clwwlknonclwlknonf1o - Metamath Proof Explorer

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Theorem clwwlknonclwlknonf1o 30784

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 2765	. . 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 2765	. . 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 2765	. . . . 5 ⊢ (1^st ‘𝑐) = (1^st ‘𝑐)
6		eqid 2765	. . . . 5 ⊢ (2^nd ‘𝑐) = (2^nd ‘𝑐)
7	5, 6, 2, 4	clwlknf1oclwwlkn 30502	. . . 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 1149	. . 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 6884	. . . . . . 7 ⊢ (𝑠 = ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐))) → (𝑠‘0) = (((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))‘0))
10	9	3ad2ant3 1153	. . . . . 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 6890	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑐 → (♯‘(1^st ‘𝑤)) = (♯‘(1^st ‘𝑐)))
12	11	eqeq1d 2767	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑐 → ((♯‘(1^st ‘𝑤)) = 𝑁 ↔ (♯‘(1^st ‘𝑐)) = 𝑁))
13	12	elrab 3652	. . . . . . . . . 10 ⊢ (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} ↔ (𝑐 ∈ (ClWalks‘𝐺) ∧ (♯‘(1^st ‘𝑐)) = 𝑁))
14		clwlkwlk 30189	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → 𝑐 ∈ (Walks‘𝐺))
15		wlkcpr 30036	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (Walks‘𝐺) ↔ (1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐))
16		eqid 2765	. . . . . . . . . . . . . . . . 17 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
17	16	wlkpwrd 30025	. . . . . . . . . . . . . . . 16 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (2^nd ‘𝑐) ∈ Word (Vtx‘𝐺))
18	17	3ad2ant1 1151	. . . . . . . . . . . . . . 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 12918	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ_≥‘1))
20		eluzfz2 13576	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ (ℤ_≥‘1) → 𝑁 ∈ (1...𝑁))
21	19, 20	sylbi 220	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (1...𝑁))
22		fzelp1 13621	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ (1...𝑁) → 𝑁 ∈ (1...(𝑁 + 1)))
23	21, 22	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (1...(𝑁 + 1)))
24	23	3ad2ant3 1153	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ (1...(𝑁 + 1)))
25	24	3ad2ant3 1153	. . . . . . . . . . . . . . . . 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 23	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘(1^st ‘𝑐)) = 𝑁 → (♯‘(1^st ‘𝑐)) = 𝑁)
27		oveq1 7426	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘(1^st ‘𝑐)) = 𝑁 → ((♯‘(1^st ‘𝑐)) + 1) = (𝑁 + 1))
28	27	oveq2d 7435	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘(1^st ‘𝑐)) = 𝑁 → (1...((♯‘(1^st ‘𝑐)) + 1)) = (1...(𝑁 + 1)))
29	26, 28	eleq12d 2859	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘(1^st ‘𝑐)) = 𝑁 → ((♯‘(1^st ‘𝑐)) ∈ (1...((♯‘(1^st ‘𝑐)) + 1)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
30	29	3ad2ant2 1152	. . . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . . 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 30026	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (♯‘(2^nd ‘𝑐)) = ((♯‘(1^st ‘𝑐)) + 1))
33	32	oveq2d 7435	. . . . . . . . . . . . . . . . . 18 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (1...(♯‘(2^nd ‘𝑐))) = (1...((♯‘(1^st ‘𝑐)) + 1)))
34	33	eleq2d 2851	. . . . . . . . . . . . . . . . 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 1151	. . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . 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 521	. . . . . . . . . . . . . 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 1137	. . . . . . . . . . . . 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 220	. . . . . . . . . . . 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 18	. . . . . . . . . . 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 412	. . . . . . . . . 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 220	. . . . . . . . 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 413	. . . . . . . 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 14751	. . . . . . . 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 18	. . . . . . 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 1150	. . . . . 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 2800	. . . . 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 2767	. . . 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 1947	. . . . 5 ⊢ Ⅎ𝑤((2^nd ‘𝑐)‘0) = 𝑋
50		fveq2 6885	. . . . . . 7 ⊢ (𝑤 = 𝑐 → (2^nd ‘𝑤) = (2^nd ‘𝑐))
51	50	fveq1d 6887	. . . . . 6 ⊢ (𝑤 = 𝑐 → ((2^nd ‘𝑤)‘0) = ((2^nd ‘𝑐)‘0))
52	51	eqeq1d 2767	. . . . 5 ⊢ (𝑤 = 𝑐 → (((2^nd ‘𝑤)‘0) = 𝑋 ↔ ((2^nd ‘𝑐)‘0) = 𝑋))
53	49, 52	sbiev 2349	. . . 4 ⊢ ([𝑐 / 𝑤]((2^nd ‘𝑤)‘0) = 𝑋 ↔ ((2^nd ‘𝑐)‘0) = 𝑋)
54	48, 53	bitr4di 292	. . 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 7128	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝐹:𝑊–1-1-onto→{𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋})
56		clwwlknon 30508	. . 3 ⊢ (𝑋(ClWWalksNOn‘𝐺)𝑁) = {𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋}
57		f1oeq3 6814	. . 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 237	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝐹:𝑊–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 [wsb 2099 ∈ wcel 2146 {crab 3418 class class class wbr 5111 ↦ cmpt 5194 –1-1-onto→wf1o 6539 ‘cfv 6540 (class class class)co 7419 1^st c1st 7990 2^nd c2nd 7991 0cc0 11115 1c1 11116 + caddc 11118 ℕcn 12248 ℤ_≥cuz 12878 ...cfz 13551 ♯chash 14384 Word cword 14568 prefix cpfx 14730 Vtxcvtx 29401 USPGraphcuspgr 29556 Walkscwlks 30004 ClWalkscclwlks 30184 ClWWalksN cclwwlkn 30442 ClWWalksNOncclwwlknon 30505

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-ifp 1079 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-oadd 8463 df-er 8700 df-map 8832 df-pm 8833 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-dju 9903 df-card 9941 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-nn 12249 df-2 12318 df-n0 12520 df-xnn0 12593 df-z 12607 df-uz 12879 df-rp 13033 df-fz 13552 df-fzo 13700 df-hash 14385 df-word 14569 df-lsw 14618 df-concat 14626 df-s1 14653 df-substr 14699 df-pfx 14731 df-edg 29453 df-uhgr 29463 df-upgr 29487 df-uspgr 29558 df-wlks 30007 df-clwlks 30185 df-clwwlk 30400 df-clwwlkn 30443 df-clwwlknon 30506

This theorem is used by: clwwlknonclwlknonen 30785 dlwwlknondlwlknonf1o 30787

W3C validator