clwwlknonclwlknonf1o - Metamath Proof Explorer

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Theorem clwwlknonclwlknonf1o 28140

Description: 𝐹 is a bijection between the two representations of closed walks of a fixed positive length on a fixed vertex. (Contributed by AV, 26-May-2022.) (Proof shortened by AV, 7-Aug-2022.) (Revised by AV, 1-Nov-2022.)

**Hypotheses**
Ref	Expression
clwwlknonclwlknonf1o.v	⊢ 𝑉 = (Vtx‘𝐺)
clwwlknonclwlknonf1o.w	⊢ 𝑊 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)}
clwwlknonclwlknonf1o.f	⊢ 𝐹 = (𝑐 ∈ 𝑊 ↦ ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐))))

**Assertion**
Ref	Expression
clwwlknonclwlknonf1o	⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝐹:𝑊–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁))

Distinct variable groups: 𝐺,𝑐,𝑤 𝑁,𝑐,𝑤 𝑉,𝑐 𝑊,𝑐 𝑋,𝑐,𝑤 Allowed substitution hints: 𝐹(𝑤,𝑐) 𝑉(𝑤) 𝑊(𝑤)

Proof of Theorem clwwlknonclwlknonf1o Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clwwlknonclwlknonf1o.w	. . 3 ⊢ 𝑊 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)}
2		eqid 2821	. . 3 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} = {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁}
3		clwwlknonclwlknonf1o.f	. . 3 ⊢ 𝐹 = (𝑐 ∈ 𝑊 ↦ ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐))))
4		eqid 2821	. . 3 ⊢ (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} ↦ ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))) = (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} ↦ ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐))))
5		eqid 2821	. . . . 5 ⊢ (1^st ‘𝑐) = (1^st ‘𝑐)
6		eqid 2821	. . . . 5 ⊢ (2^nd ‘𝑐) = (2^nd ‘𝑐)
7	5, 6, 2, 4	clwlknf1oclwwlkn 27862	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} ↦ ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))):{𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁}–1-1-onto→(𝑁 ClWWalksN 𝐺))
8	7	3adant2 1127	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} ↦ ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))):{𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁}–1-1-onto→(𝑁 ClWWalksN 𝐺))
9		fveq1 6668	. . . . . . 7 ⊢ (𝑠 = ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐))) → (𝑠‘0) = (((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))‘0))
10	9	3ad2ant3 1131	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} ∧ 𝑠 = ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))) → (𝑠‘0) = (((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))‘0))
11		2fveq3 6674	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑐 → (♯‘(1^st ‘𝑤)) = (♯‘(1^st ‘𝑐)))
12	11	eqeq1d 2823	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑐 → ((♯‘(1^st ‘𝑤)) = 𝑁 ↔ (♯‘(1^st ‘𝑐)) = 𝑁))
13	12	elrab 3679	. . . . . . . . . 10 ⊢ (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} ↔ (𝑐 ∈ (ClWalks‘𝐺) ∧ (♯‘(1^st ‘𝑐)) = 𝑁))
14		clwlkwlk 27555	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → 𝑐 ∈ (Walks‘𝐺))
15		wlkcpr 27409	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (Walks‘𝐺) ↔ (1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐))
16		eqid 2821	. . . . . . . . . . . . . . . . 17 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
17	16	wlkpwrd 27398	. . . . . . . . . . . . . . . 16 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (2^nd ‘𝑐) ∈ Word (Vtx‘𝐺))
18	17	3ad2ant1 1129	. . . . . . . . . . . . . . 15 ⊢ (((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) ∧ (♯‘(1^st ‘𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (2^nd ‘𝑐) ∈ Word (Vtx‘𝐺))
19		elnnuz 12281	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ_≥‘1))
20		eluzfz2 12914	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ (ℤ_≥‘1) → 𝑁 ∈ (1...𝑁))
21	19, 20	sylbi 219	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (1...𝑁))
