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Theorem genpdisj 7497

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 ∈ (1^st ‘𝑤) ∧ 𝑧 ∈ (1^st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝑤) ∧ 𝑧 ∈ (2^nd ‘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 ∈ (1^st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2^nd ‘(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 ∈ (1^st ‘𝑤) ∧ 𝑧 ∈ (1^st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝑤) ∧ 𝑧 ∈ (2^nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
2		genpelvl.2	. . . . . . . . 9 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)
3	1, 2	genpelvl 7486	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)) ↔ ∃𝑎 ∈ (1^st ‘𝐴)∃𝑏 ∈ (1^st ‘𝐵)𝑞 = (𝑎𝐺𝑏)))
4		r2ex 2495	. . . . . . . 8 ⊢ (∃𝑎 ∈ (1^st ‘𝐴)∃𝑏 ∈ (1^st ‘𝐵)𝑞 = (𝑎𝐺𝑏) ↔ ∃𝑎∃𝑏((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑏 ∈ (1^st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)))
5	3, 4	bitrdi 196	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)) ↔ ∃𝑎∃𝑏((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑏 ∈ (1^st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏))))
6	1, 2	genpelvu 7487	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑞 ∈ (2^nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑐 ∈ (2^nd ‘𝐴)∃𝑑 ∈ (2^nd ‘𝐵)𝑞 = (𝑐𝐺𝑑)))
7		r2ex 2495	. . . . . . . 8 ⊢ (∃𝑐 ∈ (2^nd ‘𝐴)∃𝑑 ∈ (2^nd ‘𝐵)𝑞 = (𝑐𝐺𝑑) ↔ ∃𝑐∃𝑑((𝑐 ∈ (2^nd ‘𝐴) ∧ 𝑑 ∈ (2^nd ‘𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑)))
8	6, 7	bitrdi 196	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑞 ∈ (2^nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑐∃𝑑((𝑐 ∈ (2^nd ‘𝐴) ∧ 𝑑 ∈ (2^nd ‘𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑))))
9	5, 8	anbi12d 473	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2^nd ‘(𝐴𝐹𝐵))) ↔ (∃𝑎∃𝑏((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑏 ∈ (1^st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ∃𝑐∃𝑑((𝑐 ∈ (2^nd ‘𝐴) ∧ 𝑑 ∈ (2^nd ‘𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑)))))
10		ee4anv 1932	. . . . . 6 ⊢ (∃𝑎∃𝑏∃𝑐∃𝑑(((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑏 ∈ (1^st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2^nd ‘𝐴) ∧ 𝑑 ∈ (2^nd ‘𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑))) ↔ (∃𝑎∃𝑏((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑏 ∈ (1^st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ∃𝑐∃𝑑((𝑐 ∈ (2^nd ‘𝐴) ∧ 𝑑 ∈ (2^nd ‘𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑))))
11	9, 10	bitr4di 198	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2^nd ‘(𝐴𝐹𝐵))) ↔ ∃𝑎∃𝑏∃𝑐∃𝑑(((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑏 ∈ (1^st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2^nd ‘𝐴) ∧ 𝑑 ∈ (2^nd ‘𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑)))))
12	11	biimpa 296	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2^nd ‘(𝐴𝐹𝐵)))) → ∃𝑎∃𝑏∃𝑐∃𝑑(((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑏 ∈ (1^st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2^nd ‘𝐴) ∧ 𝑑 ∈ (2^nd ‘𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑))))
13		an4 586	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑐 ∈ (2^nd ‘𝐴)) ∧ (𝑏 ∈ (1^st ‘𝐵) ∧ 𝑑 ∈ (2^nd ‘𝐵))) ↔ ((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑏 ∈ (1^st ‘𝐵)) ∧ (𝑐 ∈ (2^nd ‘𝐴) ∧ 𝑑 ∈ (2^nd ‘𝐵))))
