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 Description: Lemma for addlocpr 6385. The (𝐷 +Q 𝐸)
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
addlocprlemgt (φ → ((𝐷 +Q 𝐸) <Q 𝑄𝑅 (2nd ‘(A +P B))))

StepHypRef Expression
1 addlocprlem.a . . . . . . 7 (φA P)
2 addlocprlem.b . . . . . . 7 (φB P)
3 addlocprlem.qr . . . . . . 7 (φ𝑄 <Q 𝑅)
4 addlocprlem.p . . . . . . 7 (φ𝑃 Q)
5 addlocprlem.qppr . . . . . . 7 (φ → (𝑄 +Q (𝑃 +Q 𝑃)) = 𝑅)
6 addlocprlem.dlo . . . . . . 7 (φ𝐷 (1stA))
7 addlocprlem.uup . . . . . . 7 (φ𝑈 (2ndA))
8 addlocprlem.du . . . . . . 7 (φ𝑈 <Q (𝐷 +Q 𝑃))
9 addlocprlem.elo . . . . . . 7 (φ𝐸 (1stB))
10 addlocprlem.tup . . . . . . 7 (φ𝑇 (2ndB))
11 addlocprlem.et . . . . . . 7 (φ𝑇 <Q (𝐸 +Q 𝑃))
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11addlocprlemeqgt 6381 . . . . . 6 (φ → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
1312adantr 261 . . . . 5 ((φ (𝐷 +Q 𝐸) <Q 𝑄) → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
14 prop 6323 . . . . . . . . . . . 12 (A P → ⟨(1stA), (2ndA)⟩ P)
151, 14syl 14 . . . . . . . . . . 11 (φ → ⟨(1stA), (2ndA)⟩ P)
16 elprnql 6329 . . . . . . . . . . 11 ((⟨(1stA), (2ndA)⟩ P 𝐷 (1stA)) → 𝐷 Q)
1715, 6, 16syl2anc 393 . . . . . . . . . 10 (φ𝐷 Q)
18 prop 6323 . . . . . . . . . . . 12 (B P → ⟨(1stB), (2ndB)⟩ P)
192, 18syl 14 . . . . . . . . . . 11 (φ → ⟨(1stB), (2ndB)⟩ P)
20 elprnql 6329 . . . . . . . . . . 11 ((⟨(1stB), (2ndB)⟩ P 𝐸 (1stB)) → 𝐸 Q)
2119, 9, 20syl2anc 393 . . . . . . . . . 10 (φ𝐸 Q)
22 addclnq 6228 . . . . . . . . . 10 ((𝐷 Q 𝐸 Q) → (𝐷 +Q 𝐸) Q)
2317, 21, 22syl2anc 393 . . . . . . . . 9 (φ → (𝐷 +Q 𝐸) Q)
24 ltrelnq 6218 . . . . . . . . . . . 12 <Q ⊆ (Q × Q)
2524brel 4315 . . . . . . . . . . 11 (𝑄 <Q 𝑅 → (𝑄 Q 𝑅 Q))
263, 25syl 14 . . . . . . . . . 10 (φ → (𝑄 Q 𝑅 Q))
2726simpld 105 . . . . . . . . 9 (φ𝑄 Q)
28 addclnq 6228 . . . . . . . . . 10 ((𝑃 Q 𝑃 Q) → (𝑃 +Q 𝑃) Q)
294, 4, 28syl2anc 393 . . . . . . . . 9 (φ → (𝑃 +Q 𝑃) Q)
30 ltanqg 6253 . . . . . . . . 9 (((𝐷 +Q 𝐸) Q 𝑄 Q (𝑃 +Q 𝑃) Q) → ((𝐷 +Q 𝐸) <Q 𝑄 ↔ ((𝑃 +Q 𝑃) +Q (𝐷 +Q 𝐸)) <Q ((𝑃 +Q 𝑃) +Q 𝑄)))
3123, 27, 29, 30syl3anc 1119 . . . . . . . 8 (φ → ((𝐷 +Q 𝐸) <Q 𝑄 ↔ ((𝑃 +Q 𝑃) +Q (𝐷 +Q 𝐸)) <Q ((𝑃 +Q 𝑃) +Q 𝑄)))
