Theorem metakunt12 40985

Description: C is the right inverse for A. (Contributed by metakunt, 25-May-2024.)

Hypotheses

Ref	Expression
metakunt12.1	⊢ (𝜑 → 𝑀 ∈ ℕ)
metakunt12.2	⊢ (𝜑 → 𝐼 ∈ ℕ)
metakunt12.3	⊢ (𝜑 → 𝐼 ≤ 𝑀)
metakunt12.4	⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))
metakunt12.5	⊢ 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))
metakunt12.6	⊢ (𝜑 → 𝑋 ∈ (1...𝑀))

Assertion

Ref	Expression
metakunt12	⊢ ((𝜑 ∧ ¬ (𝑋 = 𝑀 ∨ 𝑋 < 𝐼)) → (𝐴‘(𝐶‘𝑋)) = 𝑋)

Distinct variable groups:   𝑥,𝐶   𝑥,𝐼   𝑦,𝐼   𝑥,𝑀   𝑦,𝑀   𝑥,𝑋   𝑦,𝑋   𝜑,𝑥   𝜑,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐶(𝑦)

Proof of Theorem metakunt12

Step	Hyp	Ref	Expression
1		ioran 983	. 2 ⊢ (¬ (𝑋 = 𝑀 ∨ 𝑋 < 𝐼) ↔ (¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼))
2		metakunt12.4	. . . . 5 ⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))
3	2	a1i 11	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)))))
4		eqeq1 2737	. . . . . . 7 ⊢ (𝑥 = (𝐶‘𝑋) → (𝑥 = 𝐼 ↔ (𝐶‘𝑋) = 𝐼))
5		breq1 5151	. . . . . . . 8 ⊢ (𝑥 = (𝐶‘𝑋) → (𝑥 < 𝐼 ↔ (𝐶‘𝑋) < 𝐼))
6		id 22	. . . . . . . 8 ⊢ (𝑥 = (𝐶‘𝑋) → 𝑥 = (𝐶‘𝑋))
7		oveq1 7413	. . . . . . . 8 ⊢ (𝑥 = (𝐶‘𝑋) → (𝑥 − 1) = ((𝐶‘𝑋) − 1))
8	5, 6, 7	ifbieq12d 4556	. . . . . . 7 ⊢ (𝑥 = (𝐶‘𝑋) → if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)) = if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1)))
9	4, 8	ifbieq2d 4554	. . . . . 6 ⊢ (𝑥 = (𝐶‘𝑋) → if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) = if((𝐶‘𝑋) = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1))))
10	9	adantl 483	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) ∧ 𝑥 = (𝐶‘𝑋)) → if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) = if((𝐶‘𝑋) = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1))))
11		metakunt12.5	. . . . . . . . . 10 ⊢ 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))
12	11	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1)))))
13		eqeq1 2737	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑋 → (𝑦 = 𝑀 ↔ 𝑋 = 𝑀))
14		breq1 5151	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑋 → (𝑦 < 𝐼 ↔ 𝑋 < 𝐼))
15		id 22	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑋 → 𝑦 = 𝑋)
16		oveq1 7413	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑋 → (𝑦 + 1) = (𝑋 + 1))
17	14, 15, 16	ifbieq12d 4556	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑋 → if(𝑦 < 𝐼, 𝑦, (𝑦 + 1)) = if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)))
18	13, 17	ifbieq2d 4554	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑋 → if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))) = if(𝑋 = 𝑀, 𝐼, if(𝑋 < 𝐼, 𝑋, (𝑋 + 1))))
19	18	adantl 483	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) ∧ 𝑦 = 𝑋) → if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))) = if(𝑋 = 𝑀, 𝐼, if(𝑋 < 𝐼, 𝑋, (𝑋 + 1))))
20		simp2 1138	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → ¬ 𝑋 = 𝑀)
21		iffalse 4537	. . . . . . . . . . . . 13 ⊢ (¬ 𝑋 = 𝑀 → if(𝑋 = 𝑀, 𝐼, if(𝑋 < 𝐼, 𝑋, (𝑋 + 1))) = if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)))
22	20, 21	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → if(𝑋 = 𝑀, 𝐼, if(𝑋 < 𝐼, 𝑋, (𝑋 + 1))) = if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)))
