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 Description: Lemma for addnqpr 6542. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
Assertion
Ref Expression
addnqprlemrl ((A Q B Q) → (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 6530 . . . . . 6 (A Q → ⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ P)
2 nqprlu 6530 . . . . . 6 (B Q → ⟨{𝑙𝑙 <Q B}, {uB <Q u}⟩ P)
3 df-iplp 6451 . . . . . . 7 +P = (x P, y P ↦ ⟨{f Qg Q Q (g (1stx) (1sty) f = (g +Q ))}, {f Qg Q Q (g (2ndx) (2ndy) f = (g +Q ))}⟩)
4 addclnq 6359 . . . . . . 7 ((g Q Q) → (g +Q ) Q)
53, 4genpelvl 6495 . . . . . 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 ((A Q B Q) → (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 (((A Q B Q) 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 2554 . . . . . . . . . . . . 13 𝑠 V
9 breq1 3758 . . . . . . . . . . . . 13 (𝑙 = 𝑠 → (𝑙 <Q A𝑠 <Q A))
10 ltnqex 6531 . . . . . . . . . . . . . 14 {𝑙𝑙 <Q A} V
11 gtnqex 6532 . . . . . . . . . . . . . 14 {uA <Q u} V
1210, 11op1st 5715 . . . . . . . . . . . . 13 (1st ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) = {𝑙𝑙 <Q A}
138, 9, 12elab2 2684 . . . . . . . . . . . 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 ((((A Q B Q) 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 (((((A Q B Q) 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 2554 . . . . . . . . . . . . 13 𝑡 V
18 breq1 3758 . . . . . . . . . . . . 13 (𝑙 = 𝑡 → (𝑙 <Q B𝑡 <Q B))
19 ltnqex 6531 . . . . . . . . . . . . . 14 {𝑙𝑙 <Q B} V
20 gtnqex 6532 . . . . . . . . . . . . . 14 {uB <Q u} V
2119, 20op1st 5715 . . . . . . . . . . . . 13 (1st ‘⟨{𝑙𝑙 <Q B}, {uB <Q u}⟩) = {𝑙𝑙 <Q B}
2217, 18, 21elab2 2684 . . . . . . . . . . . 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 ((((A Q B Q) 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 (((((A Q B Q) 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 6349 . . . . . . . . . . . 12 <Q ⊆ (Q × Q)
2726brel 4335 . . . . . . . . . . 11 (𝑠 <Q A → (𝑠 Q A Q))
2816, 27syl 14 . . . . . . . . . 10 (((((A Q B Q) 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 Q))
2926brel 4335 . . . . . . . . . . 11 (𝑡 <Q B → (𝑡 Q B Q))
3025, 29syl 14 . . . . . . . . . 10 (((((A Q B Q) 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 Q))
31 lt2addnq 6388 . . . . . . . . . 10 (((𝑠 Q A Q) (𝑡 Q B Q)) → ((𝑠 <Q A 𝑡 <Q B) → (𝑠 +Q 𝑡) <Q (A +Q B)))
3228, 30, 31syl2anc 391 . . . . . . . . 9 (((((A Q B Q) 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 𝑡 <Q B) → (𝑠 +Q 𝑡) <Q (A +Q B)))
3316, 25, 32mp2and 409 . . . . . . . 8 (((((A Q B Q) 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 3758 . . . . . . . . 9 (𝑟 = (𝑠 +Q 𝑡) → (𝑟 <Q (A +Q B) ↔ (𝑠 +Q 𝑡) <Q (A +Q B)))
3534adantl 262 . . . . . . . 8 (((((A Q B Q) 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 (((((A Q B Q) 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 2554 . . . . . . . 8 𝑟 V
38 breq1 3758 . . . . . . . 8 (𝑙 = 𝑟 → (𝑙 <Q (A +Q B) ↔ 𝑟 <Q (A +Q B)))
39 ltnqex 6531 . . . . . . . . 9 {𝑙𝑙 <Q (A +Q B)} V
40 gtnqex 6532 . . . . . . . . 9 {u ∣ (A +Q B) <Q u} V
4139, 40op1st 5715 . . . . . . . 8 (1st ‘⟨{𝑙𝑙 <Q (A +Q B)}, {u ∣ (A +Q B) <Q u}⟩) = {𝑙𝑙 <Q (A +Q B)}
4237, 38, 41elab2 2684 . . . . . . 7 (𝑟 (1st ‘⟨{𝑙𝑙 <Q (A +Q B)}, {u ∣ (A +Q B) <Q u}⟩) ↔ 𝑟 <Q (A +Q B))
4336, 42sylibr 137 . . . . . 6 (((((A Q B Q) 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 ((((A Q B Q) 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 2434 . . . 4 (((A Q B Q) 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 (((A Q B Q) 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 ((A Q B Q) → (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 2945 1 ((A Q B Q) → (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 1242   ∈ wcel 1390  {cab 2023  ∃wrex 2301   ⊆ wss 2911  ⟨cop 3370   class class class wbr 3755  ‘cfv 4845  (class class class)co 5455  1st c1st 5707  Qcnq 6264   +Q cplq 6266
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