Theorem genpdisj 7786

Description: The lower and upper cuts produced by addition or multiplication on positive reals are disjoint. (Contributed by Jim Kingdon, 15-Oct-2019.)

Hypotheses

Ref	Expression
genpelvl.1	⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
genpelvl.2	⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)
genpdisj.ord	⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <Q 𝑦 ↔ (𝑧𝐺𝑥) <Q (𝑧𝐺𝑦)))
genpdisj.com	⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))

Assertion

Ref	Expression
genpdisj	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑞,𝐴   𝑥,𝐵,𝑦,𝑧,𝑤,𝑣,𝑞   𝑥,𝐺,𝑦,𝑧,𝑤,𝑣,𝑞   𝐹,𝑞

Allowed substitution hints:   𝐹(𝑥,𝑦,𝑧,𝑤,𝑣)

Proof of Theorem genpdisj

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

Step	Hyp	Ref	Expression
1		genpelvl.1	. . . . . . . . 9 ⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
2		genpelvl.2	. . . . . . . . 9 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)
3	1, 2	genpelvl 7775	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑎 ∈ (1st ‘𝐴)∃𝑏 ∈ (1st ‘𝐵)𝑞 = (𝑎𝐺𝑏)))
4		r2ex 2553	. . . . . . . 8 ⊢ (∃𝑎 ∈ (1st ‘𝐴)∃𝑏 ∈ (1st ‘𝐵)𝑞 = (𝑎𝐺𝑏) ↔ ∃𝑎∃𝑏((𝑎 ∈ (1st ‘𝐴) ∧ 𝑏 ∈ (1st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)))
5	3, 4	bitrdi 196	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑎∃𝑏((𝑎 ∈ (1st ‘𝐴) ∧ 𝑏 ∈ (1st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏))))
6	1, 2	genpelvu 7776	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑐 ∈ (2nd ‘𝐴)∃𝑑 ∈ (2nd ‘𝐵)𝑞 = (𝑐𝐺𝑑)))
7		r2ex 2553	. . . . . . . 8 ⊢ (∃𝑐 ∈ (2nd ‘𝐴)∃𝑑 ∈ (2nd ‘𝐵)𝑞 = (𝑐𝐺𝑑) ↔ ∃𝑐∃𝑑((𝑐 ∈ (2nd ‘𝐴) ∧ 𝑑 ∈ (2nd ‘𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑)))
8	6, 7	bitrdi 196	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑐∃𝑑((𝑐 ∈ (2nd ‘𝐴) ∧ 𝑑 ∈ (2nd ‘𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑))))
9	5, 8	anbi12d 473	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) ↔ (∃𝑎∃𝑏((𝑎 ∈ (1st ‘𝐴) ∧ 𝑏 ∈ (1st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ∃𝑐∃𝑑((𝑐 ∈ (2nd ‘𝐴) ∧ 𝑑 ∈ (2nd ‘𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑)))))
10		ee4anv 1987	. . . . . 6 ⊢ (∃𝑎∃𝑏∃𝑐∃𝑑(((𝑎 ∈ (1st ‘𝐴) ∧ 𝑏 ∈ (1st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2nd ‘𝐴) ∧ 𝑑 ∈ (2nd ‘𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑))) ↔ (∃𝑎∃𝑏((𝑎 ∈ (1st ‘𝐴) ∧ 𝑏 ∈ (1st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ∃𝑐∃𝑑((𝑐 ∈ (2nd ‘𝐴) ∧ 𝑑 ∈ (2nd ‘𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑))))
11	9, 10	bitr4di 198	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) ↔ ∃𝑎∃𝑏∃𝑐∃𝑑(((𝑎 ∈ (1st ‘𝐴) ∧ 𝑏 ∈ (1st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2nd ‘𝐴) ∧ 𝑑 ∈ (2nd ‘𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑)))))
12	11	biimpa 296	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) → ∃𝑎∃𝑏∃𝑐∃𝑑(((𝑎 ∈ (1st ‘𝐴) ∧ 𝑏 ∈ (1st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2nd ‘𝐴) ∧ 𝑑 ∈ (2nd ‘𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑))))