23		simp3 1139	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → ¬ 𝑋 < 𝐼)
24		iffalse 4537	. . . . . . . . . . . . 13 ⊢ (¬ 𝑋 < 𝐼 → if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)) = (𝑋 + 1))
25	23, 24	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)) = (𝑋 + 1))
26	22, 25	eqtrd 2773	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → if(𝑋 = 𝑀, 𝐼, if(𝑋 < 𝐼, 𝑋, (𝑋 + 1))) = (𝑋 + 1))
27	26	adantr 482	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) ∧ 𝑦 = 𝑋) → if(𝑋 = 𝑀, 𝐼, if(𝑋 < 𝐼, 𝑋, (𝑋 + 1))) = (𝑋 + 1))
28	19, 27	eqtrd 2773	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) ∧ 𝑦 = 𝑋) → if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))) = (𝑋 + 1))
29		metakunt12.6	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (1...𝑀))
30	29	3ad2ant1 1134	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → 𝑋 ∈ (1...𝑀))
31	29	elfzelzd 13499	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ ℤ)
32	31	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → 𝑋 ∈ ℤ)
33	32	peano2zd 12666	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → (𝑋 + 1) ∈ ℤ)
34	12, 28, 30, 33	fvmptd 7003	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → (𝐶‘𝑋) = (𝑋 + 1))
35		eqeq1 2737	. . . . . . . . 9 ⊢ ((𝐶‘𝑋) = (𝑋 + 1) → ((𝐶‘𝑋) = 𝐼 ↔ (𝑋 + 1) = 𝐼))
36		breq1 5151	. . . . . . . . . 10 ⊢ ((𝐶‘𝑋) = (𝑋 + 1) → ((𝐶‘𝑋) < 𝐼 ↔ (𝑋 + 1) < 𝐼))
37		id 22	. . . . . . . . . 10 ⊢ ((𝐶‘𝑋) = (𝑋 + 1) → (𝐶‘𝑋) = (𝑋 + 1))
38		oveq1 7413	. . . . . . . . . 10 ⊢ ((𝐶‘𝑋) = (𝑋 + 1) → ((𝐶‘𝑋) − 1) = ((𝑋 + 1) − 1))
39	36, 37, 38	ifbieq12d 4556	. . . . . . . . 9 ⊢ ((𝐶‘𝑋) = (𝑋 + 1) → if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1)) = if((𝑋 + 1) < 𝐼, (𝑋 + 1), ((𝑋 + 1) − 1)))
40	35, 39	ifbieq2d 4554	. . . . . . . 8 ⊢ ((𝐶‘𝑋) = (𝑋 + 1) → if((𝐶‘𝑋) = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1))) = if((𝑋 + 1) = 𝐼, 𝑀, if((𝑋 + 1) < 𝐼, (𝑋 + 1), ((𝑋 + 1) − 1))))
41	34, 40	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → if((𝐶‘𝑋) = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1))) = if((𝑋 + 1) = 𝐼, 𝑀, if((𝑋 + 1) < 𝐼, (𝑋 + 1), ((𝑋 + 1) − 1))))
42		metakunt12.2	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐼 ∈ ℕ)
43	42	nnred 12224	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐼 ∈ ℝ)
44	43	3ad2ant1 1134	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → 𝐼 ∈ ℝ)
45	32	zred 12663	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → 𝑋 ∈ ℝ)
46	33	zred 12663	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → (𝑋 + 1) ∈ ℝ)
47	44, 45	lenltd 11357	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → (𝐼 ≤ 𝑋 ↔ ¬ 𝑋 < 𝐼))
48	23, 47	mpbird 257	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → 𝐼 ≤ 𝑋)
49	45	ltp1d 12141	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → 𝑋 < (𝑋 + 1))
50	44, 45, 46, 48, 49	lelttrd 11369	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → 𝐼 < (𝑋 + 1))
51	44, 50	ltned 11347	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → 𝐼 ≠ (𝑋 + 1))
52	51	necomd 2997	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → (𝑋 + 1) ≠ 𝐼)
