ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  mulnqprlemrl GIF version

Theorem mulnqprlemrl 7574
Description: Lemma for mulnqpr 7578. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
Assertion
Ref Expression
mulnqprlemrl ((𝐴Q𝐵Q) → (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ⊆ (1st ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩))
Distinct variable groups:   𝐴,𝑙,𝑢   𝐵,𝑙,𝑢

Proof of Theorem mulnqprlemrl
Dummy variables 𝑓 𝑔 𝑟 𝑠 𝑡 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nqprlu 7548 . . . . . 6 (𝐴Q → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ∈ P)
2 nqprlu 7548 . . . . . 6 (𝐵Q → ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩ ∈ P)
3 df-imp 7470 . . . . . . 7 ·P = (𝑥P, 𝑦P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑥) ∧ ∈ (1st𝑦) ∧ 𝑓 = (𝑔 ·Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑥) ∧ ∈ (2nd𝑦) ∧ 𝑓 = (𝑔 ·Q ))}⟩)
4 mulclnq 7377 . . . . . . 7 ((𝑔QQ) → (𝑔 ·Q ) ∈ Q)
53, 4genpelvl 7513 . . . . . 6 ((⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ∈ P ∧ ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩ ∈ P) → (𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ↔ ∃𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)∃𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)𝑟 = (𝑠 ·Q 𝑡)))
61, 2, 5syl2an 289 . . . . 5 ((𝐴Q𝐵Q) → (𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ↔ ∃𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)∃𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)𝑟 = (𝑠 ·Q 𝑡)))
76biimpa 296 . . . 4 (((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) → ∃𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)∃𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)𝑟 = (𝑠 ·Q 𝑡))
8 vex 2742 . . . . . . . . . . . . 13 𝑠 ∈ V
9 breq1 4008 . . . . . . . . . . . . 13 (𝑙 = 𝑠 → (𝑙 <Q 𝐴𝑠 <Q 𝐴))
10 ltnqex 7550 . . . . . . . . . . . . . 14 {𝑙𝑙 <Q 𝐴} ∈ V
11 gtnqex 7551 . . . . . . . . . . . . . 14 {𝑢𝐴 <Q 𝑢} ∈ V
1210, 11op1st 6149 . . . . . . . . . . . . 13 (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) = {𝑙𝑙 <Q 𝐴}
138, 9, 12elab2 2887 . . . . . . . . . . . 12 (𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ↔ 𝑠 <Q 𝐴)
1413biimpi 120 . . . . . . . . . . 11 (𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) → 𝑠 <Q 𝐴)
1514ad2antrl 490 . . . . . . . . . 10 ((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) → 𝑠 <Q 𝐴)
1615adantr 276 . . . . . . . . 9 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 ·Q 𝑡)) → 𝑠 <Q 𝐴)
17 vex 2742 . . . . . . . . . . . . 13 𝑡 ∈ V
18 breq1 4008 . . . . . . . . . . . . 13 (𝑙 = 𝑡 → (𝑙 <Q 𝐵𝑡 <Q 𝐵))
19 ltnqex 7550 . . . . . . . . . . . . . 14 {𝑙𝑙 <Q 𝐵} ∈ V
20 gtnqex 7551 . . . . . . . . . . . . . 14 {𝑢𝐵 <Q 𝑢} ∈ V
2119, 20op1st 6149 . . . . . . . . . . . . 13 (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩) = {𝑙𝑙 <Q 𝐵}
2217, 18, 21elab2 2887 . . . . . . . . . . . 12 (𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩) ↔ 𝑡 <Q 𝐵)
2322biimpi 120 . . . . . . . . . . 11 (𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩) → 𝑡 <Q 𝐵)
2423ad2antll 491 . . . . . . . . . 10 ((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) → 𝑡 <Q 𝐵)
