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 Description: Lemma for addlocpr 7351. The 𝑄 = (𝐷 +Q 𝐸) case. (Contributed by Jim Kingdon, 6-Dec-2019.)
Hypotheses
Ref Expression
addlocprlem.qppr (𝜑 → (𝑄 +Q (𝑃 +Q 𝑃)) = 𝑅)
addlocprlem.du (𝜑𝑈 <Q (𝐷 +Q 𝑃))
addlocprlem.et (𝜑𝑇 <Q (𝐸 +Q 𝑃))
Assertion
Ref Expression
addlocprlemeq (𝜑 → (𝑄 = (𝐷 +Q 𝐸) → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))

StepHypRef Expression
1 addlocprlem.a . . . . . 6 (𝜑𝐴P)
2 addlocprlem.b . . . . . 6 (𝜑𝐵P)
3 addlocprlem.qr . . . . . 6 (𝜑𝑄 <Q 𝑅)
4 addlocprlem.p . . . . . 6 (𝜑𝑃Q)
5 addlocprlem.qppr . . . . . 6 (𝜑 → (𝑄 +Q (𝑃 +Q 𝑃)) = 𝑅)
6 addlocprlem.dlo . . . . . 6 (𝜑𝐷 ∈ (1st𝐴))
7 addlocprlem.uup . . . . . 6 (𝜑𝑈 ∈ (2nd𝐴))
8 addlocprlem.du . . . . . 6 (𝜑𝑈 <Q (𝐷 +Q 𝑃))
9 addlocprlem.elo . . . . . 6 (𝜑𝐸 ∈ (1st𝐵))
10 addlocprlem.tup . . . . . 6 (𝜑𝑇 ∈ (2nd𝐵))
11 addlocprlem.et . . . . . 6 (𝜑𝑇 <Q (𝐸 +Q 𝑃))
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11addlocprlemeqgt 7347 . . . . 5 (𝜑 → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
1312adantr 274 . . . 4 ((𝜑𝑄 = (𝐷 +Q 𝐸)) → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
14 oveq1 5781 . . . . 5 (𝑄 = (𝐷 +Q 𝐸) → (𝑄 +Q (𝑃 +Q 𝑃)) = ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
155, 14sylan9req 2193 . . . 4 ((𝜑𝑄 = (𝐷 +Q 𝐸)) → 𝑅 = ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
1613, 15breqtrrd 3956 . . 3 ((𝜑𝑄 = (𝐷 +Q 𝐸)) → (𝑈 +Q 𝑇) <Q 𝑅)
171, 7jca 304 . . . . 5 (𝜑 → (𝐴P𝑈 ∈ (2nd𝐴)))
182, 10jca 304 . . . . 5 (𝜑 → (𝐵P𝑇 ∈ (2nd𝐵)))
19 ltrelnq 7180 . . . . . . . 8 <Q ⊆ (Q × Q)
2019brel 4591 . . . . . . 7 (𝑄 <Q 𝑅 → (𝑄Q𝑅Q))
2120simprd 113 . . . . . 6 (𝑄 <Q 𝑅𝑅Q)
223, 21syl 14 . . . . 5 (𝜑𝑅Q)
23 addnqpru 7345 . . . . 5 ((((𝐴P𝑈 ∈ (2nd𝐴)) ∧ (𝐵P𝑇 ∈ (2nd𝐵))) ∧ 𝑅Q) → ((𝑈 +Q 𝑇) <Q 𝑅𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))
2417, 18, 22, 23syl21anc 1215 . . . 4 (𝜑 → ((𝑈 +Q 𝑇) <Q 𝑅𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))
2524adantr 274 . . 3 ((𝜑𝑄 = (𝐷 +Q 𝐸)) → ((𝑈 +Q 𝑇) <Q 𝑅𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))
2616, 25mpd 13 . 2 ((𝜑𝑄 = (𝐷 +Q 𝐸)) → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵)))
2726ex 114 1 (𝜑 → (𝑄 = (𝐷 +Q 𝐸) → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480   class class class wbr 3929  ‘cfv 5123  (class class class)co 5774  1st c1st 6036  2nd c2nd 6037  Qcnq 7095   +Q cplq 7097
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