
67	66	eleq1i 2828	. . . . . . . . . 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 1135	. . . . . . . 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 2837	. . . . 5 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → {(lastS‘𝑃), (𝑃‘0)} ∈ (Edg‘𝐺))
73	19, 54, 72	3jca 1129	. . . 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 688	. 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 12228	. . 3 ⊢ 3 ∈ ℕ
78	12, 29	isclwwlknx 30094	. . 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 848 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 Vcvv 3441 ⊆ wss 3902 {cpr 4583 Fn wfn 6488 ⟶wf 6489 –1-1-onto→wf1o 6492 ‘cfv 6493 (class class class)co 7360 0cc0 11030 1c1 11031 + caddc 11033 − cmin 11368 ℕcn 12149 2c2 12204 3c3 12205 ℕ₀cn0 12405 ..^cfzo 13574 ♯chash 14257 Word cword 14440 lastSclsw 14489 Vtxcvtx 29052 Edgcedg 29103 ClWWalksN cclwwlkn 30082 GrTrianglescgrtri 48219

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 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 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 3062 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-3o 8401 df-oadd 8403 df-er 8637 df-map 8769 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-dju 9817 df-card 9855 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12150 df-2 12212 df-3 12213 df-n0 12406 df-xnn0 12479 df-z 12493 df-uz 12756 df-fz 13428 df-fzo 13575 df-hash 14258 df-word 14441 df-lsw 14490 df-clwwlk 30040 df-clwwlkn 30083 df-grtri 48220

This theorem is referenced by: (None)

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