Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  addlocprlemeqgt Structured version   GIF version

 Description: Lemma for addlocpr 6512. This is a step used in both the 𝑄 = (𝐷 +Q 𝐸) and (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
addlocprlemeqgt (φ → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))

Dummy variables f g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addlocprlem.du . . 3 (φ𝑈 <Q (𝐷 +Q 𝑃))
2 addlocprlem.et . . 3 (φ𝑇 <Q (𝐸 +Q 𝑃))
3 addlocprlem.a . . . . . 6 (φA P)
4 prop 6450 . . . . . 6 (A P → ⟨(1stA), (2ndA)⟩ P)
53, 4syl 14 . . . . 5 (φ → ⟨(1stA), (2ndA)⟩ P)
6 addlocprlem.uup . . . . 5 (φ𝑈 (2ndA))
7 elprnqu 6457 . . . . 5 ((⟨(1stA), (2ndA)⟩ P 𝑈 (2ndA)) → 𝑈 Q)
85, 6, 7syl2anc 391 . . . 4 (φ𝑈 Q)
9 addlocprlem.dlo . . . . . 6 (φ𝐷 (1stA))
10 elprnql 6456 . . . . . 6 ((⟨(1stA), (2ndA)⟩ P 𝐷 (1stA)) → 𝐷 Q)
115, 9, 10syl2anc 391 . . . . 5 (φ𝐷 Q)
12 addlocprlem.p . . . . 5 (φ𝑃 Q)
13 addclnq 6352 . . . . 5 ((𝐷 Q 𝑃 Q) → (𝐷 +Q 𝑃) Q)
1411, 12, 13syl2anc 391 . . . 4 (φ → (𝐷 +Q 𝑃) Q)
15 addlocprlem.b . . . . . 6 (φB P)
16 prop 6450 . . . . . 6 (B P → ⟨(1stB), (2ndB)⟩ P)
1715, 16syl 14 . . . . 5 (φ → ⟨(1stB), (2ndB)⟩ P)
18 addlocprlem.tup . . . . 5 (φ𝑇 (2ndB))
19 elprnqu 6457 . . . . 5 ((⟨(1stB), (2ndB)⟩ P 𝑇 (2ndB)) → 𝑇 Q)
2017, 18, 19syl2anc 391 . . . 4 (φ𝑇 Q)
21 addlocprlem.elo . . . . . 6 (φ𝐸 (1stB))
22 elprnql 6456 . . . . . 6 ((⟨(1stB), (2ndB)⟩ P 𝐸 (1stB)) → 𝐸 Q)
2317, 21, 22syl2anc 391 . . . . 5 (φ𝐸 Q)
24 addclnq 6352 . . . . 5 ((𝐸 Q 𝑃 Q) → (𝐸 +Q 𝑃) Q)
2523, 12, 24syl2anc 391 . . . 4 (φ → (𝐸 +Q 𝑃) Q)
26 lt2addnq 6381 . . . 4 (((𝑈 Q (𝐷 +Q 𝑃) Q) (𝑇 Q (𝐸 +Q 𝑃) Q)) → ((𝑈 <Q (𝐷 +Q 𝑃) 𝑇 <Q (𝐸 +Q 𝑃)) → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝑃) +Q (𝐸 +Q 𝑃))))
278, 14, 20, 25, 26syl22anc 1135 . . 3 (φ → ((𝑈 <Q (𝐷 +Q 𝑃) 𝑇 <Q (𝐸 +Q 𝑃)) → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝑃) +Q (𝐸 +Q 𝑃))))
281, 2, 27mp2and 409 . 2 (φ → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝑃) +Q (𝐸 +Q 𝑃)))
29 addcomnqg 6358 . . . 4 ((f Q g Q) → (f +Q g) = (g +Q f))
3029adantl 262 . . 3 ((φ (f Q g Q)) → (f +Q g) = (g +Q f))
31 addassnqg 6359 . . . 4 ((f Q g Q Q) → ((f +Q g) +Q ) = (f +Q (g +Q )))
3231adantl 262 . . 3 ((φ (f Q g Q Q)) → ((f +Q g) +Q ) = (f +Q (g +Q )))
33 addclnq 6352 . . . 4 ((f Q g Q) → (f +Q g) Q)
3433adantl 262 . . 3 ((φ (f Q g Q)) → (f +Q g) Q)
3511, 12, 23, 30, 32, 12, 34caov4d 5624 . 2 (φ → ((𝐷 +Q 𝑃) +Q (𝐸 +Q 𝑃)) = ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
3628, 35breqtrd 3778 1 (φ → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  ⟨cop 3369   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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