Theorem mulnqprlemru 7794

Description: Lemma for mulnqpr 7797. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.)

Assertion

Ref	Expression
mulnqprlemru	⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)) ⊆ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩))

Distinct variable groups:   𝐴,𝑙,𝑢   𝐵,𝑙,𝑢

Proof of Theorem mulnqprlemru

Dummy variables 𝑓 𝑔 ℎ 𝑟 𝑠 𝑡 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nqprlu 7767	. . . . . 6 ⊢ (𝐴 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ∈ P)
2		nqprlu 7767	. . . . . 6 ⊢ (𝐵 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩ ∈ P)
3		df-imp 7689	. . . . . . 7 ⊢ ·P = (𝑥 ∈ P, 𝑦 ∈ P ↦ ⟨{𝑓 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (1st ‘𝑥) ∧ ℎ ∈ (1st ‘𝑦) ∧ 𝑓 = (𝑔 ·Q ℎ))}, {𝑓 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (2nd ‘𝑥) ∧ ℎ ∈ (2nd ‘𝑦) ∧ 𝑓 = (𝑔 ·Q ℎ))}⟩)
4		mulclnq 7596	. . . . . . 7 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 ·Q ℎ) ∈ Q)
5	3, 4	genpelvu 7733	. . . . . 6 ⊢ ((⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ∈ P ∧ ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩ ∈ P) → (𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)) ↔ ∃𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩)∃𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)𝑟 = (𝑠 ·Q 𝑡)))
6	1, 2, 5	syl2an 289	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)) ↔ ∃𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩)∃𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)𝑟 = (𝑠 ·Q 𝑡)))
7	6	biimpa 296	. . . 4 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) → ∃𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩)∃𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)𝑟 = (𝑠 ·Q 𝑡))
8		vex 2805	. . . . . . . . . . . . 13 ⊢ 𝑠 ∈ V
9		breq2 4092	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑠 → (𝐴 <Q 𝑢 ↔ 𝐴 <Q 𝑠))
10		ltnqex 7769	. . . . . . . . . . . . . 14 ⊢ {𝑙 ∣ 𝑙 <Q 𝐴} ∈ V
11		gtnqex 7770	. . . . . . . . . . . . . 14 ⊢ {𝑢 ∣ 𝐴 <Q 𝑢} ∈ V
12	10, 11	op2nd 6310	. . . . . . . . . . . . 13 ⊢ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) = {𝑢 ∣ 𝐴 <Q 𝑢}
13	8, 9, 12	elab2 2954	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ↔ 𝐴 <Q 𝑠)
14	13	biimpi 120	. . . . . . . . . . 11 ⊢ (𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) → 𝐴 <Q 𝑠)
15	14	ad2antrl 490	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) → 𝐴 <Q 𝑠)
16	15	adantr 276	. . . . . . . . 9 ⊢ (((((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 ·Q 𝑡)) → 𝐴 <Q 𝑠)
17		vex 2805	. . . . . . . . . . . . 13 ⊢ 𝑡 ∈ V
18		breq2 4092	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑡 → (𝐵 <Q 𝑢 ↔ 𝐵 <Q 𝑡))
19		ltnqex 7769	. . . . . . . . . . . . . 14 ⊢ {𝑙 ∣ 𝑙 <Q 𝐵} ∈ V
20		gtnqex 7770	. . . . . . . . . . . . . 14 ⊢ {𝑢 ∣ 𝐵 <Q 𝑢} ∈ V
21	19, 20	op2nd 6310	. . . . . . . . . . . . 13 ⊢ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩) = {𝑢 ∣ 𝐵 <Q 𝑢}
22	17, 18, 21	elab2 2954	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩) ↔ 𝐵 <Q 𝑡)
23	22	biimpi 120	. . . . . . . . . . 11 ⊢ (𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩) → 𝐵 <Q 𝑡)
24	23	ad2antll 491	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) → 𝐵 <Q 𝑡)
25	24	adantr 276	. . . . . . . . 9 ⊢ (((((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 ·Q 𝑡)) → 𝐵 <Q 𝑡)
26		ltrelnq 7585	. . . . . . . . . . . 12 ⊢ <Q ⊆ (Q × Q)
