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

 Description: The sum of two fractions is greater than one of them. (Contributed by NM, 14-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.)
Assertion
Ref Expression
ltaddnq ((A Q B Q) → A <Q (A +Q B))

Dummy variables 𝑟 𝑠 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1lt2nq 6389 . . . . . . 7 1Q <Q (1Q +Q 1Q)
2 1nq 6350 . . . . . . . 8 1Q Q
3 addclnq 6359 . . . . . . . . 9 ((1Q Q 1Q Q) → (1Q +Q 1Q) Q)
42, 2, 3mp2an 402 . . . . . . . 8 (1Q +Q 1Q) Q
5 ltmnqg 6385 . . . . . . . 8 ((1Q Q (1Q +Q 1Q) Q B Q) → (1Q <Q (1Q +Q 1Q) ↔ (B ·Q 1Q) <Q (B ·Q (1Q +Q 1Q))))
62, 4, 5mp3an12 1221 . . . . . . 7 (B Q → (1Q <Q (1Q +Q 1Q) ↔ (B ·Q 1Q) <Q (B ·Q (1Q +Q 1Q))))
71, 6mpbii 136 . . . . . 6 (B Q → (B ·Q 1Q) <Q (B ·Q (1Q +Q 1Q)))
8 mulidnq 6373 . . . . . 6 (B Q → (B ·Q 1Q) = B)
9 distrnqg 6371 . . . . . . . 8 ((B Q 1Q Q 1Q Q) → (B ·Q (1Q +Q 1Q)) = ((B ·Q 1Q) +Q (B ·Q 1Q)))
102, 2, 9mp3an23 1223 . . . . . . 7 (B Q → (B ·Q (1Q +Q 1Q)) = ((B ·Q 1Q) +Q (B ·Q 1Q)))
118, 8oveq12d 5473 . . . . . . 7 (B Q → ((B ·Q 1Q) +Q (B ·Q 1Q)) = (B +Q B))
1210, 11eqtrd 2069 . . . . . 6 (B Q → (B ·Q (1Q +Q 1Q)) = (B +Q B))
137, 8, 123brtr3d 3784 . . . . 5 (B QB <Q (B +Q B))
1413adantl 262 . . . 4 ((A Q B Q) → B <Q (B +Q B))
15 simpr 103 . . . . 5 ((A Q B Q) → B Q)
16 addclnq 6359 . . . . . . 7 ((B Q B Q) → (B +Q B) Q)
1716anidms 377 . . . . . 6 (B Q → (B +Q B) Q)
1817adantl 262 . . . . 5 ((A Q B Q) → (B +Q B) Q)
19 simpl 102 . . . . 5 ((A Q B Q) → A Q)
20 ltanqg 6384 . . . . 5 ((B Q (B +Q B) Q A Q) → (B <Q (B +Q B) ↔ (A +Q B) <Q (A +Q (B +Q B))))
2115, 18, 19, 20syl3anc 1134 . . . 4 ((A Q B Q) → (B <Q (B +Q B) ↔ (A +Q B) <Q (A +Q (B +Q B))))
2214, 21mpbid 135 . . 3 ((A Q B Q) → (A +Q B) <Q (A +Q (B +Q B)))
23 addcomnqg 6365 . . 3 ((A Q B Q) → (A +Q B) = (B +Q A))
24 addcomnqg 6365 . . . . 5 ((𝑟 Q 𝑠 Q) → (𝑟 +Q 𝑠) = (𝑠 +Q 𝑟))
2524adantl 262 . . . 4 (((A Q B Q) (𝑟 Q 𝑠 Q)) → (𝑟 +Q 𝑠) = (𝑠 +Q 𝑟))
26 addassnqg 6366 . . . . 5 ((𝑟 Q 𝑠 Q 𝑡 Q) → ((𝑟 +Q 𝑠) +Q 𝑡) = (𝑟 +Q (𝑠 +Q 𝑡)))
2726adantl 262 . . . 4 (((A Q B Q) (𝑟 Q 𝑠 Q 𝑡 Q)) → ((𝑟 +Q 𝑠) +Q 𝑡) = (𝑟 +Q (𝑠 +Q 𝑡)))
2819, 15, 15, 25, 27caov12d 5624 . . 3 ((A Q B Q) → (A +Q (B +Q B)) = (B +Q (A +Q B)))
2922, 23, 283brtr3d 3784 . 2 ((A Q B Q) → (B +Q A) <Q (B +Q (A +Q B)))
30 addclnq 6359 . . 3 ((A Q B Q) → (A +Q B) Q)
31 ltanqg 6384 . . 3 ((A Q (A +Q B) Q B Q) → (A <Q (A +Q B) ↔ (B +Q A) <Q (B +Q (A +Q B))))
3219, 30, 15, 31syl3anc 1134 . 2 ((A Q B Q) → (A <Q (A +Q B) ↔ (B +Q A) <Q (B +Q (A +Q B))))
3329, 32mpbird 156 1 ((A Q B Q) → A <Q (A +Q B))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 884   = wceq 1242   ∈ wcel 1390   class class class wbr 3755  (class class class)co 5455  Qcnq 6264  1Qc1q 6265   +Q cplq 6266   ·Q cmq 6267
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