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 Description: Lemma for addnqpr 6659. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
Assertion
Ref Expression
addnqprlemrl ((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

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-iplp 6566 . . . . . . 7 +P = (𝑥P, 𝑦P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑥) ∧ ∈ (1st𝑦) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑥) ∧ ∈ (2nd𝑦) ∧ 𝑓 = (𝑔 +Q ))}⟩)
4 addclnq 6473 . . . . . . 7 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
53, 4genpelvl 6610 . . . . . 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 273 . . . . 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 280 . . . 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 2560 . . . . . . . . . . . . 13 𝑠 ∈ V
9 breq1 3767 . . . . . . . . . . . . 13 (𝑙 = 𝑠 → (𝑙 <Q 𝐴𝑠 <Q 𝐴))
10 ltnqex 6647 . . . . . . . . . . . . . 14 {𝑙𝑙 <Q 𝐴} ∈ V
11 gtnqex 6648 . . . . . . . . . . . . . 14 {𝑢𝐴 <Q 𝑢} ∈ V
1210, 11op1st 5773 . . . . . . . . . . . . 13 (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) = {𝑙𝑙 <Q 𝐴}
138, 9, 12elab2 2690 . . . . . . . . . . . 12 (𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) ↔ 𝑠 <Q 𝐴)
1413biimpi 113 . . . . . . . . . . 11 (𝑠 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩) → 𝑠 <Q 𝐴)
1514ad2antrl 459 . . . . . . . . . 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 261 . . . . . . . . 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 2560 . . . . . . . . . . . . 13 𝑡 ∈ V
18 breq1 3767 . . . . . . . . . . . . 13 (𝑙 = 𝑡 → (𝑙 <Q 𝐵𝑡 <Q 𝐵))
19 ltnqex 6647 . . . . . . . . . . . . . 14 {𝑙𝑙 <Q 𝐵} ∈ V
20 gtnqex 6648 . . . . . . . . . . . . . 14 {𝑢𝐵 <Q 𝑢} ∈ V
2119, 20op1st 5773 . . . . . . . . . . . . 13 (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩) = {𝑙𝑙 <Q 𝐵}
2217, 18, 21elab2 2690 . . . . . . . . . . . 12 (𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩) ↔ 𝑡 <Q 𝐵)
2322biimpi 113 . . . . . . . . . . 11 (𝑡 ∈ (1st ‘⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩) → 𝑡 <Q 𝐵)
2423ad2antll 460 . . . . . . . . . 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 261 . . . . . . . . 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 6463 . . . . . . . . . . . 12 <Q ⊆ (Q × Q)
2726brel 4392 . . . . . . . . . . 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 4392 . . . . . . . . . . 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 lt2addnq 6502 . . . . . . . . . 10 (((𝑠Q𝐴Q) ∧ (𝑡Q𝐵Q)) → ((𝑠 <Q 𝐴𝑡 <Q 𝐵) → (𝑠 +Q 𝑡) <Q (𝐴 +Q 𝐵)))
3228, 30, 31syl2anc 391 . . . . . . . . 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 409 . . . . . . . 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 3767 . . . . . . . . 9 (𝑟 = (𝑠 +Q 𝑡) → (𝑟 <Q (𝐴 +Q 𝐵) ↔ (𝑠 +Q 𝑡) <Q (𝐴 +Q 𝐵)))
3534adantl 262 . . . . . . . 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 156 . . . . . . 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 2560 . . . . . . . 8 𝑟 ∈ V
38 breq1 3767 . . . . . . . 8 (𝑙 = 𝑟 → (𝑙 <Q (𝐴 +Q 𝐵) ↔ 𝑟 <Q (𝐴 +Q 𝐵)))
39 ltnqex 6647 . . . . . . . . 9 {𝑙𝑙 <Q (𝐴 +Q 𝐵)} ∈ V
40 gtnqex 6648 . . . . . . . . 9 {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢} ∈ V
4139, 40op1st 5773 . . . . . . . 8 (1st ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) = {𝑙𝑙 <Q (𝐴 +Q 𝐵)}
4237, 38, 41elab2 2690 . . . . . . 7 (𝑟 ∈ (1st ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) ↔ 𝑟 <Q (𝐴 +Q 𝐵))
4336, 42sylibr 137 . . . . . 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 108 . . . . 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 2440 . . . 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 108 . 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 2951 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 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  1st c1st 5765  Qcnq 6378   +Q cplq 6380
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