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

Theorem mulnqprlemru 6672
 Description: Lemma for mulnqpr 6675. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
Assertion
Ref Expression
mulnqprlemru ((𝐴Q𝐵Q) → (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ⊆ (2nd ‘⟨{𝑙𝑙 <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 𝑟 𝑠 𝑡 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nqprlu 6645 . . . . . 6 (𝐴Q → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ∈ P)
2 nqprlu 6645 . . . . . 6 (𝐵Q → ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩ ∈ P)
3 df-imp 6567 . . . . . . 7 ·P = (𝑥P, 𝑦P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑥) ∧ ∈ (1st𝑦) ∧ 𝑓 = (𝑔 ·Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑥) ∧ ∈ (2nd𝑦) ∧ 𝑓 = (𝑔 ·Q ))}⟩)
4 mulclnq 6474 . . . . . . 7 ((𝑔QQ) → (𝑔 ·Q ) ∈ Q)
53, 4genpelvu 6611 . . . . . 6 ((⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ∈ P ∧ ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩ ∈ P) → (𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ↔ ∃𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)∃𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)𝑟 = (𝑠 ·Q 𝑡)))
61, 2, 5syl2an 273 . . . . 5 ((𝐴Q𝐵Q) → (𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ↔ ∃𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)∃𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)𝑟 = (𝑠 ·Q 𝑡)))
76biimpa 280 . . . 4 (((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) → ∃𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)∃𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)𝑟 = (𝑠 ·Q 𝑡))
8 vex 2560 . . . . . . . . . . . . 13 𝑠 ∈ V
9 breq2 3768 . . . . . . . . . . . . 13 (𝑢 = 𝑠 → (𝐴 <Q 𝑢𝐴 <Q 𝑠))
10 ltnqex 6647 . . . . . . . . . . . . . 14 {𝑙𝑙 <Q 𝐴} ∈ V
11 gtnqex 6648 . . . . . . . . . . . . . 14 {𝑢𝐴 <Q 𝑢} ∈ V
1210, 11op2nd 5774 . . . . . . . . . . . . 13 (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) = {𝑢𝐴 <Q 𝑢}
138, 9, 12elab2 2690 . . . . . . . . . . . 12 (𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ↔ 𝐴 <Q 𝑠)
1413biimpi 113 . . . . . . . . . . 11 (𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) → 𝐴 <Q 𝑠)
1514ad2antrl 459 . . . . . . . . . 10 ((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) → 𝐴 <Q 𝑠)
1615adantr 261 . . . . . . . . 9 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 ·Q 𝑡)) → 𝐴 <Q 𝑠)
17 vex 2560 . . . . . . . . . . . . 13 𝑡 ∈ V
18 breq2 3768 . . . . . . . . . . . . 13 (𝑢 = 𝑡 → (𝐵 <Q 𝑢𝐵 <Q 𝑡))
19 ltnqex 6647 . . . . . . . . . . . . . 14 {𝑙𝑙 <Q 𝐵} ∈ V
20 gtnqex 6648 . . . . . . . . . . . . . 14 {𝑢𝐵 <Q 𝑢} ∈ V
2119, 20op2nd 5774 . . . . . . . . . . . . 13 (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩) = {𝑢𝐵 <Q 𝑢}
2217, 18, 21elab2 2690 . . . . . . . . . . . 12 (𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩) ↔ 𝐵 <Q 𝑡)
2322biimpi 113 . . . . . . . . . . 11 (𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩) → 𝐵 <Q 𝑡)
2423ad2antll 460 . . . . . . . . . 10 ((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) → 𝐵 <Q 𝑡)
2524adantr 261 . . . . . . . . 9 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 ·Q 𝑡)) → 𝐵 <Q 𝑡)
26 ltrelnq 6463 . . . . . . . . . . . 12 <Q ⊆ (Q × Q)
2726brel 4392 . . . . . . . . . . 11 (𝐴 <Q 𝑠 → (𝐴Q𝑠Q))
2816, 27syl 14 . . . . . . . . . 10 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 ·Q 𝑡)) → (𝐴Q𝑠Q))
2926brel 4392 . . . . . . . . . . 11 (𝐵 <Q 𝑡 → (𝐵Q𝑡Q))
3025, 29syl 14 . . . . . . . . . 10 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 ·Q 𝑡)) → (𝐵Q𝑡Q))
31 lt2mulnq 6503 . . . . . . . . . 10 (((𝐴Q𝑠Q) ∧ (𝐵Q𝑡Q)) → ((𝐴 <Q 𝑠𝐵 <Q 𝑡) → (𝐴 ·Q 𝐵) <Q (𝑠 ·Q 𝑡)))
3228, 30, 31syl2anc 391 . . . . . . . . 9 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 ·Q 𝑡)) → ((𝐴 <Q 𝑠𝐵 <Q 𝑡) → (𝐴 ·Q 𝐵) <Q (𝑠 ·Q 𝑡)))
3316, 25, 32mp2and 409 . . . . . . . 8 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 ·Q 𝑡)) → (𝐴 ·Q 𝐵) <Q (𝑠 ·Q 𝑡))
34 breq2 3768 . . . . . . . . 9 (𝑟 = (𝑠 ·Q 𝑡) → ((𝐴 ·Q 𝐵) <Q 𝑟 ↔ (𝐴 ·Q 𝐵) <Q (𝑠 ·Q 𝑡)))
3534adantl 262 . . . . . . . 8 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 ·Q 𝑡)) → ((𝐴 ·Q 𝐵) <Q 𝑟 ↔ (𝐴 ·Q 𝐵) <Q (𝑠 ·Q 𝑡)))
3633, 35mpbird 156 . . . . . . 7 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 ·Q 𝑡)) → (𝐴 ·Q 𝐵) <Q 𝑟)
37 vex 2560 . . . . . . . 8 𝑟 ∈ V
38 breq2 3768 . . . . . . . 8 (𝑢 = 𝑟 → ((𝐴 ·Q 𝐵) <Q 𝑢 ↔ (𝐴 ·Q 𝐵) <Q 𝑟))
39 ltnqex 6647 . . . . . . . . 9 {𝑙𝑙 <Q (𝐴 ·Q 𝐵)} ∈ V
40 gtnqex 6648 . . . . . . . . 9 {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢} ∈ V
4139, 40op2nd 5774 . . . . . . . 8 (2nd ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) = {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}
4237, 38, 41elab2 2690 . . . . . . 7 (𝑟 ∈ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) ↔ (𝐴 ·Q 𝐵) <Q 𝑟)
4336, 42sylibr 137 . . . . . 6 (((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 ·Q 𝑡)) → 𝑟 ∈ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩))
4443ex 108 . . . . 5 ((((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) → (𝑟 = (𝑠 ·Q 𝑡) → 𝑟 ∈ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩)))
4544rexlimdvva 2440 . . . 4 (((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) → (∃𝑠 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩)∃𝑡 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)𝑟 = (𝑠 ·Q 𝑡) → 𝑟 ∈ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩)))
467, 45mpd 13 . . 3 (((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))) → 𝑟 ∈ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩))
4746ex 108 . 2 ((𝐴Q𝐵Q) → (𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) → 𝑟 ∈ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩)))
4847ssrdv 2951 1 ((𝐴Q𝐵Q) → (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ⊆ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  {cab 2026  ∃wrex 2307   ⊆ wss 2917  ⟨cop 3378   class class class wbr 3764  ‘cfv 4902  (class class class)co 5512  2nd c2nd 5766  Qcnq 6378   ·Q cmq 6381
 Copyright terms: Public domain W3C validator