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 Description: Lemma for addlocpr 6512. 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 (φA P)
2 addlocprlem.b . . . . . 6 (φB P)
3 addlocprlem.qr . . . . . 6 (φ𝑄 <Q 𝑅)
4 addlocprlem.p . . . . . 6 (φ𝑃 Q)
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 6508 . . . . 5 (φ → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
1312adantr 261 . . . 4 ((φ 𝑄 = (𝐷 +Q 𝐸)) → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
14 oveq1 5459 . . . . 5 (𝑄 = (𝐷 +Q 𝐸) → (𝑄 +Q (𝑃 +Q 𝑃)) = ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
155, 14sylan9req 2090 . . . 4 ((φ 𝑄 = (𝐷 +Q 𝐸)) → 𝑅 = ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
1613, 15breqtrrd 3780 . . 3 ((φ 𝑄 = (𝐷 +Q 𝐸)) → (𝑈 +Q 𝑇) <Q 𝑅)
171, 7jca 290 . . . . 5 (φ → (A P 𝑈 (2ndA)))
182, 10jca 290 . . . . 5 (φ → (B P 𝑇 (2ndB)))
19 ltrelnq 6342 . . . . . . . 8 <Q ⊆ (Q × Q)
2019brel 4334 . . . . . . 7 (𝑄 <Q 𝑅 → (𝑄 Q 𝑅 Q))
2120simprd 107 . . . . . 6 (𝑄 <Q 𝑅𝑅 Q)
223, 21syl 14 . . . . 5 (φ𝑅 Q)
23 addnqpru 6506 . . . . 5 ((((A P 𝑈 (2ndA)) (B P 𝑇 (2ndB))) 𝑅 Q) → ((𝑈 +Q 𝑇) <Q 𝑅𝑅 (2nd ‘(A +P B))))
2417, 18, 22, 23syl21anc 1133 . . . 4 (φ → ((𝑈 +Q 𝑇) <Q 𝑅𝑅 (2nd ‘(A +P B))))
2524adantr 261 . . 3 ((φ 𝑄 = (𝐷 +Q 𝐸)) → ((𝑈 +Q 𝑇) <Q 𝑅𝑅 (2nd ‘(A +P B))))
2616, 25mpd 13 . 2 ((φ 𝑄 = (𝐷 +Q 𝐸)) → 𝑅 (2nd ‘(A +P B)))
2726ex 108 1 (φ → (𝑄 = (𝐷 +Q 𝐸) → 𝑅 (2nd ‘(A +P B))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390   class class class wbr 3754  ‘cfv 4844  (class class class)co 5452  1st c1st 5704  2nd c2nd 5705  Qcnq 6257   +Q cplq 6259
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