Theorem mulnqprlemru 7941

Description: Lemma for mulnqpr 7944. 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 7914	. . . . . 6 ⊢ (𝐴 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ∈ P)
2		nqprlu 7914	. . . . . 6 ⊢ (𝐵 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩ ∈ P)
3		df-imp 7836	. . . . . . 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 7743	. . . . . . 7 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 ·Q ℎ) ∈ Q)
5	3, 4	genpelvu 7880	. . . . . 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 2824	. . . . . . . . . . . . 13 ⊢ 𝑠 ∈ V
9		breq2 4134	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑠 → (𝐴 <Q 𝑢 ↔ 𝐴 <Q 𝑠))
10		ltnqex 7916	. . . . . . . . . . . . . 14 ⊢ {𝑙 ∣ 𝑙 <Q 𝐴} ∈ V
11		gtnqex 7917	. . . . . . . . . . . . . 14 ⊢ {𝑢 ∣ 𝐴 <Q 𝑢} ∈ V
12	10, 11	op2nd 6381	. . . . . . . . . . . . 13 ⊢ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) = {𝑢 ∣ 𝐴 <Q 𝑢}
13	8, 9, 12	elab2 2974	. . . . . . . . . . . 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 494	. . . . . . . . . 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 2824	. . . . . . . . . . . . 13 ⊢ 𝑡 ∈ V
18		breq2 4134	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑡 → (𝐵 <Q 𝑢 ↔ 𝐵 <Q 𝑡))
19		ltnqex 7916	. . . . . . . . . . . . . 14 ⊢ {𝑙 ∣ 𝑙 <Q 𝐵} ∈ V
20		gtnqex 7917	. . . . . . . . . . . . . 14 ⊢ {𝑢 ∣ 𝐵 <Q 𝑢} ∈ V
21	19, 20	op2nd 6381	. . . . . . . . . . . . 13 ⊢ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩) = {𝑢 ∣ 𝐵 <Q 𝑢}
22	17, 18, 21	elab2 2974	. . . . . . . . . . . 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 495	. . . . . . . . . 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 7732	. . . . . . . . . . . 12 ⊢ <Q ⊆ (Q × Q)
27	26	brel 4827	. . . . . . . . . . 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 4827	. . . . . . . . . . 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 7772	. . . . . . . . . 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 415	. . . . . . . . 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 437	. . . . . . . 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 4134	. . . . . . . . 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 2824	. . . . . . . 8 ⊢ 𝑟 ∈ V
38		breq2 4134	. . . . . . . 8 ⊢ (𝑢 = 𝑟 → ((𝐴 ·Q 𝐵) <Q 𝑢 ↔ (𝐴 ·Q 𝐵) <Q 𝑟))
39		ltnqex 7916	. . . . . . . . 9 ⊢ {𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)} ∈ V
40		gtnqex 7917	. . . . . . . . 9 ⊢ {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢} ∈ V
41	39, 40	op2nd 6381	. . . . . . . 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 2974	. . . . . . 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 2676	. . . 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 3254	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}⟩))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  {cab 2224  ∃wrex 2529   ⊆ wss 3220  ⟨cop 3712   class class class wbr 4130  ‘cfv 5377  (class class class)co 6085  2^nd c2nd 6373  Qcnq 7647   ·_Q cmq 7650   <_Q cltq 7652  Pcnp 7658   ·_P cmp 7661

This proof depends on 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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof 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 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-eprel 4434  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-1o 6687  df-oadd 6691  df-omul 6692  df-er 6807  df-ec 6809  df-qs 6813  df-ni 7671  df-pli 7672  df-mi 7673  df-lti 7674  df-plpq 7711  df-mpq 7712  df-enq 7714  df-nqqs 7715  df-plqqs 7716  df-mqqs 7717  df-1nqqs 7718  df-rq 7719  df-ltnqqs 7720  df-inp 7833  df-imp 7836

This theorem is used by:  mulnqprlemfl  7942  mulnqpr  7944