Theorem metakunt12 41489

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 980	. 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 2728	. . . . . . 7 ⊢ (𝑥 = (𝐶‘𝑋) → (𝑥 = 𝐼 ↔ (𝐶‘𝑋) = 𝐼))
5		breq1 5141	. . . . . . . 8 ⊢ (𝑥 = (𝐶‘𝑋) → (𝑥 < 𝐼 ↔ (𝐶‘𝑋) < 𝐼))
6		id 22	. . . . . . . 8 ⊢ (𝑥 = (𝐶‘𝑋) → 𝑥 = (𝐶‘𝑋))
7		oveq1 7408	. . . . . . . 8 ⊢ (𝑥 = (𝐶‘𝑋) → (𝑥 − 1) = ((𝐶‘𝑋) − 1))
8	5, 6, 7	ifbieq12d 4548	. . . . . . 7 ⊢ (𝑥 = (𝐶‘𝑋) → if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)) = if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1)))
9	4, 8	ifbieq2d 4546	. . . . . 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 481	. . . . 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 2728	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑋 → (𝑦 = 𝑀 ↔ 𝑋 = 𝑀))
14		breq1 5141	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑋 → (𝑦 < 𝐼 ↔ 𝑋 < 𝐼))
15		id 22	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑋 → 𝑦 = 𝑋)
16		oveq1 7408	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑋 → (𝑦 + 1) = (𝑋 + 1))
17	14, 15, 16	ifbieq12d 4548	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑋 → if(𝑦 < 𝐼, 𝑦, (𝑦 + 1)) = if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)))
18	13, 17	ifbieq2d 4546	. . . . . . . . . . 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 481	. . . . . . . . . 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 1134	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → ¬ 𝑋 = 𝑀)
21		iffalse 4529	. . . . . . . . . . . . 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 1135	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → ¬ 𝑋 < 𝐼)
24		iffalse 4529	. . . . . . . . . . . . 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 2764	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → if(𝑋 = 𝑀, 𝐼, if(𝑋 < 𝐼, 𝑋, (𝑋 + 1))) = (𝑋 + 1))
27	26	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) ∧ 𝑦 = 𝑋) → if(𝑋 = 𝑀, 𝐼, if(𝑋 < 𝐼, 𝑋, (𝑋 + 1))) = (𝑋 + 1))
28	19, 27	eqtrd 2764	. . . . . . . . 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 1130	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → 𝑋 ∈ (1...𝑀))
31	29	elfzelzd 13499	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ ℤ)
32	31	3ad2ant1 1130	. . . . . . . . . 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 6995	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → (𝐶‘𝑋) = (𝑋 + 1))
35		eqeq1 2728	. . . . . . . . 9 ⊢ ((𝐶‘𝑋) = (𝑋 + 1) → ((𝐶‘𝑋) = 𝐼 ↔ (𝑋 + 1) = 𝐼))
36		breq1 5141	. . . . . . . . . 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 7408	. . . . . . . . . 10 ⊢ ((𝐶‘𝑋) = (𝑋 + 1) → ((𝐶‘𝑋) − 1) = ((𝑋 + 1) − 1))
39	36, 37, 38	ifbieq12d 4548	. . . . . . . . 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 4546	. . . . . . . 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 1130	. . . . . . . . . . . 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 2988	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → (𝑋 + 1) ≠ 𝐼)
53	52	neneqd 2937	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → ¬ (𝑋 + 1) = 𝐼)
54		iffalse 4529	. . . . . . . . 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 4529	. . . . . . . . 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 2768	. . . . . . 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 2764	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → if((𝐶‘𝑋) = 𝐼, 𝑀, if((𝐶‘𝑋) < 𝐼, (𝐶‘𝑋), ((𝐶‘𝑋) − 1))) = 𝑋)
67	66	adantr 480	. . . . 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 2764	. . . 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 41479	. . . . . 6 ⊢ (𝜑 → 𝐶:(1...𝑀)⟶(1...𝑀))
72	71	3ad2ant1 1130	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → 𝐶:(1...𝑀)⟶(1...𝑀))
73	72, 30	ffvelcdmd 7077	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → (𝐶‘𝑋) ∈ (1...𝑀))
74	3, 68, 73, 30	fvmptd 6995	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼) → (𝐴‘(𝐶‘𝑋)) = 𝑋)
75	74	3expb 1117	. 2 ⊢ ((𝜑 ∧ (¬ 𝑋 = 𝑀 ∧ ¬ 𝑋 < 𝐼)) → (𝐴‘(𝐶‘𝑋)) = 𝑋)
76	1, 75	sylan2b 593	1 ⊢ ((𝜑 ∧ ¬ (𝑋 = 𝑀 ∨ 𝑋 < 𝐼)) → (𝐴‘(𝐶‘𝑋)) = 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 844   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ifcif 4520   class class class wbr 5138   ↦ cmpt 5221  ⟶wf 6529  ‘cfv 6533  (class class class)co 7401  ℝcr 11105  1c1 11107   + caddc 11109   < 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 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 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  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  41490