32 addcomnqg 6234 . . . . . . . . . 10 (((𝑃 +Q 𝑃) Q (𝐷 +Q 𝐸) Q) → ((𝑃 +Q 𝑃) +Q (𝐷 +Q 𝐸)) = ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
3329, 23, 32syl2anc 393 . . . . . . . . 9 (φ → ((𝑃 +Q 𝑃) +Q (𝐷 +Q 𝐸)) = ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
34 addcomnqg 6234 . . . . . . . . . 10 (((𝑃 +Q 𝑃) Q 𝑄 Q) → ((𝑃 +Q 𝑃) +Q 𝑄) = (𝑄 +Q (𝑃 +Q 𝑃)))
3529, 27, 34syl2anc 393 . . . . . . . . 9 (φ → ((𝑃 +Q 𝑃) +Q 𝑄) = (𝑄 +Q (𝑃 +Q 𝑃)))
3633, 35breq12d 3747 . . . . . . . 8 (φ → (((𝑃 +Q 𝑃) +Q (𝐷 +Q 𝐸)) <Q ((𝑃 +Q 𝑃) +Q 𝑄) ↔ ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) <Q (𝑄 +Q (𝑃 +Q 𝑃))))
3731, 36bitrd 177 . . . . . . 7 (φ → ((𝐷 +Q 𝐸) <Q 𝑄 ↔ ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) <Q (𝑄 +Q (𝑃 +Q 𝑃))))
3837biimpa 280 . . . . . 6 ((φ (𝐷 +Q 𝐸) <Q 𝑄) → ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) <Q (𝑄 +Q (𝑃 +Q 𝑃)))
395breq2d 3746 . . . . . . 7 (φ → (((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) <Q (𝑄 +Q (𝑃 +Q 𝑃)) ↔ ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) <Q 𝑅))
4039adantr 261 . . . . . 6 ((φ (𝐷 +Q 𝐸) <Q 𝑄) → (((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) <Q (𝑄 +Q (𝑃 +Q 𝑃)) ↔ ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) <Q 𝑅))
4138, 40mpbid 135 . . . . 5 ((φ (𝐷 +Q 𝐸) <Q 𝑄) → ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) <Q 𝑅)
4213, 41jca 290 . . . 4 ((φ (𝐷 +Q 𝐸) <Q 𝑄) → ((𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) <Q 𝑅))
43 ltsonq 6251 . . . . 5 <Q Or Q
4443, 24sotri 4643 . . . 4 (((𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) <Q 𝑅) → (𝑈 +Q 𝑇) <Q 𝑅)
4542, 44syl 14 . . 3 ((φ (𝐷 +Q 𝐸) <Q 𝑄) → (𝑈 +Q 𝑇) <Q 𝑅)
461, 7jca 290 . . . . 5 (φ → (A P 𝑈 (2ndA)))
472, 10jca 290 . . . . 5 (φ → (B P 𝑇 (2ndB)))
4826simprd 107 . . . . 5 (φ𝑅 Q)
49 addnqpru 6379 . . . . 5 ((((A P 𝑈 (2ndA)) (B P 𝑇 (2ndB))) 𝑅 Q) → ((𝑈 +Q 𝑇) <Q 𝑅𝑅 (2nd ‘(A +P B))))
5046, 47, 48, 49syl21anc 1118 . . . 4 (φ → ((𝑈 +Q 𝑇) <Q 𝑅𝑅 (2nd ‘(A +P B))))
5150adantr 261 . . 3 ((φ (𝐷 +Q 𝐸) <Q 𝑄) → ((𝑈 +Q 𝑇) <Q 𝑅𝑅 (2nd ‘(A +P B))))
5245, 51mpd 13 . 2 ((φ (𝐷 +Q 𝐸) <Q 𝑄) → 𝑅 (2nd ‘(A +P B)))
5352ex 108 1 (φ → ((𝐷 +Q 𝐸) <Q 𝑄𝑅 (2nd ‘(A +P B))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = 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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