Theorem reprdifc 35055

Description: Express the representations as a sum of integers in a difference of sets using conditions on each of the indices. (Contributed by Thierry Arnoux, 27-Dec-2021.)

Hypotheses

Ref	Expression
reprdifc.c	⊢ 𝐶 = {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑥) ∈ 𝐵}
reprdifc.a	⊢ (𝜑 → 𝐴 ⊆ ℕ)
reprdifc.b	⊢ (𝜑 → 𝐵 ⊆ ℕ)
reprdifc.m	⊢ (𝜑 → 𝑀 ∈ ℕ0)
reprdifc.s	⊢ (𝜑 → 𝑆 ∈ ℕ0)

Assertion

Ref	Expression
reprdifc	⊢ (𝜑 → ((𝐴(repr‘𝑆)𝑀) ∖ (𝐵(repr‘𝑆)𝑀)) = ∪ 𝑥 ∈ (0..^𝑆)𝐶)

Distinct variable groups:   𝐴,𝑐,𝑥   𝐵,𝑐,𝑥   𝑀,𝑐,𝑥   𝑆,𝑐,𝑥   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑐)   𝐶(𝑥, 𝑐)

Proof of Theorem reprdifc

Dummy variables 𝑑 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1947	. . 3 ⊢ Ⅎ𝑑𝜑
2		nfrab1 3438	. . 3 ⊢ Ⅎ𝑑{𝑑 ∈ ((𝐴 ↑m (0..^𝑆)) ∖ (𝐵 ↑m (0..^𝑆))) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀}
3		nfcv 2927	. . 3 ⊢ Ⅎ𝑑∪ 𝑥 ∈ (0..^𝑆){𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑥) ∈ 𝐵}
4		reprdifc.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ⊆ ℕ)
5		reprdifc.m	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
6	5	nn0zd 12627	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7		reprdifc.s	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ ℕ0)
8	4, 6, 7	reprval 35038	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑑 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀})
9	8	eleq2d 2851	. . . . . . . . 9 ⊢ (𝜑 → (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ↔ 𝑑 ∈ {𝑑 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀}))
10		rabid 3439	. . . . . . . . 9 ⊢ (𝑑 ∈ {𝑑 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀} ↔ (𝑑 ∈ (𝐴 ↑m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀))
11	9, 10	bitrdi 290	. . . . . . . 8 ⊢ (𝜑 → (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ↔ (𝑑 ∈ (𝐴 ↑m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀)))
12	11	anbi1d 643	. . . . . . 7 ⊢ (𝜑 → ((𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ 𝑑 ∈ (𝐵 ↑m (0..^𝑆))) ↔ ((𝑑 ∈ (𝐴 ↑m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀) ∧ ¬ 𝑑 ∈ (𝐵 ↑m (0..^𝑆)))))
13		eldif 3916	. . . . . . . . . 10 ⊢ (𝑑 ∈ ((𝐴 ↑m (0..^𝑆)) ∖ (𝐵 ↑m (0..^𝑆))) ↔ (𝑑 ∈ (𝐴 ↑m (0..^𝑆)) ∧ ¬ 𝑑 ∈ (𝐵 ↑m (0..^𝑆))))
14	13	anbi1i 636	. . . . . . . . 9 ⊢ ((𝑑 ∈ ((𝐴 ↑m (0..^𝑆)) ∖ (𝐵 ↑m (0..^𝑆))) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀) ↔ ((𝑑 ∈ (𝐴 ↑m (0..^𝑆)) ∧ ¬ 𝑑 ∈ (𝐵 ↑m (0..^𝑆))) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀))
15		an32 659	. . . . . . . . 9 ⊢ (((𝑑 ∈ (𝐴 ↑m (0..^𝑆)) ∧ ¬ 𝑑 ∈ (𝐵 ↑m (0..^𝑆))) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀) ↔ ((𝑑 ∈ (𝐴 ↑m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀) ∧ ¬ 𝑑 ∈ (𝐵 ↑m (0..^𝑆))))
16	14, 15	bitri 278	. . . . . . . 8 ⊢ ((𝑑 ∈ ((𝐴 ↑m (0..^𝑆)) ∖ (𝐵 ↑m (0..^𝑆))) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀) ↔ ((𝑑 ∈ (𝐴 ↑m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀) ∧ ¬ 𝑑 ∈ (𝐵 ↑m (0..^𝑆))))
17	16	a1i 11	. . . . . . 7 ⊢ (𝜑 → ((𝑑 ∈ ((𝐴 ↑m (0..^𝑆)) ∖ (𝐵 ↑m (0..^𝑆))) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀) ↔ ((𝑑 ∈ (𝐴 ↑m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀) ∧ ¬ 𝑑 ∈ (𝐵 ↑m (0..^𝑆)))))
18	12, 17	bitr4d 285	. . . . . 6 ⊢ (𝜑 → ((𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ 𝑑 ∈ (𝐵 ↑m (0..^𝑆))) ↔ (𝑑 ∈ ((𝐴 ↑m (0..^𝑆)) ∖ (𝐵 ↑m (0..^𝑆))) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀)))