14		prop 7449	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
15		prltlu 7461	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑎 ∈ (1^st ‘𝐴) ∧ 𝑐 ∈ (2^nd ‘𝐴)) → 𝑎 <_Q 𝑐)
16	15	3expib 1206	. . . . . . . . . . . . . . . 16 ⊢ (⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P → ((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑐 ∈ (2^nd ‘𝐴)) → 𝑎 <_Q 𝑐))
17	14, 16	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ P → ((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑐 ∈ (2^nd ‘𝐴)) → 𝑎 <_Q 𝑐))
18		prop 7449	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
19		prltlu 7461	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ 𝑏 ∈ (1^st ‘𝐵) ∧ 𝑑 ∈ (2^nd ‘𝐵)) → 𝑏 <_Q 𝑑)
20	19	3expib 1206	. . . . . . . . . . . . . . . 16 ⊢ (⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P → ((𝑏 ∈ (1^st ‘𝐵) ∧ 𝑑 ∈ (2^nd ‘𝐵)) → 𝑏 <_Q 𝑑))
21	18, 20	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ P → ((𝑏 ∈ (1^st ‘𝐵) ∧ 𝑑 ∈ (2^nd ‘𝐵)) → 𝑏 <_Q 𝑑))
22	17, 21	im2anan9 598	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑐 ∈ (2^nd ‘𝐴)) ∧ (𝑏 ∈ (1^st ‘𝐵) ∧ 𝑑 ∈ (2^nd ‘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 7484	. . . . . . . . . . . . . 14 ⊢ ((𝑎 <_Q 𝑐 ∧ 𝑏 <_Q 𝑑) → (𝑎𝐺𝑏) <_Q (𝑐𝐺𝑑))
26	22, 25	syl6 33	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑐 ∈ (2^nd ‘𝐴)) ∧ (𝑏 ∈ (1^st ‘𝐵) ∧ 𝑑 ∈ (2^nd ‘𝐵))) → (𝑎𝐺𝑏) <_Q (𝑐𝐺𝑑)))
27	13, 26	syl5bir 153	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑏 ∈ (1^st ‘𝐵)) ∧ (𝑐 ∈ (2^nd ‘𝐴) ∧ 𝑑 ∈ (2^nd ‘𝐵))) → (𝑎𝐺𝑏) <_Q (𝑐𝐺𝑑)))
28	27	imp 124	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ ((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑏 ∈ (1^st ‘𝐵)) ∧ (𝑐 ∈ (2^nd ‘𝐴) ∧ 𝑑 ∈ (2^nd ‘𝐵)))) → (𝑎𝐺𝑏) <_Q (𝑐𝐺𝑑))
29	28	adantlr 477	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2^nd ‘(𝐴𝐹𝐵)))) ∧ ((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑏 ∈ (1^st ‘𝐵)) ∧ (𝑐 ∈ (2^nd ‘𝐴) ∧ 𝑑 ∈ (2^nd ‘𝐵)))) → (𝑎𝐺𝑏) <_Q (𝑐𝐺𝑑))
30	29	adantrlr 485	. . . . . . . . 9 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2^nd ‘(𝐴𝐹𝐵)))) ∧ (((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑏 ∈ (1^st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ (𝑐 ∈ (2^nd ‘𝐴) ∧ 𝑑 ∈ (2^nd ‘𝐵)))) → (𝑎𝐺𝑏) <_Q (𝑐𝐺𝑑))
31	30	adantrrr 487	. . . . . . . 8 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2^nd ‘(𝐴𝐹𝐵)))) ∧ (((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑏 ∈ (1^st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2^nd ‘𝐴) ∧ 𝑑 ∈ (2^nd ‘𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑)))) → (𝑎𝐺𝑏) <_Q (𝑐𝐺𝑑))
32		eqtr2 2194	. . . . . . . . . . 11 ⊢ ((𝑞 = (𝑎𝐺𝑏) ∧ 𝑞 = (𝑐𝐺𝑑)) → (𝑎𝐺𝑏) = (𝑐𝐺𝑑))
33	32	ad2ant2l 508	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑏 ∈ (1^st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2^nd ‘𝐴) ∧ 𝑑 ∈ (2^nd ‘𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑))) → (𝑎𝐺𝑏) = (𝑐𝐺𝑑))
34	33	adantl 277	. . . . . . . . 9 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2^nd ‘(𝐴𝐹𝐵)))) ∧ (((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑏 ∈ (1^st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2^nd ‘𝐴) ∧ 𝑑 ∈ (2^nd ‘𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑)))) → (𝑎𝐺𝑏) = (𝑐𝐺𝑑))
