Theorem mulnqprlemru 7406

Description: Lemma for mulnqpr 7409. 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 7379	. . . . . 6 ⊢ (𝐴 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ∈ P)
2		nqprlu 7379	. . . . . 6 ⊢ (𝐵 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩ ∈ P)
3		df-imp 7301	. . . . . . 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 7208	. . . . . . 7 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 ·Q ℎ) ∈ Q)
5	3, 4	genpelvu 7345	. . . . . 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 287	. . . . 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 294	. . . 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 2692	. . . . . . . . . . . . 13 ⊢ 𝑠 ∈ V
9		breq2 3941	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑠 → (𝐴 <Q 𝑢 ↔ 𝐴 <Q 𝑠))
10		ltnqex 7381	. . . . . . . . . . . . . 14 ⊢ {𝑙 ∣ 𝑙 <Q 𝐴} ∈ V
11		gtnqex 7382	. . . . . . . . . . . . . 14 ⊢ {𝑢 ∣ 𝐴 <Q 𝑢} ∈ V
12	10, 11	op2nd 6053	. . . . . . . . . . . . 13 ⊢ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) = {𝑢 ∣ 𝐴 <Q 𝑢}
13	8, 9, 12	elab2 2836	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ↔ 𝐴 <Q 𝑠)
14	13	biimpi 119	. . . . . . . . . . 11 ⊢ (𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) → 𝐴 <Q 𝑠)
15	14	ad2antrl 482	. . . . . . . . . 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 274	. . . . . . . . 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 2692	. . . . . . . . . . . . 13 ⊢ 𝑡 ∈ V
18		breq2 3941	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑡 → (𝐵 <Q 𝑢 ↔ 𝐵 <Q 𝑡))
19		ltnqex 7381	. . . . . . . . . . . . . 14 ⊢ {𝑙 ∣ 𝑙 <Q 𝐵} ∈ V
20		gtnqex 7382	. . . . . . . . . . . . . 14 ⊢ {𝑢 ∣ 𝐵 <Q 𝑢} ∈ V
21	19, 20	op2nd 6053	. . . . . . . . . . . . 13 ⊢ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩) = {𝑢 ∣ 𝐵 <Q 𝑢}
22	17, 18, 21	elab2 2836	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩) ↔ 𝐵 <Q 𝑡)
23	22	biimpi 119	. . . . . . . . . . 11 ⊢ (𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩) → 𝐵 <Q 𝑡)
24	23	ad2antll 483	. . . . . . . . . 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 274	. . . . . . . . 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 7197	. . . . . . . . . . . 12 ⊢ <Q ⊆ (Q × Q)
27	26	brel 4599	. . . . . . . . . . 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 4599	. . . . . . . . . . 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 7237	. . . . . . . . . 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 409	. . . . . . . . 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 430	. . . . . . . 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 3941	. . . . . . . . 9 ⊢ (𝑟 = (𝑠 ·Q 𝑡) → ((𝐴 ·Q 𝐵) <Q 𝑟 ↔ (𝐴 ·Q 𝐵) <Q (𝑠 ·Q 𝑡)))
35	34	adantl 275	. . . . . . . 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 166	. . . . . . 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 2692	. . . . . . . 8 ⊢ 𝑟 ∈ V
38		breq2 3941	. . . . . . . 8 ⊢ (𝑢 = 𝑟 → ((𝐴 ·Q 𝐵) <Q 𝑢 ↔ (𝐴 ·Q 𝐵) <Q 𝑟))
39		ltnqex 7381	. . . . . . . . 9 ⊢ {𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)} ∈ V
40		gtnqex 7382	. . . . . . . . 9 ⊢ {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢} ∈ V
41	39, 40	op2nd 6053	. . . . . . . 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 2836	. . . . . . 7 ⊢ (𝑟 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) ↔ (𝐴 ·Q 𝐵) <Q 𝑟)
43	36, 42	sylibr 133	. . . . . 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 114	. . . . 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 2560	. . . 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 114	. 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 3108	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 103   ↔ wb 104   = wceq 1332   ∈ wcel 1481  {cab 2126  ∃wrex 2418   ⊆ wss 3076  ⟨cop 3535   class class class wbr 3937  ‘cfv 5131  (class class class)co 5782  2^nd c2nd 6045  Qcnq 7112   ·_Q cmq 7115   <_Q cltq 7117  Pcnp 7123   ·_P cmp 7126

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-eprel 4219  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-1o 6321  df-oadd 6325  df-omul 6326  df-er 6437  df-ec 6439  df-qs 6443  df-ni 7136  df-pli 7137  df-mi 7138  df-lti 7139  df-plpq 7176  df-mpq 7177  df-enq 7179  df-nqqs 7180  df-plqqs 7181  df-mqqs 7182  df-1nqqs 7183  df-rq 7184  df-ltnqqs 7185  df-inp 7298  df-imp 7301

This theorem is referenced by:  mulnqprlemfl  7407  mulnqpr  7409