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Theorem 1lt2nq 9996
 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 9928 . . . . . 6 1𝑜 <N (1𝑜 +N 1𝑜)
2 1pi 9906 . . . . . . 7 1𝑜N
3 mulidpi 9909 . . . . . . 7 (1𝑜N → (1𝑜 ·N 1𝑜) = 1𝑜)
42, 3ax-mp 5 . . . . . 6 (1𝑜 ·N 1𝑜) = 1𝑜
5 addclpi 9915 . . . . . . . 8 ((1𝑜N ∧ 1𝑜N) → (1𝑜 +N 1𝑜) ∈ N)
62, 2, 5mp2an 664 . . . . . . 7 (1𝑜 +N 1𝑜) ∈ N
7 mulidpi 9909 . . . . . . 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 4814 . . . . 5 (1𝑜 ·N 1𝑜) <N ((1𝑜 +N 1𝑜) ·N 1𝑜)
10 ordpipq 9965 . . . . 5 (⟨1𝑜, 1𝑜⟩ <pQ ⟨(1𝑜 +N 1𝑜), 1𝑜⟩ ↔ (1𝑜 ·N 1𝑜) <N ((1𝑜 +N 1𝑜) ·N 1𝑜))
119, 10mpbir 221 . . . 4 ⟨1𝑜, 1𝑜⟩ <pQ ⟨(1𝑜 +N 1𝑜), 1𝑜
12 df-1nq 9939 . . . 4 1Q = ⟨1𝑜, 1𝑜
1312, 12oveq12i 6804 . . . . 5 (1Q +pQ 1Q) = (⟨1𝑜, 1𝑜⟩ +pQ ⟨1𝑜, 1𝑜⟩)
14 addpipq 9960 . . . . . 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 665 . . . . 5 (⟨1𝑜, 1𝑜⟩ +pQ ⟨1𝑜, 1𝑜⟩) = ⟨((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)), (1𝑜 ·N 1𝑜)⟩
164, 4oveq12i 6804 . . . . . 6 ((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)) = (1𝑜 +N 1𝑜)
1716, 4opeq12i 4542 . . . . 5 ⟨((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)), (1𝑜 ·N 1𝑜)⟩ = ⟨(1𝑜 +N 1𝑜), 1𝑜
1813, 15, 173eqtri 2796 . . . 4 (1Q +pQ 1Q) = ⟨(1𝑜 +N 1𝑜), 1𝑜
1911, 12, 183brtr4i 4814 . . 3 1Q <pQ (1Q +pQ 1Q)
20 lterpq 9993 . . 3 (1Q <pQ (1Q +pQ 1Q) ↔ ([Q]‘1Q) <Q ([Q]‘(1Q +pQ 1Q)))
2119, 20mpbi 220 . 2 ([Q]‘1Q) <Q ([Q]‘(1Q +pQ 1Q))
22 1nq 9951 . . . 4 1QQ
23 nqerid 9956 . . . 4 (1QQ → ([Q]‘1Q) = 1Q)
2422, 23ax-mp 5 . . 3 ([Q]‘1Q) = 1Q
2524eqcomi 2779 . 2 1Q = ([Q]‘1Q)
26 addpqnq 9961 . . 3 ((1QQ ∧ 1QQ) → (1Q +Q 1Q) = ([Q]‘(1Q +pQ 1Q)))
2722, 22, 26mp2an 664 . 2 (1Q +Q 1Q) = ([Q]‘(1Q +pQ 1Q))
2821, 25, 273brtr4i 4814 1 1Q <Q (1Q +Q 1Q)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1630   ∈ wcel 2144  ⟨cop 4320   class class class wbr 4784  ‘cfv 6031  (class class class)co 6792  1𝑜c1o 7705  Ncnpi 9867   +N cpli 9868   ·N cmi 9869
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