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

 Description: Lemma for addlocpr 7364. 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 (𝜑𝐴P)
4 prop 7303 . . . . . 6 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
53, 4syl 14 . . . . 5 (𝜑 → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
6 addlocprlem.uup . . . . 5 (𝜑𝑈 ∈ (2nd𝐴))
7 elprnqu 7310 . . . . 5 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑈 ∈ (2nd𝐴)) → 𝑈Q)
85, 6, 7syl2anc 409 . . . 4 (𝜑𝑈Q)
9 addlocprlem.dlo . . . . . 6 (𝜑𝐷 ∈ (1st𝐴))
10 elprnql 7309 . . . . . 6 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝐷 ∈ (1st𝐴)) → 𝐷Q)
115, 9, 10syl2anc 409 . . . . 5 (𝜑𝐷Q)
12 addlocprlem.p . . . . 5 (𝜑𝑃Q)
13 addclnq 7203 . . . . 5 ((𝐷Q𝑃Q) → (𝐷 +Q 𝑃) ∈ Q)
1411, 12, 13syl2anc 409 . . . 4 (𝜑 → (𝐷 +Q 𝑃) ∈ Q)
15 addlocprlem.b . . . . . 6 (𝜑𝐵P)
16 prop 7303 . . . . . 6 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
1715, 16syl 14 . . . . 5 (𝜑 → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
18 addlocprlem.tup . . . . 5 (𝜑𝑇 ∈ (2nd𝐵))
19 elprnqu 7310 . . . . 5 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑇 ∈ (2nd𝐵)) → 𝑇Q)
2017, 18, 19syl2anc 409 . . . 4 (𝜑𝑇Q)
21 addlocprlem.elo . . . . . 6 (𝜑𝐸 ∈ (1st𝐵))
22 elprnql 7309 . . . . . 6 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐸 ∈ (1st𝐵)) → 𝐸Q)
2317, 21, 22syl2anc 409 . . . . 5 (𝜑𝐸Q)
24 addclnq 7203 . . . . 5 ((𝐸Q𝑃Q) → (𝐸 +Q 𝑃) ∈ Q)
2523, 12, 24syl2anc 409 . . . 4 (𝜑 → (𝐸 +Q 𝑃) ∈ Q)
26 lt2addnq 7232 . . . 4 (((𝑈Q ∧ (𝐷 +Q 𝑃) ∈ Q) ∧ (𝑇Q ∧ (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 1218 . . 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 430 . 2 (𝜑 → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝑃) +Q (𝐸 +Q 𝑃)))
29 addcomnqg 7209 . . . 4 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
3029adantl 275 . . 3 ((𝜑 ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
31 addassnqg 7210 . . . 4 ((𝑓Q𝑔QQ) → ((𝑓 +Q 𝑔) +Q ) = (𝑓 +Q (𝑔 +Q )))
3231adantl 275 . . 3 ((𝜑 ∧ (𝑓Q𝑔QQ)) → ((𝑓 +Q 𝑔) +Q ) = (𝑓 +Q (𝑔 +Q )))
33 addclnq 7203 . . . 4 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) ∈ Q)
3433adantl 275 . . 3 ((𝜑 ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) ∈ Q)
3511, 12, 23, 30, 32, 12, 34caov4d 5959 . 2 (𝜑 → ((𝐷 +Q 𝑃) +Q (𝐸 +Q 𝑃)) = ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
3628, 35breqtrd 3958 1 (𝜑 → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 963   = wceq 1332   ∈ wcel 1481  ⟨cop 3531   class class class wbr 3933  ‘cfv 5127  (class class class)co 5778  1st c1st 6040  2nd c2nd 6041  Qcnq 7108   +Q cplq 7110
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