Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  lt2addnq GIF version

Theorem lt2addnq 6388
 Description: Ordering property of addition for positive fractions. (Contributed by Jim Kingdon, 7-Dec-2019.)
Assertion
Ref Expression
lt2addnq (((A Q B Q) (𝐶 Q 𝐷 Q)) → ((A <Q B 𝐶 <Q 𝐷) → (A +Q 𝐶) <Q (B +Q 𝐷)))

Proof of Theorem lt2addnq
StepHypRef Expression
1 ltanqg 6384 . . . . . 6 ((A Q B Q 𝐶 Q) → (A <Q B ↔ (𝐶 +Q A) <Q (𝐶 +Q B)))
213expa 1103 . . . . 5 (((A Q B Q) 𝐶 Q) → (A <Q B ↔ (𝐶 +Q A) <Q (𝐶 +Q B)))
32adantrr 448 . . . 4 (((A Q B Q) (𝐶 Q 𝐷 Q)) → (A <Q B ↔ (𝐶 +Q A) <Q (𝐶 +Q B)))
4 addcomnqg 6365 . . . . . . 7 ((𝐶 Q A Q) → (𝐶 +Q A) = (A +Q 𝐶))
54ancoms 255 . . . . . 6 ((A Q 𝐶 Q) → (𝐶 +Q A) = (A +Q 𝐶))
65ad2ant2r 478 . . . . 5 (((A Q B Q) (𝐶 Q 𝐷 Q)) → (𝐶 +Q A) = (A +Q 𝐶))
7 addcomnqg 6365 . . . . . . 7 ((𝐶 Q B Q) → (𝐶 +Q B) = (B +Q 𝐶))
87ancoms 255 . . . . . 6 ((B Q 𝐶 Q) → (𝐶 +Q B) = (B +Q 𝐶))
98ad2ant2lr 479 . . . . 5 (((A Q B Q) (𝐶 Q 𝐷 Q)) → (𝐶 +Q B) = (B +Q 𝐶))
106, 9breq12d 3768 . . . 4 (((A Q B Q) (𝐶 Q 𝐷 Q)) → ((𝐶 +Q A) <Q (𝐶 +Q B) ↔ (A +Q 𝐶) <Q (B +Q 𝐶)))
113, 10bitrd 177 . . 3 (((A Q B Q) (𝐶 Q 𝐷 Q)) → (A <Q B ↔ (A +Q 𝐶) <Q (B +Q 𝐶)))
12 ltanqg 6384 . . . . . 6 ((𝐶 Q 𝐷 Q B Q) → (𝐶 <Q 𝐷 ↔ (B +Q 𝐶) <Q (B +Q 𝐷)))
13123expa 1103 . . . . 5 (((𝐶 Q 𝐷 Q) B Q) → (𝐶 <Q 𝐷 ↔ (B +Q 𝐶) <Q (B +Q 𝐷)))
1413ancoms 255 . . . 4 ((B Q (𝐶 Q 𝐷 Q)) → (𝐶 <Q 𝐷 ↔ (B +Q 𝐶) <Q (B +Q 𝐷)))
1514adantll 445 . . 3 (((A Q B Q) (𝐶 Q 𝐷 Q)) → (𝐶 <Q 𝐷 ↔ (B +Q 𝐶) <Q (B +Q 𝐷)))
1611, 15anbi12d 442 . 2 (((A Q B Q) (𝐶 Q 𝐷 Q)) → ((A <Q B 𝐶 <Q 𝐷) ↔ ((A +Q 𝐶) <Q (B +Q 𝐶) (B +Q 𝐶) <Q (B +Q 𝐷))))
17 ltsonq 6382 . . 3 <Q Or Q
18 ltrelnq 6349 . . 3 <Q ⊆ (Q × Q)
1917, 18sotri 4663 . 2 (((A +Q 𝐶) <Q (B +Q 𝐶) (B +Q 𝐶) <Q (B +Q 𝐷)) → (A +Q 𝐶) <Q (B +Q 𝐷))
2016, 19syl6bi 152 1 (((A Q B Q) (𝐶 Q 𝐷 Q)) → ((A <Q B 𝐶 <Q 𝐷) → (A +Q 𝐶) <Q (B +Q 𝐷)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390   class class class wbr 3755  (class class class)co 5455  Qcnq 6264   +Q cplq 6266
 Copyright terms: Public domain W3C validator