53	52	neneqd 2946	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → ¬ (𝑋 + 1) = 𝐼)
54		iffalse 4537	. . . . . . . . 9 ⊢ (¬ (𝑋 + 1) = 𝐼 → if((𝑋 + 1) = 𝐼, 𝑀, if((𝑋 + 1) < 𝐼, (𝑋 + 1), ((𝑋 + 1) − 1))) = if((𝑋 + 1) < 𝐼, (𝑋 + 1), ((𝑋 + 1) − 1)))
55	53, 54	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → if((𝑋 + 1) = 𝐼, 𝑀, if((𝑋 + 1) < 𝐼, (𝑋 + 1), ((𝑋 + 1) − 1))) = if((𝑋 + 1) < 𝐼, (𝑋 + 1), ((𝑋 + 1) − 1)))
56	45	lep1d 12142	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → 𝑋 ≤ (𝑋 + 1))
57	44, 45, 46, 48, 56	letrd 11368	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → 𝐼 ≤ (𝑋 + 1))
58	44, 46	lenltd 11357	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → (𝐼 ≤ (𝑋 + 1) ↔ ¬ (𝑋 + 1) < 𝐼))
59	57, 58	mpbid 231	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → ¬ (𝑋 + 1) < 𝐼)
60		iffalse 4537	. . . . . . . . 9 ⊢ (¬ (𝑋 + 1) < 𝐼 → if((𝑋 + 1) < 𝐼, (𝑋 + 1), ((𝑋 + 1) − 1)) = ((𝑋 + 1) − 1))
61	59, 60	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → if((𝑋 + 1) < 𝐼, (𝑋 + 1), ((𝑋 + 1) − 1)) = ((𝑋 + 1) − 1))
62	32	zcnd 12664	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → 𝑋 ∈ ℂ)
63		1cnd 11206	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → 1 ∈ ℂ)
64	62, 63	pncand 11569	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → ((𝑋 + 1) − 1) = 𝑋)
65	55, 61, 64	3eqtrd 2777	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → if((𝑋 + 1) = 𝐼, 𝑀, if((𝑋 + 1) < 𝐼, (𝑋 + 1), ((𝑋 + 1) − 1))) = 𝑋)
66	41, 65	eqtrd 2773	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → if((𝐶‘𝑋) = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1))) = 𝑋)
67	66	adantr 482	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) ∧ 𝑥 = (𝐶‘𝑋)) → if((𝐶‘𝑋) = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1))) = 𝑋)
68	10, 67	eqtrd 2773	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) ∧ 𝑥 = (𝐶‘𝑋)) → if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) = 𝑋)
69		metakunt12.1	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ)
70		metakunt12.3	. . . . . . 7 ⊢ (𝜑 → 𝐼 ≤ 𝑀)
71	69, 42, 70, 11	metakunt2 40975	. . . . . 6 ⊢ (𝜑 → 𝐶:(1...𝑀)⟶(1...𝑀))
72	71	3ad2ant1 1134	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → 𝐶:(1...𝑀)⟶(1...𝑀))
73	72, 30	ffvelcdmd 7085	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → (𝐶‘𝑋) ∈ (1...𝑀))
74	3, 68, 73, 30	fvmptd 7003	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → (𝐴‘(𝐶‘𝑋)) = 𝑋)
75	74	3expb 1121	. 2 ⊢ ((𝜑 ∧ (¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼)) → (𝐴‘(𝐶‘𝑋)) = 𝑋)
76	1, 75	sylan2b 595	1 ⊢ ((𝜑 ∧ ¬ (𝑋 = 𝑀 ∨ 𝑋 < 𝐼)) → (𝐴‘(𝐶‘𝑋)) = 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ifcif 4528   class class class wbr 5148   ↦ cmpt 5231  ⟶wf 6537  ‘cfv 6541  (class class class)co 7406  ℝcr 11106  1c1 11108   + caddc 11110   < clt 11245   ≤ cle 11246   − cmin 11441  ℕcn 12209  ℤcz 12555  ...cfz 13481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-er 8700  df-en 8937  df-dom 8938  df-sdom 8939  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-n0 12470  df-z 12556  df-uz 12820  df-fz 13482

This theorem is referenced by:  metakunt13  40986