13		an4 588	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ (1st ‘𝐴) ∧ 𝑐 ∈ (2nd ‘𝐴)) ∧ (𝑏 ∈ (1st ‘𝐵) ∧ 𝑑 ∈ (2nd ‘𝐵))) ↔ ((𝑎 ∈ (1st ‘𝐴) ∧ 𝑏 ∈ (1st ‘𝐵)) ∧ (𝑐 ∈ (2nd ‘𝐴) ∧ 𝑑 ∈ (2nd ‘𝐵))))
14		prop 7738	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
15		prltlu 7750	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑎 ∈ (1st ‘𝐴) ∧ 𝑐 ∈ (2nd ‘𝐴)) → 𝑎 <Q 𝑐)
16	15	3expib 1233	. . . . . . . . . . . . . . . 16 ⊢ (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P → ((𝑎 ∈ (1st ‘𝐴) ∧ 𝑐 ∈ (2nd ‘𝐴)) → 𝑎 <Q 𝑐))
17	14, 16	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ P → ((𝑎 ∈ (1st ‘𝐴) ∧ 𝑐 ∈ (2nd ‘𝐴)) → 𝑎 <Q 𝑐))
18		prop 7738	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
19		prltlu 7750	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑏 ∈ (1st ‘𝐵) ∧ 𝑑 ∈ (2nd ‘𝐵)) → 𝑏 <Q 𝑑)
20	19	3expib 1233	. . . . . . . . . . . . . . . 16 ⊢ (⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P → ((𝑏 ∈ (1st ‘𝐵) ∧ 𝑑 ∈ (2nd ‘𝐵)) → 𝑏 <Q 𝑑))
21	18, 20	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ P → ((𝑏 ∈ (1st ‘𝐵) ∧ 𝑑 ∈ (2nd ‘𝐵)) → 𝑏 <Q 𝑑))
22	17, 21	im2anan9 602	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (((𝑎 ∈ (1st ‘𝐴) ∧ 𝑐 ∈ (2nd ‘𝐴)) ∧ (𝑏 ∈ (1st ‘𝐵) ∧ 𝑑 ∈ (2nd ‘𝐵))) → (𝑎 <Q 𝑐 ∧ 𝑏 <Q 𝑑)))
23		genpdisj.ord	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <Q 𝑦 ↔ (𝑧𝐺𝑥) <Q (𝑧𝐺𝑦)))
24		genpdisj.com	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))
25	23, 24	genplt2i 7773	. . . . . . . . . . . . . 14 ⊢ ((𝑎 <Q 𝑐 ∧ 𝑏 <Q 𝑑) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
26	22, 25	syl6 33	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (((𝑎 ∈ (1st ‘𝐴) ∧ 𝑐 ∈ (2nd ‘𝐴)) ∧ (𝑏 ∈ (1st ‘𝐵) ∧ 𝑑 ∈ (2nd ‘𝐵))) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑)))
27	13, 26	biimtrrid 153	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (((𝑎 ∈ (1st ‘𝐴) ∧ 𝑏 ∈ (1st ‘𝐵)) ∧ (𝑐 ∈ (2nd ‘𝐴) ∧ 𝑑 ∈ (2nd ‘𝐵))) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑)))
28	27	imp 124	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ ((𝑎 ∈ (1st ‘𝐴) ∧ 𝑏 ∈ (1st ‘𝐵)) ∧ (𝑐 ∈ (2nd ‘𝐴) ∧ 𝑑 ∈ (2nd ‘𝐵)))) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
29	28	adantlr 477	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) ∧ ((𝑎 ∈ (1st ‘𝐴) ∧ 𝑏 ∈ (1st ‘𝐵)) ∧ (𝑐 ∈ (2nd ‘𝐴) ∧ 𝑑 ∈ (2nd ‘𝐵)))) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
30	29	adantrlr 485	. . . . . . . . 9 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) ∧ (((𝑎 ∈ (1st ‘𝐴) ∧ 𝑏 ∈ (1st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ (𝑐 ∈ (2nd ‘𝐴) ∧ 𝑑 ∈ (2nd ‘𝐵)))) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
31	30	adantrrr 487	. . . . . . . 8 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) ∧ (((𝑎 ∈ (1st ‘𝐴) ∧ 𝑏 ∈ (1st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2nd ‘𝐴) ∧ 𝑑 ∈ (2nd ‘𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑)))) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
32		eqtr2 2250	. . . . . . . . . . 11 ⊢ ((𝑞 = (𝑎𝐺𝑏) ∧ 𝑞 = (𝑐𝐺𝑑)) → (𝑎𝐺𝑏) = (𝑐𝐺𝑑))
