Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  1lt2nq GIF version

Theorem 1lt2nq 7309
 Description: One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.)
Assertion
Ref Expression
1lt2nq 1Q <Q (1Q +Q 1Q)

Proof of Theorem 1lt2nq
StepHypRef Expression
1 1lt2pi 7243 . . . . 5 1o <N (1o +N 1o)
2 1pi 7218 . . . . . 6 1oN
3 mulidpi 7221 . . . . . 6 (1oN → (1o ·N 1o) = 1o)
42, 3ax-mp 5 . . . . 5 (1o ·N 1o) = 1o
54, 4oveq12i 5830 . . . . 5 ((1o ·N 1o) +N (1o ·N 1o)) = (1o +N 1o)
61, 4, 53brtr4i 3994 . . . 4 (1o ·N 1o) <N ((1o ·N 1o) +N (1o ·N 1o))
7 mulclpi 7231 . . . . . 6 ((1oN ∧ 1oN) → (1o ·N 1o) ∈ N)
82, 2, 7mp2an 423 . . . . 5 (1o ·N 1o) ∈ N
9 addclpi 7230 . . . . . 6 (((1o ·N 1o) ∈ N ∧ (1o ·N 1o) ∈ N) → ((1o ·N 1o) +N (1o ·N 1o)) ∈ N)
108, 8, 9mp2an 423 . . . . 5 ((1o ·N 1o) +N (1o ·N 1o)) ∈ N
11 ltmpig 7242 . . . . 5 (((1o ·N 1o) ∈ N ∧ ((1o ·N 1o) +N (1o ·N 1o)) ∈ N ∧ 1oN) → ((1o ·N 1o) <N ((1o ·N 1o) +N (1o ·N 1o)) ↔ (1o ·N (1o ·N 1o)) <N (1o ·N ((1o ·N 1o) +N (1o ·N 1o)))))
128, 10, 2, 11mp3an 1319 . . . 4 ((1o ·N 1o) <N ((1o ·N 1o) +N (1o ·N 1o)) ↔ (1o ·N (1o ·N 1o)) <N (1o ·N ((1o ·N 1o) +N (1o ·N 1o))))
136, 12mpbi 144 . . 3 (1o ·N (1o ·N 1o)) <N (1o ·N ((1o ·N 1o) +N (1o ·N 1o)))
14 ordpipqqs 7277 . . . 4 (((1oN ∧ 1oN) ∧ (((1o ·N 1o) +N (1o ·N 1o)) ∈ N ∧ (1o ·N 1o) ∈ N)) → ([⟨1o, 1o⟩] ~Q <Q [⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q ↔ (1o ·N (1o ·N 1o)) <N (1o ·N ((1o ·N 1o) +N (1o ·N 1o)))))
152, 2, 10, 8, 14mp4an 424 . . 3 ([⟨1o, 1o⟩] ~Q <Q [⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q ↔ (1o ·N (1o ·N 1o)) <N (1o ·N ((1o ·N 1o) +N (1o ·N 1o))))
1613, 15mpbir 145 . 2 [⟨1o, 1o⟩] ~Q <Q [⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q
17 df-1nqqs 7254 . 2 1Q = [⟨1o, 1o⟩] ~Q
1817, 17oveq12i 5830 . . 3 (1Q +Q 1Q) = ([⟨1o, 1o⟩] ~Q +Q [⟨1o, 1o⟩] ~Q )
19 addpipqqs 7273 . . . 4 (((1oN ∧ 1oN) ∧ (1oN ∧ 1oN)) → ([⟨1o, 1o⟩] ~Q +Q [⟨1o, 1o⟩] ~Q ) = [⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q )
202, 2, 2, 2, 19mp4an 424 . . 3 ([⟨1o, 1o⟩] ~Q +Q [⟨1o, 1o⟩] ~Q ) = [⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q
2118, 20eqtri 2178 . 2 (1Q +Q 1Q) = [⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q
2216, 17, 213brtr4i 3994 1 1Q <Q (1Q +Q 1Q)
 Colors of variables: wff set class Syntax hints:   ↔ wb 104   = wceq 1335   ∈ wcel 2128  ⟨cop 3563   class class class wbr 3965  (class class class)co 5818  1oc1o 6350  [cec 6471  Ncnpi 7175   +N cpli 7176   ·N cmi 7177
 Copyright terms: Public domain W3C validator