22		fzelp1 12958	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ (1...𝑁) → 𝑁 ∈ (1...(𝑁 + 1)))
23	21, 22	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (1...(𝑁 + 1)))
24	23	3ad2ant3 1131	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ (1...(𝑁 + 1)))
25	24	3ad2ant3 1131	. . . . . . . . . . . . . . . . 17 ⊢ (((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) ∧ (♯‘(1^st ‘𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → 𝑁 ∈ (1...(𝑁 + 1)))
26		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘(1^st ‘𝑐)) = 𝑁 → (♯‘(1^st ‘𝑐)) = 𝑁)
27		oveq1 7162	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘(1^st ‘𝑐)) = 𝑁 → ((♯‘(1^st ‘𝑐)) + 1) = (𝑁 + 1))
28	27	oveq2d 7171	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘(1^st ‘𝑐)) = 𝑁 → (1...((♯‘(1^st ‘𝑐)) + 1)) = (1...(𝑁 + 1)))
29	26, 28	eleq12d 2907	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘(1^st ‘𝑐)) = 𝑁 → ((♯‘(1^st ‘𝑐)) ∈ (1...((♯‘(1^st ‘𝑐)) + 1)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
30	29	3ad2ant2 1130	. . . . . . . . . . . . . . . . 17 ⊢ (((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) ∧ (♯‘(1^st ‘𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → ((♯‘(1^st ‘𝑐)) ∈ (1...((♯‘(1^st ‘𝑐)) + 1)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
31	25, 30	mpbird 259	. . . . . . . . . . . . . . . 16 ⊢ (((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) ∧ (♯‘(1^st ‘𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (♯‘(1^st ‘𝑐)) ∈ (1...((♯‘(1^st ‘𝑐)) + 1)))
32		wlklenvp1 27399	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (♯‘(2^nd ‘𝑐)) = ((♯‘(1^st ‘𝑐)) + 1))
33	32	oveq2d 7171	. . . . . . . . . . . . . . . . . 18 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (1...(♯‘(2^nd ‘𝑐))) = (1...((♯‘(1^st ‘𝑐)) + 1)))
34	33	eleq2d 2898	. . . . . . . . . . . . . . . . 17 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → ((♯‘(1^st ‘𝑐)) ∈ (1...(♯‘(2^nd ‘𝑐))) ↔ (♯‘(1^st ‘𝑐)) ∈ (1...((♯‘(1^st ‘𝑐)) + 1))))
35	34	3ad2ant1 1129	. . . . . . . . . . . . . . . 16 ⊢ (((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) ∧ (♯‘(1^st ‘𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → ((♯‘(1^st ‘𝑐)) ∈ (1...(♯‘(2^nd ‘𝑐))) ↔ (♯‘(1^st ‘𝑐)) ∈ (1...((♯‘(1^st ‘𝑐)) + 1))))
36	31, 35	mpbird 259	. . . . . . . . . . . . . . 15 ⊢ (((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) ∧ (♯‘(1^st ‘𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (♯‘(1^st ‘𝑐)) ∈ (1...(♯‘(2^nd ‘𝑐))))
37	18, 36	jca 514	. . . . . . . . . . . . . 14 ⊢ (((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) ∧ (♯‘(1^st ‘𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → ((2^nd ‘𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1^st ‘𝑐)) ∈ (1...(♯‘(2^nd ‘𝑐)))))
38	37	3exp 1115	. . . . . . . . . . . . 13 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → ((♯‘(1^st ‘𝑐)) = 𝑁 → ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((2^nd ‘𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1^st ‘𝑐)) ∈ (1...(♯‘(2^nd ‘𝑐)))))))
39	15, 38	sylbi 219	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ (Walks‘𝐺) → ((♯‘(1^st ‘𝑐)) = 𝑁 → ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((2^nd ‘𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1^st ‘𝑐)) ∈ (1...(♯‘(2^nd ‘𝑐)))))))
40	14, 39	syl 17	. . . . . . . . . . 11 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → ((♯‘(1^st ‘𝑐)) = 𝑁 → ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((2^nd ‘𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1^st ‘𝑐)) ∈ (1...(♯‘(2^nd ‘𝑐)))))))
41	40	imp 409	. . . . . . . . . 10 ⊢ ((𝑐 ∈ (ClWalks‘𝐺) ∧ (♯‘(1^st ‘𝑐)) = 𝑁) → ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((2^nd ‘𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1^st ‘𝑐)) ∈ (1...(♯‘(2^nd ‘𝑐))))))