35		ltsonq 7372	. . . . . . . . . . 11 ⊢ <_Q Or Q
36		ltrelnq 7339	. . . . . . . . . . 11 ⊢ <_Q ⊆ (Q × Q)
37	35, 36	soirri 5015	. . . . . . . . . 10 ⊢ ¬ (𝑎𝐺𝑏) <_Q (𝑎𝐺𝑏)
38		breq2 4002	. . . . . . . . . 10 ⊢ ((𝑎𝐺𝑏) = (𝑐𝐺𝑑) → ((𝑎𝐺𝑏) <_Q (𝑎𝐺𝑏) ↔ (𝑎𝐺𝑏) <_Q (𝑐𝐺𝑑)))
39	37, 38	mtbii 674	. . . . . . . . 9 ⊢ ((𝑎𝐺𝑏) = (𝑐𝐺𝑑) → ¬ (𝑎𝐺𝑏) <_Q (𝑐𝐺𝑑))
40	34, 39	syl 14	. . . . . . . 8 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2^nd ‘(𝐴𝐹𝐵)))) ∧ (((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑏 ∈ (1^st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2^nd ‘𝐴) ∧ 𝑑 ∈ (2^nd ‘𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑)))) → ¬ (𝑎𝐺𝑏) <_Q (𝑐𝐺𝑑))
41	31, 40	pm2.21fal 1373	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2^nd ‘(𝐴𝐹𝐵)))) ∧ (((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑏 ∈ (1^st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2^nd ‘𝐴) ∧ 𝑑 ∈ (2^nd ‘𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑)))) → ⊥)
42	41	ex 115	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2^nd ‘(𝐴𝐹𝐵)))) → ((((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑏 ∈ (1^st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2^nd ‘𝐴) ∧ 𝑑 ∈ (2^nd ‘𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑))) → ⊥))
43	42	exlimdvv 1895	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2^nd ‘(𝐴𝐹𝐵)))) → (∃𝑐∃𝑑(((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑏 ∈ (1^st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2^nd ‘𝐴) ∧ 𝑑 ∈ (2^nd ‘𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑))) → ⊥))
44	43	exlimdvv 1895	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2^nd ‘(𝐴𝐹𝐵)))) → (∃𝑎∃𝑏∃𝑐∃𝑑(((𝑎 ∈ (1^st ‘𝐴) ∧ 𝑏 ∈ (1^st ‘𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2^nd ‘𝐴) ∧ 𝑑 ∈ (2^nd ‘𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑))) → ⊥))
45	12, 44	mpd 13	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2^nd ‘(𝐴𝐹𝐵)))) → ⊥)
46	45	inegd 1372	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ¬ (𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2^nd ‘(𝐴𝐹𝐵))))
47	46	ralrimivw 2549	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2^nd ‘(𝐴𝐹𝐵))))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 978 = wceq 1353 ⊥wfal 1358 ∃wex 1490 ∈ wcel 2146 ∀wral 2453 ∃wrex 2454 {crab 2457 ⟨cop 3592 class class class wbr 3998 ‘cfv 5208 (class class class)co 5865 ∈ cmpo 5867 1^st c1st 6129 2^nd c2nd 6130 Qcnq 7254 <_Q cltq 7259 Pcnp 7265

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 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-iinf 4581

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-ral 2458 df-rex 2459 df-reu 2460 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-eprel 4283 df-id 4287 df-po 4290 df-iso 4291 df-iord 4360 df-on 4362 df-suc 4365 df-iom 4584 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 df-recs 6296 df-irdg 6361 df-oadd 6411 df-omul 6412 df-er 6525 df-ec 6527 df-qs 6531 df-ni 7278 df-mi 7280 df-lti 7281 df-enq 7321 df-nqqs 7322 df-ltnqqs 7327 df-inp 7440

This theorem is referenced by: addclpr 7511 mulclpr 7546

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