Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  1lt2nq Structured version   Visualization version   GIF version

Theorem 1lt2nq 10388
 Description: One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
1lt2nq 1Q <Q (1Q +Q 1Q)

Proof of Theorem 1lt2nq
StepHypRef Expression
1 1lt2pi 10320 . . . . . 6 1o <N (1o +N 1o)
2 1pi 10298 . . . . . . 7 1oN
3 mulidpi 10301 . . . . . . 7 (1oN → (1o ·N 1o) = 1o)
42, 3ax-mp 5 . . . . . 6 (1o ·N 1o) = 1o
5 addclpi 10307 . . . . . . . 8 ((1oN ∧ 1oN) → (1o +N 1o) ∈ N)
62, 2, 5mp2an 691 . . . . . . 7 (1o +N 1o) ∈ N
7 mulidpi 10301 . . . . . . 7 ((1o +N 1o) ∈ N → ((1o +N 1o) ·N 1o) = (1o +N 1o))
86, 7ax-mp 5 . . . . . 6 ((1o +N 1o) ·N 1o) = (1o +N 1o)
91, 4, 83brtr4i 5063 . . . . 5 (1o ·N 1o) <N ((1o +N 1o) ·N 1o)
10 ordpipq 10357 . . . . 5 (⟨1o, 1o⟩ <pQ ⟨(1o +N 1o), 1o⟩ ↔ (1o ·N 1o) <N ((1o +N 1o) ·N 1o))
119, 10mpbir 234 . . . 4 ⟨1o, 1o⟩ <pQ ⟨(1o +N 1o), 1o
12 df-1nq 10331 . . . 4 1Q = ⟨1o, 1o
1312, 12oveq12i 7151 . . . . 5 (1Q +pQ 1Q) = (⟨1o, 1o⟩ +pQ ⟨1o, 1o⟩)
14 addpipq 10352 . . . . . 6 (((1oN ∧ 1oN) ∧ (1oN ∧ 1oN)) → (⟨1o, 1o⟩ +pQ ⟨1o, 1o⟩) = ⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩)
152, 2, 2, 2, 14mp4an 692 . . . . 5 (⟨1o, 1o⟩ +pQ ⟨1o, 1o⟩) = ⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩
164, 4oveq12i 7151 . . . . . 6 ((1o ·N 1o) +N (1o ·N 1o)) = (1o +N 1o)
1716, 4opeq12i 4773 . . . . 5 ⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩ = ⟨(1o +N 1o), 1o
1813, 15, 173eqtri 2828 . . . 4 (1Q +pQ 1Q) = ⟨(1o +N 1o), 1o
1911, 12, 183brtr4i 5063 . . 3 1Q <pQ (1Q +pQ 1Q)
20 lterpq 10385 . . 3 (1Q <pQ (1Q +pQ 1Q) ↔ ([Q]‘1Q) <Q ([Q]‘(1Q +pQ 1Q)))
2119, 20mpbi 233 . 2 ([Q]‘1Q) <Q ([Q]‘(1Q +pQ 1Q))
22 1nq 10343 . . . 4 1QQ
23 nqerid 10348 . . . 4 (1QQ → ([Q]‘1Q) = 1Q)
2422, 23ax-mp 5 . . 3 ([Q]‘1Q) = 1Q
2524eqcomi 2810 . 2 1Q = ([Q]‘1Q)
26 addpqnq 10353 . . 3 ((1QQ ∧ 1QQ) → (1Q +Q 1Q) = ([Q]‘(1Q +pQ 1Q)))
2722, 22, 26mp2an 691 . 2 (1Q +Q 1Q) = ([Q]‘(1Q +pQ 1Q))
2821, 25, 273brtr4i 5063 1 1Q <Q (1Q +Q 1Q)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2112  ⟨cop 4534   class class class wbr 5033  ‘cfv 6328  (class class class)co 7139  1oc1o 8082  Ncnpi 10259   +N cpli 10260   ·N cmi 10261
 Copyright terms: Public domain W3C validator