Theorem reprdifc 30833

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 1883	. . 3 ⊢ Ⅎ𝑑𝜑
2		nfrab1 3152	. . 3 ⊢ Ⅎ𝑑{𝑑 ∈ ((𝐴 ↑𝑚 (0..^𝑆)) ∖ (𝐵 ↑𝑚 (0..^𝑆))) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀}
3		nfcv 2793	. . 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 11518	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7		reprdifc.s	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ ℕ0)
8	4, 6, 7	reprval 30816	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑑 ∈ (𝐴 ↑𝑚 (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀})
9	8	eleq2d 2716	. . . . . . . . 9 ⊢ (𝜑 → (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ↔ 𝑑 ∈ {𝑑 ∈ (𝐴 ↑𝑚 (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀}))
10		rabid 3145	. . . . . . . . 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 276	. . . . . . . 8 ⊢ (𝜑 → (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ↔ (𝑑 ∈ (𝐴 ↑𝑚 (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀)))
12	11	anbi1d 741	. . . . . . 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 3617	. . . . . . . . . 10 ⊢ (𝑑 ∈ ((𝐴 ↑𝑚 (0..^𝑆)) ∖ (𝐵 ↑𝑚 (0..^𝑆))) ↔ (𝑑 ∈ (𝐴 ↑𝑚 (0..^𝑆)) ∧ ¬ 𝑑 ∈ (𝐵 ↑𝑚 (0..^𝑆))))
14	13	anbi1i 731	. . . . . . . . 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 856	. . . . . . . . 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 264	. . . . . . . 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 271	. . . . . 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 11064	. . . . . . . . . . . . . 14 ⊢ ℕ ∈ V
20	19	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℕ ∈ V)
21		reprdifc.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ⊆ ℕ)
22	20, 21	ssexd 4838	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ V)
23		ovexd 6720	. . . . . . . . . . . 12 ⊢ (𝜑 → (0..^𝑆) ∈ V)
24		elmapg 7912	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ V ∧ (0..^𝑆) ∈ V) → (𝑑 ∈ (𝐵 ↑𝑚 (0..^𝑆)) ↔ 𝑑:(0..^𝑆)⟶𝐵))
25	22, 23, 24	syl2anc 694	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑑 ∈ (𝐵 ↑𝑚 (0..^𝑆)) ↔ 𝑑:(0..^𝑆)⟶𝐵))
26	25	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (𝑑 ∈ (𝐵 ↑𝑚 (0..^𝑆)) ↔ 𝑑:(0..^𝑆)⟶𝐵))
27	4	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝐴 ⊆ ℕ)
28	6	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑀 ∈ ℤ)
29	7	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑆 ∈ ℕ0)
30		simpr 476	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑑 ∈ (𝐴(repr‘𝑆)𝑀))
31	27, 28, 29, 30	reprf 30818	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑑:(0..^𝑆)⟶𝐴)
32		ffn 6083	. . . . . . . . . . . . 13 ⊢ (𝑑:(0..^𝑆)⟶𝐴 → 𝑑 Fn (0..^𝑆))
33	31, 32	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑑 Fn (0..^𝑆))
34	33	biantrurd 528	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (∀𝑥 ∈ (0..^𝑆)(𝑑‘𝑥) ∈ 𝐵 ↔ (𝑑 Fn (0..^𝑆) ∧ ∀𝑥 ∈ (0..^𝑆)(𝑑‘𝑥) ∈ 𝐵)))
35		ffnfv 6428	. . . . . . . . . . 11 ⊢ (𝑑:(0..^𝑆)⟶𝐵 ↔ (𝑑 Fn (0..^𝑆) ∧ ∀𝑥 ∈ (0..^𝑆)(𝑑‘𝑥) ∈ 𝐵))
36	34, 35	syl6rbbr 279	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (𝑑:(0..^𝑆)⟶𝐵 ↔ ∀𝑥 ∈ (0..^𝑆)(𝑑‘𝑥) ∈ 𝐵))
37	26, 36	bitrd 268	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (𝑑 ∈ (𝐵 ↑𝑚 (0..^𝑆)) ↔ ∀𝑥 ∈ (0..^𝑆)(𝑑‘𝑥) ∈ 𝐵))
