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 Description: Lemma for addnqpr 7383. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
Assertion
Ref Expression
addnqprlemfu ((𝐴Q𝐵Q) → (2nd ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) ⊆ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)))
Distinct variable groups:   𝐴,𝑙,𝑢   𝐵,𝑙,𝑢

Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 addnqprlemrl 7379 . . . . . 6 ((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}⟩))
2 ltsonq 7220 . . . . . . . . 9 <Q Or Q
3 addclnq 7197 . . . . . . . . 9 ((𝐴Q𝐵Q) → (𝐴 +Q 𝐵) ∈ Q)
4 sonr 4239 . . . . . . . . 9 (( <Q Or Q ∧ (𝐴 +Q 𝐵) ∈ Q) → ¬ (𝐴 +Q 𝐵) <Q (𝐴 +Q 𝐵))
52, 3, 4sylancr 410 . . . . . . . 8 ((𝐴Q𝐵Q) → ¬ (𝐴 +Q 𝐵) <Q (𝐴 +Q 𝐵))
6 ltrelnq 7187 . . . . . . . . . . . 12 <Q ⊆ (Q × Q)
76brel 4591 . . . . . . . . . . 11 ((𝐴 +Q 𝐵) <Q (𝐴 +Q 𝐵) → ((𝐴 +Q 𝐵) ∈ Q ∧ (𝐴 +Q 𝐵) ∈ Q))
87simpld 111 . . . . . . . . . 10 ((𝐴 +Q 𝐵) <Q (𝐴 +Q 𝐵) → (𝐴 +Q 𝐵) ∈ Q)
9 elex 2697 . . . . . . . . . 10 ((𝐴 +Q 𝐵) ∈ Q → (𝐴 +Q 𝐵) ∈ V)
108, 9syl 14 . . . . . . . . 9 ((𝐴 +Q 𝐵) <Q (𝐴 +Q 𝐵) → (𝐴 +Q 𝐵) ∈ V)
11 breq1 3932 . . . . . . . . 9 (𝑙 = (𝐴 +Q 𝐵) → (𝑙 <Q (𝐴 +Q 𝐵) ↔ (𝐴 +Q 𝐵) <Q (𝐴 +Q 𝐵)))
1210, 11elab3 2836 . . . . . . . 8 ((𝐴 +Q 𝐵) ∈ {𝑙𝑙 <Q (𝐴 +Q 𝐵)} ↔ (𝐴 +Q 𝐵) <Q (𝐴 +Q 𝐵))
135, 12sylnibr 666 . . . . . . 7 ((𝐴Q𝐵Q) → ¬ (𝐴 +Q 𝐵) ∈ {𝑙𝑙 <Q (𝐴 +Q 𝐵)})
14 ltnqex 7371 . . . . . . . . 9 {𝑙𝑙 <Q (𝐴 +Q 𝐵)} ∈ V
15 gtnqex 7372 . . . . . . . . 9 {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢} ∈ V
1614, 15op1st 6044 . . . . . . . 8 (1st ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) = {𝑙𝑙 <Q (𝐴 +Q 𝐵)}
1716eleq2i 2206 . . . . . . 7 ((𝐴 +Q 𝐵) ∈ (1st ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) ↔ (𝐴 +Q 𝐵) ∈ {𝑙𝑙 <Q (𝐴 +Q 𝐵)})
1813, 17sylnibr 666 . . . . . 6 ((𝐴Q𝐵Q) → ¬ (𝐴 +Q 𝐵) ∈ (1st ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩))
191, 18ssneldd 3100 . . . . 5 ((𝐴Q𝐵Q) → ¬ (𝐴 +Q 𝐵) ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)))
2019adantr 274 . . . 4 (((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩)) → ¬ (𝐴 +Q 𝐵) ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)))
21 nqprlu 7369 . . . . . . . 8 (𝐴Q → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ∈ P)
22 nqprlu 7369 . . . . . . . 8 (𝐵Q → ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩ ∈ P)
23 addclpr 7359 . . . . . . . 8 ((⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ∈ P ∧ ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩ ∈ P) → (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩) ∈ P)
2421, 22, 23syl2an 287 . . . . . . 7 ((𝐴Q𝐵Q) → (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩) ∈ P)
25 prop 7297 . . . . . . 7 ((⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩) ∈ P → ⟨(1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)), (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))⟩ ∈ P)
2624, 25syl 14 . . . . . 6 ((𝐴Q𝐵Q) → ⟨(1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)), (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))⟩ ∈ P)
27 vex 2689 . . . . . . . 8 𝑟 ∈ V
28 breq2 3933 . . . . . . . 8 (𝑢 = 𝑟 → ((𝐴 +Q 𝐵) <Q 𝑢 ↔ (𝐴 +Q 𝐵) <Q 𝑟))
2914, 15op2nd 6045 . . . . . . . 8 (2nd ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) = {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}
3027, 28, 29elab2 2832 . . . . . . 7 (𝑟 ∈ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) ↔ (𝐴 +Q 𝐵) <Q 𝑟)
3130biimpi 119 . . . . . 6 (𝑟 ∈ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) → (𝐴 +Q 𝐵) <Q 𝑟)
32 prloc 7313 . . . . . 6 ((⟨(1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)), (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))⟩ ∈ P ∧ (𝐴 +Q 𝐵) <Q 𝑟) → ((𝐴 +Q 𝐵) ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ∨ 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))))
3326, 31, 32syl2an 287 . . . . 5 (((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩)) → ((𝐴 +Q 𝐵) ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ∨ 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))))
3433orcomd 718 . . . 4 (((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩)) → (𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ∨ (𝐴 +Q 𝐵) ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))))
3520, 34ecased 1327 . . 3 (((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩)) → 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)))
3635ex 114 . 2 ((𝐴Q𝐵Q) → (𝑟 ∈ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) → 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))))
3736ssrdv 3103 1 ((𝐴Q𝐵Q) → (2nd ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) ⊆ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 697   ∈ wcel 1480  {cab 2125  Vcvv 2686   ⊆ wss 3071  ⟨cop 3530   class class class wbr 3929   Or wor 4217  ‘cfv 5123  (class class class)co 5774  1st c1st 6036  2nd c2nd 6037  Qcnq 7102   +Q cplq 7104
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