Theorem reprdifc 31900

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 1915	. . 3 ⊢ Ⅎ𝑑𝜑
2		nfrab1 3386	. . 3 ⊢ Ⅎ𝑑{𝑑 ∈ ((𝐴 ↑m (0..^𝑆)) ∖ (𝐵 ↑m (0..^𝑆))) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀}
3		nfcv 2979	. . 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 12088	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7		reprdifc.s	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ ℕ0)
8	4, 6, 7	reprval 31883	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑑 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀})
9	8	eleq2d 2900	. . . . . . . . 9 ⊢ (𝜑 → (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ↔ 𝑑 ∈ {𝑑 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀}))
10		rabid 3380	. . . . . . . . 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	syl6bb 289	. . . . . . . 8 ⊢ (𝜑 → (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ↔ (𝑑 ∈ (𝐴 ↑m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀)))
12	11	anbi1d 631	. . . . . . 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 3948	. . . . . . . . . 10 ⊢ (𝑑 ∈ ((𝐴 ↑m (0..^𝑆)) ∖ (𝐵 ↑m (0..^𝑆))) ↔ (𝑑 ∈ (𝐴 ↑m (0..^𝑆)) ∧ ¬ 𝑑 ∈ (𝐵 ↑m (0..^𝑆))))
14	13	anbi1i 625	. . . . . . . . 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 644	. . . . . . . . 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 277	. . . . . . . 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 284	. . . . . 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 11646	. . . . . . . . . . . . . 14 ⊢ ℕ ∈ V
20	19	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℕ ∈ V)
21		reprdifc.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ⊆ ℕ)
22	20, 21	ssexd 5230	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ V)
23		ovexd 7193	. . . . . . . . . . . 12 ⊢ (𝜑 → (0..^𝑆) ∈ V)
24		elmapg 8421	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ V ∧ (0..^𝑆) ∈ V) → (𝑑 ∈ (𝐵 ↑m (0..^𝑆)) ↔ 𝑑:(0..^𝑆)⟶𝐵))
25	22, 23, 24	syl2anc 586	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑑 ∈ (𝐵 ↑m (0..^𝑆)) ↔ 𝑑:(0..^𝑆)⟶𝐵))
26	25	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (𝑑 ∈ (𝐵 ↑m (0..^𝑆)) ↔ 𝑑:(0..^𝑆)⟶𝐵))
27	4	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝐴 ⊆ ℕ)
28	6	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑀 ∈ ℤ)
29	7	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑆 ∈ ℕ0)
30		simpr 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑑 ∈ (𝐴(repr‘𝑆)𝑀))
31	27, 28, 29, 30	reprf 31885	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑑:(0..^𝑆)⟶𝐴)
32	31	ffnd 6517	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑑 Fn (0..^𝑆))
33	32	biantrurd 535	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (∀𝑥 ∈ (0..^𝑆)(𝑑‘𝑥) ∈ 𝐵 ↔ (𝑑 Fn (0..^𝑆) ∧ ∀𝑥 ∈ (0..^𝑆)(𝑑‘𝑥) ∈ 𝐵)))
34		ffnfv 6884	. . . . . . . . . . 11 ⊢ (𝑑:(0..^𝑆)⟶𝐵 ↔ (𝑑 Fn (0..^𝑆) ∧ ∀𝑥 ∈ (0..^𝑆)(𝑑‘𝑥) ∈ 𝐵))
35	33, 34	syl6rbbr 292	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (𝑑:(0..^𝑆)⟶𝐵 ↔ ∀𝑥 ∈ (0..^𝑆)(𝑑‘𝑥) ∈ 𝐵))
36	26, 35	bitrd 281	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (𝑑 ∈ (𝐵 ↑m (0..^𝑆)) ↔ ∀𝑥 ∈ (0..^𝑆)(𝑑‘𝑥) ∈ 𝐵))
37	36	notbid 320	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (¬ 𝑑 ∈ (𝐵 ↑m (0..^𝑆)) ↔ ¬ ∀𝑥 ∈ (0..^𝑆)(𝑑‘𝑥) ∈ 𝐵))
38		rexnal 3240	. . . . . . . 8 ⊢ (∃𝑥 ∈ (0..^𝑆) ¬ (𝑑‘𝑥) ∈ 𝐵 ↔ ¬ ∀𝑥 ∈ (0..^𝑆)(𝑑‘𝑥) ∈ 𝐵)
39	37, 38	syl6bbr 291	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (¬ 𝑑 ∈ (𝐵 ↑m (0..^𝑆)) ↔ ∃𝑥 ∈ (0..^𝑆) ¬ (𝑑‘𝑥) ∈ 𝐵))
40	39	pm5.32da 581	. . . . . 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 283	. . . . 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 6671	. . . . . . . . . 10 ⊢ (𝑐 = 𝑑 → (𝑐‘𝑥) = (𝑑‘𝑥))
43	42	eleq1d 2899	. . . . . . . . 9 ⊢ (𝑐 = 𝑑 → ((𝑐‘𝑥) ∈ 𝐵 ↔ (𝑑‘𝑥) ∈ 𝐵))
44	43	notbid 320	. . . . . . . 8 ⊢ (𝑐 = 𝑑 → (¬ (𝑐‘𝑥) ∈ 𝐵 ↔ ¬ (𝑑‘𝑥) ∈ 𝐵))
45	44	elrab 3682	. . . . . . 7 ⊢ (𝑑 ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑥) ∈ 𝐵} ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘𝑥) ∈ 𝐵))
46	45	rexbii 3249	. . . . . 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 3352	. . . . . 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 277	. . . . 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	syl6bbr 291	. . . 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 3380	. . . 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 4925	. . . 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 316	. . 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 3988	. 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 31883	. . . 4 ⊢ (𝜑 → (𝐵(repr‘𝑆)𝑀) = {𝑑 ∈ (𝐵 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀})
55	8, 54	difeq12d 4102	. . 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 30263	. . 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	syl6eq 2874	. 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 4950	. 2 ⊢ (𝜑 → ∪ 𝑥 ∈ (0..^𝑆)𝐶 = ∪ 𝑥 ∈ (0..^𝑆){𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑥) ∈ 𝐵})
61	53, 57, 60	3eqtr4d 2868	1 ⊢ (𝜑 → ((𝐴(repr‘𝑆)𝑀) ∖ (𝐵(repr‘𝑆)𝑀)) = ∪ 𝑥 ∈ (0..^𝑆)𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141  {crab 3144  Vcvv 3496   ∖ cdif 3935   ⊆ wss 3938  ∪ ciun 4921   Fn wfn 6352  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158   ↑_m cmap 8408  0cc0 10539  ℕcn 11640  ℕ₀cn0 11900  ℤcz 11984  ..^cfzo 13036  Σcsu 15044  reprcrepr 31881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-i2m1 10607  ax-1ne0 10608  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-map 8410  df-neg 10875  df-nn 11641  df-n0 11901  df-z 11985  df-seq 13373  df-sum 15045  df-repr 31882

This theorem is referenced by:  hgt750lema  31930