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Theorem genpdisj 7485

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 7474	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)) ↔ ∃𝑎 ∈ (1^st ‘𝐴)∃𝑏 ∈ (1^st ‘𝐵)𝑞 = (𝑎𝐺𝑏)))
4		r2ex 2490	. . . . . . . 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 195	. . . . . . 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 7475	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑞 ∈ (2^nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑐 ∈ (2^nd ‘𝐴)∃𝑑 ∈ (2^nd ‘𝐵)𝑞 = (𝑐𝐺𝑑)))
7		r2ex 2490	. . . . . . . 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 195	. . . . . . 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 470	. . . . . 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 1927	. . . . . 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 197	. . . . 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 294	. . . 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 581	. . . . . . . . . . . . 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 7437	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
15		prltlu 7449	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑎 ∈ (1^st ‘𝐴) ∧ 𝑐 ∈ (2^nd ‘𝐴)) → 𝑎 <_Q 𝑐)
16	15	3expib 1201	. . . . . . . . . . . . . . . 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 7437	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
19		prltlu 7449	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ 𝑏 ∈ (1^st ‘𝐵) ∧ 𝑑 ∈ (2^nd ‘𝐵)) → 𝑏 <_Q 𝑑)
20	19	3expib 1201	. . . . . . . . . . . . . . . 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 593	. . . . . . . . . . . . . 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 7472	. . . . . . . . . . . . . 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 152	. . . . . . . . . . . 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 123	. . . . . . . . . . 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 474	. . . . . . . . . 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 482	. . . . . . . . 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 484	. . . . . . . 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 2189	. . . . . . . . . . 11 ⊢ ((𝑞 = (𝑎𝐺𝑏) ∧ 𝑞 = (𝑐𝐺𝑑)) → (𝑎𝐺𝑏) = (𝑐𝐺𝑑))
33	32	ad2ant2l 505	. . . . . . . . . 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 275	. . . . . . . . 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 7360	. . . . . . . . . . 11 ⊢ <_Q Or Q
36		ltrelnq 7327	. . . . . . . . . . 11 ⊢ <_Q ⊆ (Q × Q)
37	35, 36	soirri 5005	. . . . . . . . . 10 ⊢ ¬ (𝑎𝐺𝑏) <_Q (𝑎𝐺𝑏)
38		breq2 3993	. . . . . . . . . 10 ⊢ ((𝑎𝐺𝑏) = (𝑐𝐺𝑑) → ((𝑎𝐺𝑏) <_Q (𝑎𝐺𝑏) ↔ (𝑎𝐺𝑏) <_Q (𝑐𝐺𝑑)))
39	37, 38	mtbii 669	. . . . . . . . 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 1368	. . . . . . 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 114	. . . . . 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 1890	. . . . 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 1890	. . . 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 1367	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ¬ (𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2^nd ‘(𝐴𝐹𝐵))))
47	46	ralrimivw 2544	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 103 ↔ wb 104 ∧ w3a 973 = wceq 1348 ⊥wfal 1353 ∃wex 1485 ∈ wcel 2141 ∀wral 2448 ∃wrex 2449 {crab 2452 ⟨cop 3586 class class class wbr 3989 ‘cfv 5198 (class class class)co 5853 ∈ cmpo 5855 1^st c1st 6117 2^nd c2nd 6118 Qcnq 7242 <_Q cltq 7247 Pcnp 7253

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-eprel 4274 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-oadd 6399 df-omul 6400 df-er 6513 df-ec 6515 df-qs 6519 df-ni 7266 df-mi 7268 df-lti 7269 df-enq 7309 df-nqqs 7310 df-ltnqqs 7315 df-inp 7428

This theorem is referenced by: addclpr 7499 mulclpr 7534

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