Theorem reprdifc 34811

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 1921	. . 3 ⊢ Ⅎ𝑑𝜑
2		nfrab1 3411	. . 3 ⊢ Ⅎ𝑑{𝑑 ∈ ((𝐴 ↑m (0..^𝑆)) ∖ (𝐵 ↑m (0..^𝑆))) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀}
3		nfcv 2901	. . 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 12540	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7		reprdifc.s	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ ℕ0)
8	4, 6, 7	reprval 34794	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑑 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀})
9	8	eleq2d 2825	. . . . . . . . 9 ⊢ (𝜑 → (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ↔ 𝑑 ∈ {𝑑 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀}))
10		rabid 3412	. . . . . . . . 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 288	. . . . . . . 8 ⊢ (𝜑 → (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ↔ (𝑑 ∈ (𝐴 ↑m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀)))
12	11	anbi1d 637	. . . . . . 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 3893	. . . . . . . . . 10 ⊢ (𝑑 ∈ ((𝐴 ↑m (0..^𝑆)) ∖ (𝐵 ↑m (0..^𝑆))) ↔ (𝑑 ∈ (𝐴 ↑m (0..^𝑆)) ∧ ¬ 𝑑 ∈ (𝐵 ↑m (0..^𝑆))))
14	13	anbi1i 630	. . . . . . . . 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 652	. . . . . . . . 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 276	. . . . . . . 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 283	. . . . . 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 12171	. . . . . . . . . . . . . 14 ⊢ ℕ ∈ V
20	19	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℕ ∈ V)
21		reprdifc.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ⊆ ℕ)
22	20, 21	ssexd 5252	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ V)
23		ovexd 7391	. . . . . . . . . . . 12 ⊢ (𝜑 → (0..^𝑆) ∈ V)
24		elmapg 8776	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ V ∧ (0..^𝑆) ∈ V) → (𝑑 ∈ (𝐵 ↑m (0..^𝑆)) ↔ 𝑑:(0..^𝑆)⟶𝐵))
25	22, 23, 24	syl2anc 590	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑑 ∈ (𝐵 ↑m (0..^𝑆)) ↔ 𝑑:(0..^𝑆)⟶𝐵))
26	25	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (𝑑 ∈ (𝐵 ↑m (0..^𝑆)) ↔ 𝑑:(0..^𝑆)⟶𝐵))
27		ffnfv 7060	. . . . . . . . . . 11 ⊢ (𝑑:(0..^𝑆)⟶𝐵 ↔ (𝑑 Fn (0..^𝑆) ∧ ∀𝑥 ∈ (0..^𝑆)(𝑑‘𝑥) ∈ 𝐵))
28	4	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝐴 ⊆ ℕ)
29	6	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑀 ∈ ℤ)
30	7	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑆 ∈ ℕ0)
31		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑑 ∈ (𝐴(repr‘𝑆)𝑀))
32	28, 29, 30, 31	reprf 34796	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑑:(0..^𝑆)⟶𝐴)
33	32	ffnd 6656	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑑 Fn (0..^𝑆))
34	33	biantrurd 537	. . . . . . . . . . 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 291	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (𝑑:(0..^𝑆)⟶𝐵 ↔ ∀𝑥 ∈ (0..^𝑆)(𝑑‘𝑥) ∈ 𝐵))
36	26, 35	bitrd 280	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (𝑑 ∈ (𝐵 ↑m (0..^𝑆)) ↔ ∀𝑥 ∈ (0..^𝑆)(𝑑‘𝑥) ∈ 𝐵))
37	36	notbid 319	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (¬ 𝑑 ∈ (𝐵 ↑m (0..^𝑆)) ↔ ¬ ∀𝑥 ∈ (0..^𝑆)(𝑑‘𝑥) ∈ 𝐵))
38		rexnal 3091	. . . . . . . 8 ⊢ (∃𝑥 ∈ (0..^𝑆) ¬ (𝑑‘𝑥) ∈ 𝐵 ↔ ¬ ∀𝑥 ∈ (0..^𝑆)(𝑑‘𝑥) ∈ 𝐵)
39	37, 38	bitr4di 290	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (¬ 𝑑 ∈ (𝐵 ↑m (0..^𝑆)) ↔ ∃𝑥 ∈ (0..^𝑆) ¬ (𝑑‘𝑥) ∈ 𝐵))
40	39	pm5.32da 584	. . . . . 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 282	. . . . 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 6826	. . . . . . . . . 10 ⊢ (𝑐 = 𝑑 → (𝑐‘𝑥) = (𝑑‘𝑥))
43	42	eleq1d 2824	. . . . . . . . 9 ⊢ (𝑐 = 𝑑 → ((𝑐‘𝑥) ∈ 𝐵 ↔ (𝑑‘𝑥) ∈ 𝐵))
44	43	notbid 319	. . . . . . . 8 ⊢ (𝑐 = 𝑑 → (¬ (𝑐‘𝑥) ∈ 𝐵 ↔ ¬ (𝑑‘𝑥) ∈ 𝐵))
45	44	elrab 3629	. . . . . . 7 ⊢ (𝑑 ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑥) ∈ 𝐵} ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘𝑥) ∈ 𝐵))
46	45	rexbii 3086	. . . . . 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 3171	. . . . . 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 276	. . . . 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 290	. . . 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 3412	. . . 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 315	. . 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 3934	. 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 34794	. . . 4 ⊢ (𝜑 → (𝐵(repr‘𝑆)𝑀) = {𝑑 ∈ (𝐵 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀})
55	8, 54	difeq12d 4058	. . 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 32585	. . 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 2790	. 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 4952	. 2 ⊢ (𝜑 → ∪ 𝑥 ∈ (0..^𝑆)𝐶 = ∪ 𝑥 ∈ (0..^𝑆){𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑥) ∈ 𝐵})
61	53, 57, 60	3eqtr4d 2784	1 ⊢ (𝜑 → ((𝐴(repr‘𝑆)𝑀) ∖ (𝐵(repr‘𝑆)𝑀)) = ∪ 𝑥 ∈ (0..^𝑆)𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ∃wrex 3063  {crab 3391  Vcvv 3431   ∖ cdif 3880   ⊆ wss 3883  ∪ ciun 4921   Fn wfn 6480  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356   ↑_m cmap 8763  0cc0 11029  ℕcn 12165  ℕ₀cn0 12428  ℤcz 12515  ..^cfzo 13599  Σcsu 15639  reprcrepr 34792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-i2m1 11097  ax-1ne0 11098  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-map 8765  df-neg 11371  df-nn 12166  df-n0 12429  df-z 12516  df-seq 13955  df-sum 15640  df-repr 34793

This theorem is referenced by:  hgt750lema  34841