clwwlknonclwlknonf1o - Metamath Proof Explorer

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Theorem clwwlknonclwlknonf1o 30363

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 2733	. . 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 2733	. . 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 2733	. . . . 5 ⊢ (1^st ‘𝑐) = (1^st ‘𝑐)
6		eqid 2733	. . . . 5 ⊢ (2^nd ‘𝑐) = (2^nd ‘𝑐)
7	5, 6, 2, 4	clwlknf1oclwwlkn 30085	. . . 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 1131	. . 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 6830	. . . . . . 7 ⊢ (𝑠 = ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐))) → (𝑠‘0) = (((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))‘0))
10	9	3ad2ant3 1135	. . . . . 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 6836	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑐 → (♯‘(1^st ‘𝑤)) = (♯‘(1^st ‘𝑐)))
12	11	eqeq1d 2735	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑐 → ((♯‘(1^st ‘𝑤)) = 𝑁 ↔ (♯‘(1^st ‘𝑐)) = 𝑁))
13	12	elrab 3643	. . . . . . . . . 10 ⊢ (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} ↔ (𝑐 ∈ (ClWalks‘𝐺) ∧ (♯‘(1^st ‘𝑐)) = 𝑁))
14		clwlkwlk 29774	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → 𝑐 ∈ (Walks‘𝐺))
15		wlkcpr 29628	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (Walks‘𝐺) ↔ (1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐))
16		eqid 2733	. . . . . . . . . . . . . . . . 17 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
17	16	wlkpwrd 29617	. . . . . . . . . . . . . . . 16 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (2^nd ‘𝑐) ∈ Word (Vtx‘𝐺))
18	17	3ad2ant1 1133	. . . . . . . . . . . . . . 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 12782	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ_≥‘1))
20		eluzfz2 13439	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ (ℤ_≥‘1) → 𝑁 ∈ (1...𝑁))
21	19, 20	sylbi 217	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (1...𝑁))
22		fzelp1 13483	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ (1...𝑁) → 𝑁 ∈ (1...(𝑁 + 1)))
23	21, 22	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (1...(𝑁 + 1)))
24	23	3ad2ant3 1135	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ (1...(𝑁 + 1)))
25	24	3ad2ant3 1135	. . . . . . . . . . . . . . . . 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 7362	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘(1^st ‘𝑐)) = 𝑁 → ((♯‘(1^st ‘𝑐)) + 1) = (𝑁 + 1))
28	27	oveq2d 7371	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘(1^st ‘𝑐)) = 𝑁 → (1...((♯‘(1^st ‘𝑐)) + 1)) = (1...(𝑁 + 1)))
29	26, 28	eleq12d 2827	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘(1^st ‘𝑐)) = 𝑁 → ((♯‘(1^st ‘𝑐)) ∈ (1...((♯‘(1^st ‘𝑐)) + 1)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
30	29	3ad2ant2 1134	. . . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . . 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 29618	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (♯‘(2^nd ‘𝑐)) = ((♯‘(1^st ‘𝑐)) + 1))
33	32	oveq2d 7371	. . . . . . . . . . . . . . . . . 18 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (1...(♯‘(2^nd ‘𝑐))) = (1...((♯‘(1^st ‘𝑐)) + 1)))
34	33	eleq2d 2819	. . . . . . . . . . . . . . . . 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 1133	. . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . 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 511	. . . . . . . . . . . . . 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 1119	. . . . . . . . . . . . 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 217	. . . . . . . . . . . 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 406	. . . . . . . . . 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 217	. . . . . . . . 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 407	. . . . . . . 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 14606	. . . . . . . 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 1132	. . . . . 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 2768	. . . . 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 2735	. . . 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 1915	. . . . 5 ⊢ Ⅎ𝑤((2^nd ‘𝑐)‘0) = 𝑋
50		fveq2 6831	. . . . . . 7 ⊢ (𝑤 = 𝑐 → (2^nd ‘𝑤) = (2^nd ‘𝑐))
51	50	fveq1d 6833	. . . . . 6 ⊢ (𝑤 = 𝑐 → ((2^nd ‘𝑤)‘0) = ((2^nd ‘𝑐)‘0))
52	51	eqeq1d 2735	. . . . 5 ⊢ (𝑤 = 𝑐 → (((2^nd ‘𝑤)‘0) = 𝑋 ↔ ((2^nd ‘𝑐)‘0) = 𝑋))
53	49, 52	sbiev 2317	. . . 4 ⊢ ([𝑐 / 𝑤]((2^nd ‘𝑤)‘0) = 𝑋 ↔ ((2^nd ‘𝑐)‘0) = 𝑋)
54	48, 53	bitr4di 289	. . 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 7070	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝐹:𝑊–1-1-onto→{𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋})
56		clwwlknon 30091	. . 3 ⊢ (𝑋(ClWWalksNOn‘𝐺)𝑁) = {𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋}
57		f1oeq3 6761	. . 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 234	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 [wsb 2067 ∈ wcel 2113 {crab 3396 class class class wbr 5095 ↦ cmpt 5176 –1-1-onto→wf1o 6488 ‘cfv 6489 (class class class)co 7355 1^st c1st 7928 2^nd c2nd 7929 0cc0 11017 1c1 11018 + caddc 11020 ℕcn 12136 ℤ_≥cuz 12742 ...cfz 13414 ♯chash 14244 Word cword 14427 prefix cpfx 14585 Vtxcvtx 28995 USPGraphcuspgr 29147 Walkscwlks 29596 ClWalkscclwlks 29769 ClWWalksN cclwwlkn 30025 ClWWalksNOncclwwlknon 30088

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-om 7806 df-1st 7930 df-2nd 7931 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-oadd 8398 df-er 8631 df-map 8761 df-pm 8762 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-dju 9805 df-card 9843 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-nn 12137 df-2 12199 df-n0 12393 df-xnn0 12466 df-z 12480 df-uz 12743 df-rp 12897 df-fz 13415 df-fzo 13562 df-hash 14245 df-word 14428 df-lsw 14477 df-concat 14485 df-s1 14511 df-substr 14556 df-pfx 14586 df-edg 29047 df-uhgr 29057 df-upgr 29081 df-uspgr 29149 df-wlks 29599 df-clwlks 29770 df-clwwlk 29983 df-clwwlkn 30026 df-clwwlknon 30089

This theorem is referenced by: clwwlknonclwlknonen 30364 dlwwlknondlwlknonf1o 30366

W3C validator