Theorem reprdifc 32273

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 1922	. . 3 ⊢ Ⅎ𝑑𝜑
2		nfrab1 3286	. . 3 ⊢ Ⅎ𝑑{𝑑 ∈ ((𝐴 ↑m (0..^𝑆)) ∖ (𝐵 ↑m (0..^𝑆))) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀}
3		nfcv 2897	. . 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 12245	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7		reprdifc.s	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ ℕ0)
8	4, 6, 7	reprval 32256	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑑 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀})
9	8	eleq2d 2816	. . . . . . . . 9 ⊢ (𝜑 → (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ↔ 𝑑 ∈ {𝑑 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀}))
10		rabid 3280	. . . . . . . . 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 633	. . . . . . 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 3863	. . . . . . . . . 10 ⊢ (𝑑 ∈ ((𝐴 ↑m (0..^𝑆)) ∖ (𝐵 ↑m (0..^𝑆))) ↔ (𝑑 ∈ (𝐴 ↑m (0..^𝑆)) ∧ ¬ 𝑑 ∈ (𝐵 ↑m (0..^𝑆))))
14	13	anbi1i 627	. . . . . . . . 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 646	. . . . . . . . 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 11801	. . . . . . . . . . . . . 14 ⊢ ℕ ∈ V
20	19	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℕ ∈ V)
21		reprdifc.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ⊆ ℕ)
22	20, 21	ssexd 5202	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ V)
23		ovexd 7226	. . . . . . . . . . . 12 ⊢ (𝜑 → (0..^𝑆) ∈ V)
24		elmapg 8499	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ V ∧ (0..^𝑆) ∈ V) → (𝑑 ∈ (𝐵 ↑m (0..^𝑆)) ↔ 𝑑:(0..^𝑆)⟶𝐵))
25	22, 23, 24	syl2anc 587	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑑 ∈ (𝐵 ↑m (0..^𝑆)) ↔ 𝑑:(0..^𝑆)⟶𝐵))
26	25	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (𝑑 ∈ (𝐵 ↑m (0..^𝑆)) ↔ 𝑑:(0..^𝑆)⟶𝐵))
27		ffnfv 6913	. . . . . . . . . . 11 ⊢ (𝑑:(0..^𝑆)⟶𝐵 ↔ (𝑑 Fn (0..^𝑆) ∧ ∀𝑥 ∈ (0..^𝑆)(𝑑‘𝑥) ∈ 𝐵))
28	4	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝐴 ⊆ ℕ)
29	6	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑀 ∈ ℤ)
30	7	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑆 ∈ ℕ0)
31		simpr 488	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑑 ∈ (𝐴(repr‘𝑆)𝑀))
32	28, 29, 30, 31	reprf 32258	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑑:(0..^𝑆)⟶𝐴)
33	32	ffnd 6524	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑑 Fn (0..^𝑆))
34	33	biantrurd 536	. . . . . . . . . . 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 3150	. . . . . . . 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 582	. . . . . 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 6694	. . . . . . . . . 10 ⊢ (𝑐 = 𝑑 → (𝑐‘𝑥) = (𝑑‘𝑥))
43	42	eleq1d 2815	. . . . . . . . 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 3591	. . . . . . 7 ⊢ (𝑑 ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑥) ∈ 𝐵} ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘𝑥) ∈ 𝐵))
46	45	rexbii 3160	. . . . . 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 3253	. . . . . 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 3280	. . . 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 4894	. . . 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 3906	. 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 32256	. . . 4 ⊢ (𝜑 → (𝐵(repr‘𝑆)𝑀) = {𝑑 ∈ (𝐵 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀})
55	8, 54	difeq12d 4024	. . 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 30518	. . 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 2787	. 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 4919	. 2 ⊢ (𝜑 → ∪ 𝑥 ∈ (0..^𝑆)𝐶 = ∪ 𝑥 ∈ (0..^𝑆){𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑥) ∈ 𝐵})
61	53, 57, 60	3eqtr4d 2781	1 ⊢ (𝜑 → ((𝐴(repr‘𝑆)𝑀) ∖ (𝐵(repr‘𝑆)𝑀)) = ∪ 𝑥 ∈ (0..^𝑆)𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2112  ∀wral 3051  ∃wrex 3052  {crab 3055  Vcvv 3398   ∖ cdif 3850   ⊆ wss 3853  ∪ ciun 4890   Fn wfn 6353  ⟶wf 6354  ‘cfv 6358  (class class class)co 7191   ↑_m cmap 8486  0cc0 10694  ℕcn 11795  ℕ₀cn0 12055  ℤcz 12141  ..^cfzo 13203  Σcsu 15214  reprcrepr 32254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501  ax-cnex 10750  ax-resscn 10751  ax-1cn 10752  ax-icn 10753  ax-addcl 10754  ax-addrcl 10755  ax-mulcl 10756  ax-mulrcl 10757  ax-i2m1 10762  ax-1ne0 10763  ax-rnegex 10765  ax-rrecex 10766  ax-cnre 10767

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6140  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7194  df-oprab 7195  df-mpo 7196  df-om 7623  df-1st 7739  df-2nd 7740  df-wrecs 8025  df-recs 8086  df-rdg 8124  df-map 8488  df-neg 11030  df-nn 11796  df-n0 12056  df-z 12142  df-seq 13540  df-sum 15215  df-repr 32255

This theorem is referenced by:  hgt750lema  32303