2524adantr 276 . . . . . . . . 9 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 ·Q 𝑡)) → 𝑡 <Q 𝐵)
26 ltrelnq 7366 . . . . . . . . . . . 12 <Q ⊆ (Q × Q)
2726brel 4680 . . . . . . . . . . 11 (𝑠 <Q 𝐴 → (𝑠Q𝐴Q))
2816, 27syl 14 . . . . . . . . . 10 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 ·Q 𝑡)) → (𝑠Q𝐴Q))
2926brel 4680 . . . . . . . . . . 11 (𝑡 <Q 𝐵 → (𝑡Q𝐵Q))
3025, 29syl 14 . . . . . . . . . 10 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 ·Q 𝑡)) → (𝑡Q𝐵Q))
31 lt2mulnq 7406 . . . . . . . . . 10 (((𝑠Q𝐴Q) ∧ (𝑡Q𝐵Q)) → ((𝑠 <Q 𝐴𝑡 <Q 𝐵) → (𝑠 ·Q 𝑡) <Q (𝐴 ·Q 𝐵)))
3228, 30, 31syl2anc 411 . . . . . . . . 9 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 ·Q 𝑡)) → ((𝑠 <Q 𝐴𝑡 <Q 𝐵) → (𝑠 ·Q 𝑡) <Q (𝐴 ·Q 𝐵)))
3316, 25, 32mp2and 433 . . . . . . . 8 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 ·Q 𝑡)) → (𝑠 ·Q 𝑡) <Q (𝐴 ·Q 𝐵))
34 breq1 4008 . . . . . . . . 9 (𝑟 = (𝑠 ·Q 𝑡) → (𝑟 <Q (𝐴 ·Q 𝐵) ↔ (𝑠 ·Q 𝑡) <Q (𝐴 ·Q 𝐵)))
3534adantl 277 . . . . . . . 8 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 ·Q 𝑡)) → (𝑟 <Q (𝐴 ·Q 𝐵) ↔ (𝑠 ·Q 𝑡) <Q (𝐴 ·Q 𝐵)))
3633, 35mpbird 167 . . . . . . 7 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 ·Q 𝑡)) → 𝑟 <Q (𝐴 ·Q 𝐵))
37 vex 2742 . . . . . . . 8 𝑟 ∈ V
38 breq1 4008 . . . . . . . 8 (𝑙 = 𝑟 → (𝑙 <Q (𝐴 ·Q 𝐵) ↔ 𝑟 <Q (𝐴 ·Q 𝐵)))
39 ltnqex 7550 . . . . . . . . 9 {𝑙𝑙 <Q (𝐴 ·Q 𝐵)} ∈ V
40 gtnqex 7551 . . . . . . . . 9 {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢} ∈ V
4139, 40op1st 6149 . . . . . . . 8 (1st ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) = {𝑙𝑙 <Q (𝐴 ·Q 𝐵)}
4237, 38, 41elab2 2887 . . . . . . 7 (𝑟 ∈ (1st ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) ↔ 𝑟 <Q (𝐴 ·Q 𝐵))
4336, 42sylibr 134 . . . . . 6 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 ·Q 𝑡)) → 𝑟 ∈ (1st ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩))
4443ex 115 . . . . 5 ((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) → (𝑟 = (𝑠 ·Q 𝑡) → 𝑟 ∈ (1st ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩)))
4544rexlimdvva 2602 . . . 4 (((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) → (∃𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)∃𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)𝑟 = (𝑠 ·Q 𝑡) → 𝑟 ∈ (1st ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩)))
467, 45mpd 13 . . 3 (((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) → 𝑟 ∈ (1st ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩))
4746ex 115 . 2 ((𝐴Q𝐵Q) → (𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) → 𝑟 ∈ (1st ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩)))
4847ssrdv 3163 1 ((𝐴Q𝐵Q) → (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ⊆ (1st ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  {cab 2163  wrex 2456  wss 3131  cop 3597   class class class wbr 4005  cfv 5218  (class class class)co 5877  1st c1st 6141  Qcnq 7281   ·Q cmq 7284   <Q cltq 7286  Pcnp 7292   ·P cmp 7295
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354  df-inp 7467  df-imp 7470
This theorem is referenced by:  mulnqprlemfu  7577  mulnqpr  7578
  Copyright terms: Public domain W3C validator