33	32	ad2ant2l 508	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ (1st ‘𝐴) ∧ 𝑏 ∈ (1st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2nd ‘𝐴) ∧ 𝑑 ∈ (2nd ‘𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑))) → (𝑎𝐺𝑏) = (𝑐𝐺𝑑))
34	33	adantl 277	. . . . . . . . 9 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) ∧ (((𝑎 ∈ (1st ‘𝐴) ∧ 𝑏 ∈ (1st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2nd ‘𝐴) ∧ 𝑑 ∈ (2nd ‘𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑)))) → (𝑎𝐺𝑏) = (𝑐𝐺𝑑))
35		ltsonq 7661	. . . . . . . . . . 11 ⊢ <Q Or Q
36		ltrelnq 7628	. . . . . . . . . . 11 ⊢ <Q ⊆ (Q × Q)
37	35, 36	soirri 5138	. . . . . . . . . 10 ⊢ ¬ (𝑎𝐺𝑏) <Q (𝑎𝐺𝑏)
38		breq2 4097	. . . . . . . . . 10 ⊢ ((𝑎𝐺𝑏) = (𝑐𝐺𝑑) → ((𝑎𝐺𝑏) <Q (𝑎𝐺𝑏) ↔ (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑)))
39	37, 38	mtbii 681	. . . . . . . . 9 ⊢ ((𝑎𝐺𝑏) = (𝑐𝐺𝑑) → ¬ (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
40	34, 39	syl 14	. . . . . . . 8 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) ∧ (((𝑎 ∈ (1st ‘𝐴) ∧ 𝑏 ∈ (1st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2nd ‘𝐴) ∧ 𝑑 ∈ (2nd ‘𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑)))) → ¬ (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
41	31, 40	pm2.21fal 1418	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) ∧ (((𝑎 ∈ (1st ‘𝐴) ∧ 𝑏 ∈ (1st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2nd ‘𝐴) ∧ 𝑑 ∈ (2nd ‘𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑)))) → ⊥)
42	41	ex 115	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) → ((((𝑎 ∈ (1st ‘𝐴) ∧ 𝑏 ∈ (1st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2nd ‘𝐴) ∧ 𝑑 ∈ (2nd ‘𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑))) → ⊥))
43	42	exlimdvv 1946	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) → (∃𝑐∃𝑑(((𝑎 ∈ (1st ‘𝐴) ∧ 𝑏 ∈ (1st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2nd ‘𝐴) ∧ 𝑑 ∈ (2nd ‘𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑))) → ⊥))
44	43	exlimdvv 1946	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) → (∃𝑎∃𝑏∃𝑐∃𝑑(((𝑎 ∈ (1st ‘𝐴) ∧ 𝑏 ∈ (1st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2nd ‘𝐴) ∧ 𝑑 ∈ (2nd ‘𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑))) → ⊥))
45	12, 44	mpd 13	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) → ⊥)
46	45	inegd 1417	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ¬ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))))
47	46	ralrimivw 2607	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398  ⊥wfal 1403  ∃wex 1541   ∈ wcel 2202  ∀wral 2511  ∃wrex 2512  {crab 2515  ⟨cop 3676   class class class wbr 4093  ‘cfv 5333  (class class class)co 6028   ∈ cmpo 6030  1^st c1st 6310  2^nd c2nd 6311  Qcnq 7543   <_Q cltq 7548  Pcnp 7554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7567  df-mi 7569  df-lti 7570  df-enq 7610  df-nqqs 7611  df-ltnqqs 7616  df-inp 7729

This theorem is referenced by:  addclpr  7800  mulclpr  7835