42	13, 41	sylbi 219	. . . . . . . . 9 ⊢ (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} → ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((2^nd ‘𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1^st ‘𝑐)) ∈ (1...(♯‘(2^nd ‘𝑐))))))
43	42	impcom 410	. . . . . . . 8 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁}) → ((2^nd ‘𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1^st ‘𝑐)) ∈ (1...(♯‘(2^nd ‘𝑐)))))
44		pfxfv0 14053	. . . . . . . 8 ⊢ (((2^nd ‘𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1^st ‘𝑐)) ∈ (1...(♯‘(2^nd ‘𝑐)))) → (((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))‘0) = ((2^nd ‘𝑐)‘0))
45	43, 44	syl 17	. . . . . . 7 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁}) → (((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))‘0) = ((2^nd ‘𝑐)‘0))
46	45	3adant3 1128	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} ∧ 𝑠 = ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))) → (((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))‘0) = ((2^nd ‘𝑐)‘0))
47	10, 46	eqtrd 2856	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} ∧ 𝑠 = ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))) → (𝑠‘0) = ((2^nd ‘𝑐)‘0))
48	47	eqeq1d 2823	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} ∧ 𝑠 = ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))) → ((𝑠‘0) = 𝑋 ↔ ((2^nd ‘𝑐)‘0) = 𝑋))
49		nfv 1911	. . . . 5 ⊢ Ⅎ𝑤((2^nd ‘𝑐)‘0) = 𝑋
50		fveq2 6669	. . . . . . 7 ⊢ (𝑤 = 𝑐 → (2^nd ‘𝑤) = (2^nd ‘𝑐))
51	50	fveq1d 6671	. . . . . 6 ⊢ (𝑤 = 𝑐 → ((2^nd ‘𝑤)‘0) = ((2^nd ‘𝑐)‘0))
52	51	eqeq1d 2823	. . . . 5 ⊢ (𝑤 = 𝑐 → (((2^nd ‘𝑤)‘0) = 𝑋 ↔ ((2^nd ‘𝑐)‘0) = 𝑋))
53	49, 52	sbiev 2326	. . . 4 ⊢ ([𝑐 / 𝑤]((2^nd ‘𝑤)‘0) = 𝑋 ↔ ((2^nd ‘𝑐)‘0) = 𝑋)
54	48, 53	syl6bbr 291	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} ∧ 𝑠 = ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))) → ((𝑠‘0) = 𝑋 ↔ [𝑐 / 𝑤]((2^nd ‘𝑤)‘0) = 𝑋))
55	1, 2, 3, 4, 8, 54	f1ossf1o 6889	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝐹:𝑊–1-1-onto→{𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋})
56		clwwlknon 27868	. . 3 ⊢ (𝑋(ClWWalksNOn‘𝐺)𝑁) = {𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋}
57		f1oeq3 6605	. . 3 ⊢ ((𝑋(ClWWalksNOn‘𝐺)𝑁) = {𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋} → (𝐹:𝑊–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ 𝐹:𝑊–1-1-onto→{𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋}))
58	56, 57	ax-mp 5	. 2 ⊢ (𝐹:𝑊–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ 𝐹:𝑊–1-1-onto→{𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋})
59	55, 58	sylibr 236	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝐹:𝑊–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 [wsb 2065 ∈ wcel 2110 {crab 3142 class class class wbr 5065 ↦ cmpt 5145 –1-1-onto→wf1o 6353 ‘cfv 6354 (class class class)co 7155 1^st c1st 7686 2^nd c2nd 7687 0cc0 10536 1c1 10537 + caddc 10539 ℕcn 11637 ℤ_≥cuz 12242 ...cfz 12891 ♯chash 13689 Word cword 13860 prefix cpfx 14031 Vtxcvtx 26780 USPGraphcuspgr 26932 Walkscwlks 27377 ClWalkscclwlks 27550 ClWWalksN cclwwlkn 27801 ClWWalksNOncclwwlknon 27865

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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-pm 8408 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-dju 9329 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-n0 11897 df-xnn0 11967 df-z 11981 df-uz 12243 df-rp 12389 df-fz 12892 df-fzo 13033 df-hash 13690 df-word 13861 df-lsw 13914 df-concat 13922 df-s1 13949 df-substr 14002 df-pfx 14032 df-edg 26832 df-uhgr 26842 df-upgr 26866 df-uspgr 26934 df-wlks 27380 df-clwlks 27551 df-clwwlk 27759 df-clwwlkn 27802 df-clwwlknon 27866

This theorem is referenced by: clwwlknonclwlknonen 28141 dlwwlknondlwlknonf1o 28143

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