grtriclwlk3 - Mathbox for Alexander van der Vekens

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Theorem grtriclwlk3 47944

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 6801	. . . . 5 ⊢ (𝑃:(0..^3)–1-1-onto→𝑇 → 𝑃 Fn (0..^3))
3		hashfn 14340	. . . . 5 ⊢ (𝑃 Fn (0..^3) → (♯‘𝑃) = (♯‘(0..^3)))
4	1, 2, 3	3syl 18	. . . 4 ⊢ (𝜑 → (♯‘𝑃) = (♯‘(0..^3)))
5		3nn0 12460	. . . . 5 ⊢ 3 ∈ ℕ₀
6		hashfzo0 14395	. . . . 5 ⊢ (3 ∈ ℕ₀ → (♯‘(0..^3)) = 3)
7	5, 6	mp1i 13	. . . 4 ⊢ (𝜑 → (♯‘(0..^3)) = 3)
8	4, 7	eqtrd 2764	. . 3 ⊢ (𝜑 → (♯‘𝑃) = 3)
9		f1of 6800	. . . . . . . . 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 2729	. . . . . . . . . 10 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
13	12	grtrissvtx 47943	. . . . . . . . 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 6704	. . . . . 6 ⊢ ((𝑃:(0..^3)⟶𝑇 ∧ 𝑇 ⊆ (Vtx‘𝐺)) → 𝑃:(0..^3)⟶(Vtx‘𝐺))
18		iswrdi 14482	. . . . . 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 7394	. . . . . . . . . . . 12 ⊢ ((♯‘𝑃) = 3 → ((♯‘𝑃) − 1) = (3 − 1))
21		3m1e2 12309	. . . . . . . . . . . 12 ⊢ (3 − 1) = 2
22	20, 21	eqtrdi 2780	. . . . . . . . . . 11 ⊢ ((♯‘𝑃) = 3 → ((♯‘𝑃) − 1) = 2)
23	22	oveq2d 7403	. . . . . . . . . 10 ⊢ ((♯‘𝑃) = 3 → (0..^((♯‘𝑃) − 1)) = (0..^2))
24		fzo0to2pr 13711	. . . . . . . . . 10 ⊢ (0..^2) = {0, 1}
25	23, 24	eqtrdi 2780	. . . . . . . . 9 ⊢ ((♯‘𝑃) = 3 → (0..^((♯‘𝑃) − 1)) = {0, 1})
26	25	eleq2d 2814	. . . . . . . 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 2729	. . . . . . . . . . . . 13 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
30	12, 29	grtrif1o 47941	. . . . . . . . . . . 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 1136	. . . . . . . . . . . 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 6858	. . . . . . . . . . . 12 ⊢ (𝑖 = 0 → (𝑃‘𝑖) = (𝑃‘0))
35		fv0p1e1 12304	. . . . . . . . . . . 12 ⊢ (𝑖 = 0 → (𝑃‘(𝑖 + 1)) = (𝑃‘1))
36	34, 35	preq12d 4705	. . . . . . . . . . 11 ⊢ (𝑖 = 0 → {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} = {(𝑃‘0), (𝑃‘1)})
37	36	eleq1d 2813	. . . . . . . . . 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 1138	. . . . . . . . . . . 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 6858	. . . . . . . . . . . 12 ⊢ (𝑖 = 1 → (𝑃‘𝑖) = (𝑃‘1))
43		oveq1 7394	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 1 → (𝑖 + 1) = (1 + 1))
44		1p1e2 12306	. . . . . . . . . . . . . 14 ⊢ (1 + 1) = 2
45	43, 44	eqtrdi 2780	. . . . . . . . . . . . 13 ⊢ (𝑖 = 1 → (𝑖 + 1) = 2)
46	45	fveq2d 6862	. . . . . . . . . . . 12 ⊢ (𝑖 = 1 → (𝑃‘(𝑖 + 1)) = (𝑃‘2))
47	42, 46	preq12d 4705	. . . . . . . . . . 11 ⊢ (𝑖 = 1 → {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} = {(𝑃‘1), (𝑃‘2)})
48	47	eleq1d 2813	. . . . . . . . . 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 4613	. . . . . . . 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 3124	. . . . 5 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))
55		ovexd 7422	. . . . . . . . . . . 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 7200	. . . . . . . . . . 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 14529	. . . . . . . . 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 6862	. . . . . . . . 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 2764	. . . . . . 7 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → (lastS‘𝑃) = (𝑃‘2))
65	64	preq1d 4703	. . . . . 6 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → {(lastS‘𝑃), (𝑃‘0)} = {(𝑃‘2), (𝑃‘0)})
66		prcom 4696	. . . . . . . . . . 11 ⊢ {(𝑃‘0), (𝑃‘2)} = {(𝑃‘2), (𝑃‘0)}
67	66	eleq1i 2819	. . . . . . . . . 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 1134	. . . . . . . 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 2828	. . . . 5 ⊢ ((𝜑 ∧ (♯‘𝑃) = 3) → {(lastS‘𝑃), (𝑃‘0)} ∈ (Edg‘𝐺))
73	19, 54, 72	3jca 1128	. . . 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 12265	. . 3 ⊢ 3 ∈ ℕ
78	12, 29	isclwwlknx 29965	. . 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 1540 ∈ wcel 2109 ∀wral 3044 Vcvv 3447 ⊆ wss 3914 {cpr 4591 Fn wfn 6506 ⟶wf 6507 –1-1-onto→wf1o 6510 ‘cfv 6511 (class class class)co 7387 0cc0 11068 1c1 11069 + caddc 11071 − cmin 11405 ℕcn 12186 2c2 12241 3c3 12242 ℕ₀cn0 12442 ..^cfzo 13615 ♯chash 14295 Word cword 14478 lastSclsw 14527 Vtxcvtx 28923 Edgcedg 28974 ClWWalksN cclwwlkn 29953 GrTrianglescgrtri 47936

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-3o 8436 df-oadd 8438 df-er 8671 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-dju 9854 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-n0 12443 df-xnn0 12516 df-z 12530 df-uz 12794 df-fz 13469 df-fzo 13616 df-hash 14296 df-word 14479 df-lsw 14528 df-clwwlk 29911 df-clwwlkn 29954 df-grtri 47937

This theorem is referenced by: (None)

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