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Theorem genpdisj 7880

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 7869	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)) ↔ ∃𝑎 ∈ (1^st ‘𝐴)∃𝑏 ∈ (1^st ‘𝐵)𝑞 = (𝑎𝐺𝑏)))
4		r2ex 2570	. . . . . . . 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 7870	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑞 ∈ (2^nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑐 ∈ (2^nd ‘𝐴)∃𝑑 ∈ (2^nd ‘𝐵)𝑞 = (𝑐𝐺𝑑)))
7		r2ex 2570	. . . . . . . 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 477	. . . . . 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 1994	. . . . . 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 592	. . . . . . . . . . . . 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 7832	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
15		prltlu 7844	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑎 ∈ (1^st ‘𝐴) ∧ 𝑐 ∈ (2^nd ‘𝐴)) → 𝑎 <_Q 𝑐)
16	15	3expib 1237	. . . . . . . . . . . . . . . 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 7832	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
19		prltlu 7844	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ 𝑏 ∈ (1^st ‘𝐵) ∧ 𝑑 ∈ (2^nd ‘𝐵)) → 𝑏 <_Q 𝑑)
20	19	3expib 1237	. . . . . . . . . . . . . . . 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 606	. . . . . . . . . . . . . 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 7867	. . . . . . . . . . . . . 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	biimtrrid 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 481	. . . . . . . . . 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 489	. . . . . . . . 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 491	. . . . . . . 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 2257	. . . . . . . . . . 11 ⊢ ((𝑞 = (𝑎𝐺𝑏) ∧ 𝑞 = (𝑐𝐺𝑑)) → (𝑎𝐺𝑏) = (𝑐𝐺𝑑))
33	32	ad2ant2l 512	. . . . . . . . . 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 7755	. . . . . . . . . . 11 ⊢ <_Q Or Q
36		ltrelnq 7722	. . . . . . . . . . 11 ⊢ <_Q ⊆ (Q × Q)
37	35, 36	soirri 5177	. . . . . . . . . 10 ⊢ ¬ (𝑎𝐺𝑏) <_Q (𝑎𝐺𝑏)
38		breq2 4129	. . . . . . . . . 10 ⊢ ((𝑎𝐺𝑏) = (𝑐𝐺𝑑) → ((𝑎𝐺𝑏) <_Q (𝑎𝐺𝑏) ↔ (𝑎𝐺𝑏) <_Q (𝑐𝐺𝑑)))
39	37, 38	mtbii 685	. . . . . . . . 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 1422	. . . . . . 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 1953	. . . . 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 1953	. . . 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 1421	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ¬ (𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2^nd ‘(𝐴𝐹𝐵))))
47	46	ralrimivw 2624	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 1009 = wceq 1402 ⊥wfal 1407 ∃wex 1545 ∈ wcel 2209 ∀wral 2528 ∃wrex 2529 {crab 2532 ⟨cop 3708 class class class wbr 4125 ‘cfv 5372 (class class class)co 6075 ∈ cmpo 6077 1^st c1st 6362 2^nd c2nd 6363 Qcnq 7637 <_Q cltq 7642 Pcnp 7648

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4241 ax-sep 4244 ax-nul 4254 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-iinf 4730

This theorem depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-int 3966 df-iun 4009 df-br 4126 df-opab 4188 df-mpt 4189 df-tr 4225 df-eprel 4429 df-id 4433 df-po 4436 df-iso 4437 df-iord 4506 df-on 4508 df-suc 4511 df-iom 4733 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-f1 5377 df-fo 5378 df-f1o 5379 df-fv 5380 df-ov 6078 df-oprab 6079 df-mpo 6080 df-1st 6364 df-2nd 6365 df-recs 6566 df-irdg 6631 df-oadd 6681 df-omul 6682 df-er 6797 df-ec 6799 df-qs 6803 df-ni 7661 df-mi 7663 df-lti 7664 df-enq 7704 df-nqqs 7705 df-ltnqqs 7710 df-inp 7823

This theorem is referenced by: addclpr 7894 mulclpr 7929

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