19		nnex 12250	. . . . . . . . . . . . . 14 ⊢ ℕ ∈ V
20	19	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℕ ∈ V)
21		reprdifc.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ⊆ ℕ)
22	20, 21	ssexd 5297	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ V)
23		ovexd 7451	. . . . . . . . . . . 12 ⊢ (𝜑 → (0..^𝑆) ∈ V)
24		elmapg 8838	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ V ∧ (0..^𝑆) ∈ V) → (𝑑 ∈ (𝐵 ↑m (0..^𝑆)) ↔ 𝑑:(0..^𝑆)⟶𝐵))
25	22, 23, 24	syl2anc 596	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑑 ∈ (𝐵 ↑m (0..^𝑆)) ↔ 𝑑:(0..^𝑆)⟶𝐵))
26	25	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (𝑑 ∈ (𝐵 ↑m (0..^𝑆)) ↔ 𝑑:(0..^𝑆)⟶𝐵))
27		ffnfv 7118	. . . . . . . . . . 11 ⊢ (𝑑:(0..^𝑆)⟶𝐵 ↔ (𝑑 Fn (0..^𝑆) ∧ ∀𝑥 ∈ (0..^𝑆)(𝑑‘𝑥) ∈ 𝐵))
28	4	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝐴 ⊆ ℕ)
29	6	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑀 ∈ ℤ)
30	7	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑆 ∈ ℕ0)
31		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑑 ∈ (𝐴(repr‘𝑆)𝑀))
32	28, 29, 30, 31	reprf 35040	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑑:(0..^𝑆)⟶𝐴)
33	32	ffnd 6710	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑑 Fn (0..^𝑆))
34	33	biantrurd 542	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (∀𝑥 ∈ (0..^𝑆)(𝑑‘𝑥) ∈ 𝐵 ↔ (𝑑 Fn (0..^𝑆) ∧ ∀𝑥 ∈ (0..^𝑆)(𝑑‘𝑥) ∈ 𝐵)))
35	27, 34	bitr4id 293	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (𝑑:(0..^𝑆)⟶𝐵 ↔ ∀𝑥 ∈ (0..^𝑆)(𝑑‘𝑥) ∈ 𝐵))
36	26, 35	bitrd 282	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (𝑑 ∈ (𝐵 ↑m (0..^𝑆)) ↔ ∀𝑥 ∈ (0..^𝑆)(𝑑‘𝑥) ∈ 𝐵))
37	36	notbid 321	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (¬ 𝑑 ∈ (𝐵 ↑m (0..^𝑆)) ↔ ¬ ∀𝑥 ∈ (0..^𝑆)(𝑑‘𝑥) ∈ 𝐵))
38		rexnal 3119	. . . . . . . 8 ⊢ (∃𝑥 ∈ (0..^𝑆) ¬ (𝑑‘𝑥) ∈ 𝐵 ↔ ¬ ∀𝑥 ∈ (0..^𝑆)(𝑑‘𝑥) ∈ 𝐵)
39	37, 38	bitr4di 292	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (¬ 𝑑 ∈ (𝐵 ↑m (0..^𝑆)) ↔ ∃𝑥 ∈ (0..^𝑆) ¬ (𝑑‘𝑥) ∈ 𝐵))
40	39	pm5.32da 590	. . . . . 6 ⊢ (𝜑 → ((𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ 𝑑 ∈ (𝐵 ↑m (0..^𝑆))) ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ∃𝑥 ∈ (0..^𝑆) ¬ (𝑑‘𝑥) ∈ 𝐵)))
41	18, 40	bitr3d 284	. . . . 5 ⊢ (𝜑 → ((𝑑 ∈ ((𝐴 ↑m (0..^𝑆)) ∖ (𝐵 ↑m (0..^𝑆))) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀) ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ∃𝑥 ∈ (0..^𝑆) ¬ (𝑑‘𝑥) ∈ 𝐵)))
42		fveq1 6884	. . . . . . . . . 10 ⊢ (𝑐 = 𝑑 → (𝑐‘𝑥) = (𝑑‘𝑥))
43	42	eleq1d 2850	. . . . . . . . 9 ⊢ (𝑐 = 𝑑 → ((𝑐‘𝑥) ∈ 𝐵 ↔ (𝑑‘𝑥) ∈ 𝐵))
44	43	notbid 321	. . . . . . . 8 ⊢ (𝑐 = 𝑑 → (¬ (𝑐‘𝑥) ∈ 𝐵 ↔ ¬ (𝑑‘𝑥) ∈ 𝐵))
45	44	elrab 3652	. . . . . . 7 ⊢ (𝑑 ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑥) ∈ 𝐵} ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘𝑥) ∈ 𝐵))
46	45	rexbii 3114	. . . . . 6 ⊢ (∃𝑥 ∈ (0..^𝑆)𝑑 ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑥) ∈ 𝐵} ↔ ∃𝑥 ∈ (0..^𝑆)(𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘𝑥) ∈ 𝐵))
47		r19.42v 3199	. . . . . 6 ⊢ (∃𝑥 ∈ (0..^𝑆)(𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘𝑥) ∈ 𝐵) ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ∃𝑥 ∈ (0..^𝑆) ¬ (𝑑‘𝑥) ∈ 𝐵))