27	26	brel 4778	. . . . . . . . . . 11 ⊢ (𝐴 <Q 𝑠 → (𝐴 ∈ Q ∧ 𝑠 ∈ Q))
28	16, 27	syl 14	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 ·Q 𝑡)) → (𝐴 ∈ Q ∧ 𝑠 ∈ Q))
29	26	brel 4778	. . . . . . . . . . 11 ⊢ (𝐵 <Q 𝑡 → (𝐵 ∈ Q ∧ 𝑡 ∈ Q))
30	25, 29	syl 14	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 ·Q 𝑡)) → (𝐵 ∈ Q ∧ 𝑡 ∈ Q))
31		lt2mulnq 7625	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Q ∧ 𝑠 ∈ Q) ∧ (𝐵 ∈ Q ∧ 𝑡 ∈ Q)) → ((𝐴 <Q 𝑠 ∧ 𝐵 <Q 𝑡) → (𝐴 ·Q 𝐵) <Q (𝑠 ·Q 𝑡)))
32	28, 30, 31	syl2anc 411	. . . . . . . . 9 ⊢ (((((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 ·Q 𝑡)) → ((𝐴 <Q 𝑠 ∧ 𝐵 <Q 𝑡) → (𝐴 ·Q 𝐵) <Q (𝑠 ·Q 𝑡)))
33	16, 25, 32	mp2and 433	. . . . . . . 8 ⊢ (((((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 ·Q 𝑡)) → (𝐴 ·Q 𝐵) <Q (𝑠 ·Q 𝑡))
34		breq2 4092	. . . . . . . . 9 ⊢ (𝑟 = (𝑠 ·Q 𝑡) → ((𝐴 ·Q 𝐵) <Q 𝑟 ↔ (𝐴 ·Q 𝐵) <Q (𝑠 ·Q 𝑡)))
35	34	adantl 277	. . . . . . . 8 ⊢ (((((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 ·Q 𝑡)) → ((𝐴 ·Q 𝐵) <Q 𝑟 ↔ (𝐴 ·Q 𝐵) <Q (𝑠 ·Q 𝑡)))
36	33, 35	mpbird 167	. . . . . . 7 ⊢ (((((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 ·Q 𝑡)) → (𝐴 ·Q 𝐵) <Q 𝑟)
37		vex 2805	. . . . . . . 8 ⊢ 𝑟 ∈ V
38		breq2 4092	. . . . . . . 8 ⊢ (𝑢 = 𝑟 → ((𝐴 ·Q 𝐵) <Q 𝑢 ↔ (𝐴 ·Q 𝐵) <Q 𝑟))
39		ltnqex 7769	. . . . . . . . 9 ⊢ {𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)} ∈ V
40		gtnqex 7770	. . . . . . . . 9 ⊢ {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢} ∈ V
41	39, 40	op2nd 6310	. . . . . . . 8 ⊢ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) = {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}
42	37, 38, 41	elab2 2954	. . . . . . 7 ⊢ (𝑟 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) ↔ (𝐴 ·Q 𝐵) <Q 𝑟)
43	36, 42	sylibr 134	. . . . . 6 ⊢ (((((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 ·Q 𝑡)) → 𝑟 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩))
44	43	ex 115	. . . . 5 ⊢ ((((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) → (𝑟 = (𝑠 ·Q 𝑡) → 𝑟 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩)))
45	44	rexlimdvva 2658	. . . 4 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) → (∃𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩)∃𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)𝑟 = (𝑠 ·Q 𝑡) → 𝑟 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩)))
46	7, 45	mpd 13	. . 3 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) → 𝑟 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩))
47	46	ex 115	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)) → 𝑟 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩)))
48	47	ssrdv 3233	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)) ⊆ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1397   ∈ wcel 2202  {cab 2217  ∃wrex 2511   ⊆ wss 3200  ⟨cop 3672   class class class wbr 4088  ‘cfv 5326  (class class class)co 6018  2^nd c2nd 6302  Qcnq 7500   ·_Q cmq 7503   <_Q cltq 7505  Pcnp 7511   ·_P cmp 7514

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-1o 6582  df-oadd 6586  df-omul 6587  df-er 6702  df-ec 6704  df-qs 6708  df-ni 7524  df-pli 7525  df-mi 7526  df-lti 7527  df-plpq 7564  df-mpq 7565  df-enq 7567  df-nqqs 7568  df-plqqs 7569  df-mqqs 7570  df-1nqqs 7571  df-rq 7572  df-ltnqqs 7573  df-inp 7686  df-imp 7689

This theorem is referenced by:  mulnqprlemfl  7795  mulnqpr  7797