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 Description: Lemma for addlocpr 6385. The result, in deduction form. (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
addlocprlem (φ → (𝑄 (1st ‘(A +P B)) 𝑅 (2nd ‘(A +P B))))

StepHypRef Expression
1 addlocprlem.qr . . . 4 (φ𝑄 <Q 𝑅)
2 ltrelnq 6218 . . . . . 6 <Q ⊆ (Q × Q)
32brel 4315 . . . . 5 (𝑄 <Q 𝑅 → (𝑄 Q 𝑅 Q))
43simpld 105 . . . 4 (𝑄 <Q 𝑅𝑄 Q)
51, 4syl 14 . . 3 (φ𝑄 Q)
6 addlocprlem.a . . . . . 6 (φA P)
7 prop 6323 . . . . . 6 (A P → ⟨(1stA), (2ndA)⟩ P)
86, 7syl 14 . . . . 5 (φ → ⟨(1stA), (2ndA)⟩ P)
9 addlocprlem.dlo . . . . 5 (φ𝐷 (1stA))
10 elprnql 6329 . . . . 5 ((⟨(1stA), (2ndA)⟩ P 𝐷 (1stA)) → 𝐷 Q)
118, 9, 10syl2anc 393 . . . 4 (φ𝐷 Q)
12 addlocprlem.b . . . . . 6 (φB P)
13 prop 6323 . . . . . 6 (B P → ⟨(1stB), (2ndB)⟩ P)
1412, 13syl 14 . . . . 5 (φ → ⟨(1stB), (2ndB)⟩ P)
15 addlocprlem.elo . . . . 5 (φ𝐸 (1stB))
16 elprnql 6329 . . . . 5 ((⟨(1stB), (2ndB)⟩ P 𝐸 (1stB)) → 𝐸 Q)
1714, 15, 16syl2anc 393 . . . 4 (φ𝐸 Q)
18 addclnq 6228 . . . 4 ((𝐷 Q 𝐸 Q) → (𝐷 +Q 𝐸) Q)
1911, 17, 18syl2anc 393 . . 3 (φ → (𝐷 +Q 𝐸) Q)
20 nqtri3or 6249 . . 3 ((𝑄 Q (𝐷 +Q 𝐸) Q) → (𝑄 <Q (𝐷 +Q 𝐸) 𝑄 = (𝐷 +Q 𝐸) (𝐷 +Q 𝐸) <Q 𝑄))
215, 19, 20syl2anc 393 . 2 (φ → (𝑄 <Q (𝐷 +Q 𝐸) 𝑄 = (𝐷 +Q 𝐸) (𝐷 +Q 𝐸) <Q 𝑄))
22 addlocprlem.p . . . . 5 (φ𝑃 Q)
23 addlocprlem.qppr . . . . 5 (φ → (𝑄 +Q (𝑃 +Q 𝑃)) = 𝑅)
24 addlocprlem.uup . . . . 5 (φ𝑈 (2ndA))
25 addlocprlem.du . . . . 5 (φ𝑈 <Q (𝐷 +Q 𝑃))
26 addlocprlem.tup . . . . 5 (φ𝑇 (2ndB))
27 addlocprlem.et . . . . 5 (φ𝑇 <Q (𝐸 +Q 𝑃))
286, 12, 1, 22, 23, 9, 24, 25, 15, 26, 27addlocprlemlt 6380 . . . 4 (φ → (𝑄 <Q (𝐷 +Q 𝐸) → 𝑄 (1st ‘(A +P B))))
29 orc 620 . . . 4 (𝑄 (1st ‘(A +P B)) → (𝑄 (1st ‘(A +P B)) 𝑅 (2nd ‘(A +P B))))
3028, 29syl6 29 . . 3 (φ → (𝑄 <Q (𝐷 +Q 𝐸) → (𝑄 (1st ‘(A +P B)) 𝑅 (2nd ‘(A +P B)))))
316, 12, 1, 22, 23, 9, 24, 25, 15, 26, 27addlocprlemeq 6382 . . . 4 (φ → (𝑄 = (𝐷 +Q 𝐸) → 𝑅 (2nd ‘(A +P B))))
32 olc 619 . . . 4 (𝑅 (2nd ‘(A +P B)) → (𝑄 (1st ‘(A +P B)) 𝑅 (2nd ‘(A +P B))))
3331, 32syl6 29 . . 3 (φ → (𝑄 = (𝐷 +Q 𝐸) → (𝑄 (1st ‘(A +P B)) 𝑅 (2nd ‘(A +P B)))))
346, 12, 1, 22, 23, 9, 24, 25, 15, 26, 27addlocprlemgt 6383 . . . 4 (φ → ((𝐷 +Q 𝐸) <Q 𝑄𝑅 (2nd ‘(A +P B))))
3534, 32syl6 29 . . 3 (φ → ((𝐷 +Q 𝐸) <Q 𝑄 → (𝑄 (1st ‘(A +P B)) 𝑅 (2nd ‘(A +P B)))))
3630, 33, 353jaod 1183 . 2 (φ → ((𝑄 <Q (𝐷 +Q 𝐸) 𝑄 = (𝐷 +Q 𝐸) (𝐷 +Q 𝐸) <Q 𝑄) → (𝑄 (1st ‘(A +P B)) 𝑅 (2nd ‘(A +P B)))))
3721, 36mpd 13 1 (φ → (𝑄 (1st ‘(A +P B)) 𝑅 (2nd ‘(A +P B))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∨ wo 616   ∨ w3o 870   = wceq 1226   ∈ wcel 1370  ⟨cop 3349   class class class wbr 3734  ‘cfv 4825  (class class class)co 5432  1st c1st 5684  2nd c2nd 5685  Qcnq 6134   +Q cplq 6136
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