48	46, 47	bitri 278	. . . . 5 ⊢ (∃𝑥 ∈ (0..^𝑆)𝑑 ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑥) ∈ 𝐵} ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ∃𝑥 ∈ (0..^𝑆) ¬ (𝑑‘𝑥) ∈ 𝐵))
49	41, 48	bitr4di 292	. . . 4 ⊢ (𝜑 → ((𝑑 ∈ ((𝐴 ↑m (0..^𝑆)) ∖ (𝐵 ↑m (0..^𝑆))) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀) ↔ ∃𝑥 ∈ (0..^𝑆)𝑑 ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑥) ∈ 𝐵}))
50		rabid 3439	. . . 4 ⊢ (𝑑 ∈ {𝑑 ∈ ((𝐴 ↑m (0..^𝑆)) ∖ (𝐵 ↑m (0..^𝑆))) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀} ↔ (𝑑 ∈ ((𝐴 ↑m (0..^𝑆)) ∖ (𝐵 ↑m (0..^𝑆))) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀))
51		eliun 4962	. . . 4 ⊢ (𝑑 ∈ ∪ 𝑥 ∈ (0..^𝑆){𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑥) ∈ 𝐵} ↔ ∃𝑥 ∈ (0..^𝑆)𝑑 ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑥) ∈ 𝐵})
52	49, 50, 51	3bitr4g 317	. . 3 ⊢ (𝜑 → (𝑑 ∈ {𝑑 ∈ ((𝐴 ↑m (0..^𝑆)) ∖ (𝐵 ↑m (0..^𝑆))) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀} ↔ 𝑑 ∈ ∪ 𝑥 ∈ (0..^𝑆){𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑥) ∈ 𝐵}))
53	1, 2, 3, 52	eqrd 3957	. 2 ⊢ (𝜑 → {𝑑 ∈ ((𝐴 ↑m (0..^𝑆)) ∖ (𝐵 ↑m (0..^𝑆))) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀} = ∪ 𝑥 ∈ (0..^𝑆){𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑥) ∈ 𝐵})
54	21, 6, 7	reprval 35038	. . . 4 ⊢ (𝜑 → (𝐵(repr‘𝑆)𝑀) = {𝑑 ∈ (𝐵 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀})
55	8, 54	difeq12d 4082	. . 3 ⊢ (𝜑 → ((𝐴(repr‘𝑆)𝑀) ∖ (𝐵(repr‘𝑆)𝑀)) = ({𝑑 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀} ∖ {𝑑 ∈ (𝐵 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀}))
56		difrab2 32891	. . 3 ⊢ ({𝑑 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀} ∖ {𝑑 ∈ (𝐵 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀}) = {𝑑 ∈ ((𝐴 ↑m (0..^𝑆)) ∖ (𝐵 ↑m (0..^𝑆))) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀}
57	55, 56	eqtrdi 2816	. 2 ⊢ (𝜑 → ((𝐴(repr‘𝑆)𝑀) ∖ (𝐵(repr‘𝑆)𝑀)) = {𝑑 ∈ ((𝐴 ↑m (0..^𝑆)) ∖ (𝐵 ↑m (0..^𝑆))) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀})
58		reprdifc.c	. . . 4 ⊢ 𝐶 = {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑥) ∈ 𝐵}
59	58	a1i 11	. . 3 ⊢ (𝜑 → 𝐶 = {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑥) ∈ 𝐵})
60	59	iuneq2d 4989	. 2 ⊢ (𝜑 → ∪ 𝑥 ∈ (0..^𝑆)𝐶 = ∪ 𝑥 ∈ (0..^𝑆){𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑥) ∈ 𝐵})
61	53, 57, 60	3eqtr4d 2810	1 ⊢ (𝜑 → ((𝐴(repr‘𝑆)𝑀) ∖ (𝐵(repr‘𝑆)𝑀)) = ∪ 𝑥 ∈ (0..^𝑆)𝐶)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091  {crab 3418  Vcvv 3457   ∖ cdif 3903   ⊆ wss 3906  ∪ ciun 4958   Fn wfn 6535  ⟶wf 6536  ‘cfv 6540  (class class class)co 7416   ↑_m cmap 8826  0cc0 11111  ℕcn 12244  ℕ₀cn0 12515  ℤcz 12602  ..^cfzo 13695  Σcsu 15757  reprcrepr 35036

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 7738  ax-cnex 11167  ax-resscn 11168  ax-1cn 11169  ax-icn 11170  ax-addcl 11171  ax-addrcl 11172  ax-mulcl 11173  ax-mulrcl 11174  ax-i2m1 11179  ax-1ne0 11180  ax-rnegex 11182  ax-rrecex 11183  ax-cnre 11184

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  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-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-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-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-map 8828  df-neg 11455  df-nn 12245  df-n0 12516  df-z 12603  df-seq 14052  df-sum 15758  df-repr 35037

This theorem is used by:  hgt750lema  35085