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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   <N clti 9870   +pQ cplpq 9871   <pQ cltpq 9873  Qcnq 9875  1Qc1q 9876  [Q]cerq 9877   +Q cplq 9878   <Q cltq 9881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-omul 7717  df-er 7895  df-ni 9895  df-pli 9896  df-mi 9897  df-lti 9898  df-plpq 9931  df-ltpq 9933  df-enq 9934  df-nq 9935  df-erq 9936  df-plq 9937  df-1nq 9939  df-ltnq 9941
This theorem is referenced by:  ltaddnq  9997
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