clwwlknonclwlknonf1o - Metamath Proof Explorer

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Theorem clwwlknonclwlknonf1o 30228

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 2725	. . 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 2725	. . 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 2725	. . . . 5 ⊢ (1^st ‘𝑐) = (1^st ‘𝑐)
6		eqid 2725	. . . . 5 ⊢ (2^nd ‘𝑐) = (2^nd ‘𝑐)
7	5, 6, 2, 4	clwlknf1oclwwlkn 29950	. . . 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 1128	. . 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 6893	. . . . . . 7 ⊢ (𝑠 = ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐))) → (𝑠‘0) = (((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))‘0))
10	9	3ad2ant3 1132	. . . . . 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 6899	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑐 → (♯‘(1^st ‘𝑤)) = (♯‘(1^st ‘𝑐)))
12	11	eqeq1d 2727	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑐 → ((♯‘(1^st ‘𝑤)) = 𝑁 ↔ (♯‘(1^st ‘𝑐)) = 𝑁))
13	12	elrab 3680	. . . . . . . . . 10 ⊢ (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} ↔ (𝑐 ∈ (ClWalks‘𝐺) ∧ (♯‘(1^st ‘𝑐)) = 𝑁))
14		clwlkwlk 29645	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → 𝑐 ∈ (Walks‘𝐺))
15		wlkcpr 29499	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (Walks‘𝐺) ↔ (1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐))
16		eqid 2725	. . . . . . . . . . . . . . . . 17 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
17	16	wlkpwrd 29487	. . . . . . . . . . . . . . . 16 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (2^nd ‘𝑐) ∈ Word (Vtx‘𝐺))
18	17	3ad2ant1 1130	. . . . . . . . . . . . . . 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 12896	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ_≥‘1))
20		eluzfz2 13541	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ (ℤ_≥‘1) → 𝑁 ∈ (1...𝑁))
21	19, 20	sylbi 216	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (1...𝑁))
22		fzelp1 13585	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ (1...𝑁) → 𝑁 ∈ (1...(𝑁 + 1)))
23	21, 22	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (1...(𝑁 + 1)))
24	23	3ad2ant3 1132	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ (1...(𝑁 + 1)))
25	24	3ad2ant3 1132	. . . . . . . . . . . . . . . . 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 7424	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘(1^st ‘𝑐)) = 𝑁 → ((♯‘(1^st ‘𝑐)) + 1) = (𝑁 + 1))
28	27	oveq2d 7433	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘(1^st ‘𝑐)) = 𝑁 → (1...((♯‘(1^st ‘𝑐)) + 1)) = (1...(𝑁 + 1)))
29	26, 28	eleq12d 2819	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘(1^st ‘𝑐)) = 𝑁 → ((♯‘(1^st ‘𝑐)) ∈ (1...((♯‘(1^st ‘𝑐)) + 1)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
30	29	3ad2ant2 1131	. . . . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . . . . 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 29488	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (♯‘(2^nd ‘𝑐)) = ((♯‘(1^st ‘𝑐)) + 1))
33	32	oveq2d 7433	. . . . . . . . . . . . . . . . . 18 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (1...(♯‘(2^nd ‘𝑐))) = (1...((♯‘(1^st ‘𝑐)) + 1)))
34	33	eleq2d 2811	. . . . . . . . . . . . . . . . 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 1130	. . . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . . . 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 510	. . . . . . . . . . . . . 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 1116	. . . . . . . . . . . . 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 216	. . . . . . . . . . . 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 405	. . . . . . . . . 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 216	. . . . . . . . 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 406	. . . . . . . 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 14674	. . . . . . . 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 1129	. . . . . 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 2765	. . . . 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 2727	. . . 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 1909	. . . . 5 ⊢ Ⅎ𝑤((2^nd ‘𝑐)‘0) = 𝑋
50		fveq2 6894	. . . . . . 7 ⊢ (𝑤 = 𝑐 → (2^nd ‘𝑤) = (2^nd ‘𝑐))
51	50	fveq1d 6896	. . . . . 6 ⊢ (𝑤 = 𝑐 → ((2^nd ‘𝑤)‘0) = ((2^nd ‘𝑐)‘0))
52	51	eqeq1d 2727	. . . . 5 ⊢ (𝑤 = 𝑐 → (((2^nd ‘𝑤)‘0) = 𝑋 ↔ ((2^nd ‘𝑐)‘0) = 𝑋))
53	49, 52	sbiev 2303	. . . 4 ⊢ ([𝑐 / 𝑤]((2^nd ‘𝑤)‘0) = 𝑋 ↔ ((2^nd ‘𝑐)‘0) = 𝑋)
54	48, 53	bitr4di 288	. . 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 7135	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝐹:𝑊–1-1-onto→{𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋})
56		clwwlknon 29956	. . 3 ⊢ (𝑋(ClWWalksNOn‘𝐺)𝑁) = {𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋}
57		f1oeq3 6826	. . 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 233	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 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 [wsb 2059 ∈ wcel 2098 {crab 3419 class class class wbr 5148 ↦ cmpt 5231 –1-1-onto→wf1o 6546 ‘cfv 6547 (class class class)co 7417 1^st c1st 7990 2^nd c2nd 7991 0cc0 11138 1c1 11139 + caddc 11141 ℕcn 12242 ℤ_≥cuz 12852 ...cfz 13516 ♯chash 14321 Word cword 14496 prefix cpfx 14652 Vtxcvtx 28865 USPGraphcuspgr 29017 Walkscwlks 29466 ClWalkscclwlks 29640 ClWWalksN cclwwlkn 29890 ClWWalksNOncclwwlknon 29953

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5364 ax-pr 5428 ax-un 7739 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ifp 1061 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3965 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6499 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-riota 7373 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7870 df-1st 7992 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-oadd 8489 df-er 8723 df-map 8845 df-pm 8846 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-dju 9924 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-n0 12503 df-xnn0 12575 df-z 12589 df-uz 12853 df-rp 13007 df-fz 13517 df-fzo 13660 df-hash 14322 df-word 14497 df-lsw 14545 df-concat 14553 df-s1 14578 df-substr 14623 df-pfx 14653 df-edg 28917 df-uhgr 28927 df-upgr 28951 df-uspgr 29019 df-wlks 29469 df-clwlks 29641 df-clwwlk 29848 df-clwwlkn 29891 df-clwwlknon 29954

This theorem is referenced by: clwwlknonclwlknonen 30229 dlwwlknondlwlknonf1o 30231

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