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Theorem 1lt2nq 10074
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 10006 . . . . . 6 1𝑜 <N (1𝑜 +N 1𝑜)
2 1pi 9984 . . . . . . 7 1𝑜N
3 mulidpi 9987 . . . . . . 7 (1𝑜N → (1𝑜 ·N 1𝑜) = 1𝑜)
42, 3ax-mp 5 . . . . . 6 (1𝑜 ·N 1𝑜) = 1𝑜
5 addclpi 9993 . . . . . . . 8 ((1𝑜N ∧ 1𝑜N) → (1𝑜 +N 1𝑜) ∈ N)
62, 2, 5mp2an 675 . . . . . . 7 (1𝑜 +N 1𝑜) ∈ N
7 mulidpi 9987 . . . . . . 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 4867 . . . . 5 (1𝑜 ·N 1𝑜) <N ((1𝑜 +N 1𝑜) ·N 1𝑜)
10 ordpipq 10043 . . . . 5 (⟨1𝑜, 1𝑜⟩ <pQ ⟨(1𝑜 +N 1𝑜), 1𝑜⟩ ↔ (1𝑜 ·N 1𝑜) <N ((1𝑜 +N 1𝑜) ·N 1𝑜))
119, 10mpbir 222 . . . 4 ⟨1𝑜, 1𝑜⟩ <pQ ⟨(1𝑜 +N 1𝑜), 1𝑜
12 df-1nq 10017 . . . 4 1Q = ⟨1𝑜, 1𝑜
1312, 12oveq12i 6880 . . . . 5 (1Q +pQ 1Q) = (⟨1𝑜, 1𝑜⟩ +pQ ⟨1𝑜, 1𝑜⟩)
14 addpipq 10038 . . . . . 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 676 . . . . 5 (⟨1𝑜, 1𝑜⟩ +pQ ⟨1𝑜, 1𝑜⟩) = ⟨((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)), (1𝑜 ·N 1𝑜)⟩
164, 4oveq12i 6880 . . . . . 6 ((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)) = (1𝑜 +N 1𝑜)
1716, 4opeq12i 4593 . . . . 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 4867 . . 3 1Q <pQ (1Q +pQ 1Q)
20 lterpq 10071 . . 3 (1Q <pQ (1Q +pQ 1Q) ↔ ([Q]‘1Q) <Q ([Q]‘(1Q +pQ 1Q)))
2119, 20mpbi 221 . 2 ([Q]‘1Q) <Q ([Q]‘(1Q +pQ 1Q))
22 1nq 10029 . . . 4 1QQ
23 nqerid 10034 . . . 4 (1QQ → ([Q]‘1Q) = 1Q)
2422, 23ax-mp 5 . . 3 ([Q]‘1Q) = 1Q
2524eqcomi 2811 . 2 1Q = ([Q]‘1Q)
26 addpqnq 10039 . . 3 ((1QQ ∧ 1QQ) → (1Q +Q 1Q) = ([Q]‘(1Q +pQ 1Q)))
2722, 22, 26mp2an 675 . 2 (1Q +Q 1Q) = ([Q]‘(1Q +pQ 1Q))
2821, 25, 273brtr4i 4867 1 1Q <Q (1Q +Q 1Q)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1637  wcel 2155  cop 4370   class class class wbr 4837  cfv 6095  (class class class)co 6868  1𝑜c1o 7783  Ncnpi 9945   +N cpli 9946   ·N cmi 9947   <N clti 9948   +pQ cplpq 9949   <pQ cltpq 9951  Qcnq 9953  1Qc1q 9954  [Q]cerq 9955   +Q cplq 9956   <Q cltq 9959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090  ax-un 7173
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ne 2975  df-ral 3097  df-rex 3098  df-reu 3099  df-rmo 3100  df-rab 3101  df-v 3389  df-sbc 3628  df-csb 3723  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-pss 3779  df-nul 4111  df-if 4274  df-pw 4347  df-sn 4365  df-pr 4367  df-tp 4369  df-op 4371  df-uni 4624  df-iun 4707  df-br 4838  df-opab 4900  df-mpt 4917  df-tr 4940  df-id 5213  df-eprel 5218  df-po 5226  df-so 5227  df-fr 5264  df-we 5266  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318  df-pred 5887  df-ord 5933  df-on 5934  df-lim 5935  df-suc 5936  df-iota 6058  df-fun 6097  df-fn 6098  df-f 6099  df-f1 6100  df-fo 6101  df-f1o 6102  df-fv 6103  df-ov 6871  df-oprab 6872  df-mpt2 6873  df-om 7290  df-1st 7392  df-2nd 7393  df-wrecs 7636  df-recs 7698  df-rdg 7736  df-1o 7790  df-oadd 7794  df-omul 7795  df-er 7973  df-ni 9973  df-pli 9974  df-mi 9975  df-lti 9976  df-plpq 10009  df-ltpq 10011  df-enq 10012  df-nq 10013  df-erq 10014  df-plq 10015  df-1nq 10017  df-ltnq 10019
This theorem is referenced by:  ltaddnq  10075
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