38	37	notbid 307	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (¬ 𝑑 ∈ (𝐵 ↑𝑚 (0..^𝑆)) ↔ ¬ ∀𝑥 ∈ (0..^𝑆)(𝑑‘𝑥) ∈ 𝐵))
39		rexnal 3024	. . . . . . . 8 ⊢ (∃𝑥 ∈ (0..^𝑆) ¬ (𝑑‘𝑥) ∈ 𝐵 ↔ ¬ ∀𝑥 ∈ (0..^𝑆)(𝑑‘𝑥) ∈ 𝐵)
40	38, 39	syl6bbr 278	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (¬ 𝑑 ∈ (𝐵 ↑𝑚 (0..^𝑆)) ↔ ∃𝑥 ∈ (0..^𝑆) ¬ (𝑑‘𝑥) ∈ 𝐵))
41	40	pm5.32da 674	. . . . . 6 ⊢ (𝜑 → ((𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ 𝑑 ∈ (𝐵 ↑𝑚 (0..^𝑆))) ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ∃𝑥 ∈ (0..^𝑆) ¬ (𝑑‘𝑥) ∈ 𝐵)))
42	18, 41	bitr3d 270	. . . . 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)))
43		fveq1 6228	. . . . . . . . . 10 ⊢ (𝑐 = 𝑑 → (𝑐‘𝑥) = (𝑑‘𝑥))
44	43	eleq1d 2715	. . . . . . . . 9 ⊢ (𝑐 = 𝑑 → ((𝑐‘𝑥) ∈ 𝐵 ↔ (𝑑‘𝑥) ∈ 𝐵))
45	44	notbid 307	. . . . . . . 8 ⊢ (𝑐 = 𝑑 → (¬ (𝑐‘𝑥) ∈ 𝐵 ↔ ¬ (𝑑‘𝑥) ∈ 𝐵))
46	45	elrab 3396	. . . . . . 7 ⊢ (𝑑 ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑥) ∈ 𝐵} ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘𝑥) ∈ 𝐵))
47	46	rexbii 3070	. . . . . 6 ⊢ (∃𝑥 ∈ (0..^𝑆)𝑑 ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑥) ∈ 𝐵} ↔ ∃𝑥 ∈ (0..^𝑆)(𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘𝑥) ∈ 𝐵))
48		r19.42v 3121	. . . . . 6 ⊢ (∃𝑥 ∈ (0..^𝑆)(𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘𝑥) ∈ 𝐵) ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ∃𝑥 ∈ (0..^𝑆) ¬ (𝑑‘𝑥) ∈ 𝐵))
49	47, 48	bitri 264	. . . . 5 ⊢ (∃𝑥 ∈ (0..^𝑆)𝑑 ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑥) ∈ 𝐵} ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ∃𝑥 ∈ (0..^𝑆) ¬ (𝑑‘𝑥) ∈ 𝐵))
50	42, 49	syl6bbr 278	. . . 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}))
51		rabid 3145	. . . 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))
52		eliun 4556	. . . 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})
53	50, 51, 52	3bitr4g 303	. . 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}))
54	1, 2, 3, 53	eqrd 3655	. 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})
55	21, 6, 7	reprval 30816	. . . 4 ⊢ (𝜑 → (𝐵(repr‘𝑆)𝑀) = {𝑑 ∈ (𝐵 ↑𝑚 (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀})
56	8, 55	difeq12d 3762	. . 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}))
57		difrab2 29465	. . 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}
58	56, 57	syl6eq 2701	. 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})
59		reprdifc.c	. . . 4 ⊢ 𝐶 = {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑥) ∈ 𝐵}
60	59	a1i 11	. . 3 ⊢ (𝜑 → 𝐶 = {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑥) ∈ 𝐵})
61	60	iuneq2d 4579	. 2 ⊢ (𝜑 → ∪ 𝑥 ∈ (0..^𝑆)𝐶 = ∪ 𝑥 ∈ (0..^𝑆){𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑥) ∈ 𝐵})
62	54, 58, 61	3eqtr4d 2695	1 ⊢ (𝜑 → ((𝐴(repr‘𝑆)𝑀) ∖ (𝐵(repr‘𝑆)𝑀)) = ∪ 𝑥 ∈ (0..^𝑆)𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942  {crab 2945  Vcvv 3231   ∖ cdif 3604   ⊆ wss 3607  ∪ ciun 4552   Fn wfn 5921  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690   ↑_𝑚 cmap 7899  0cc0 9974  ℕcn 11058  ℕ₀cn0 11330  ℤcz 11415  ..^cfzo 12504  Σcsu 14460  reprcrepr 30814

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-seq 12842  df-sum 14461  df-repr 30815

This theorem is referenced by:  hgt750lema  30863