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Theorem 1lt2nq 9674
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 9606 . . . . . 6 1𝑜 <N (1𝑜 +N 1𝑜)
2 1pi 9584 . . . . . . 7 1𝑜N
3 mulidpi 9587 . . . . . . 7 (1𝑜N → (1𝑜 ·N 1𝑜) = 1𝑜)
42, 3ax-mp 5 . . . . . 6 (1𝑜 ·N 1𝑜) = 1𝑜
5 addclpi 9593 . . . . . . . 8 ((1𝑜N ∧ 1𝑜N) → (1𝑜 +N 1𝑜) ∈ N)
62, 2, 5mp2an 704 . . . . . . 7 (1𝑜 +N 1𝑜) ∈ N
7 mulidpi 9587 . . . . . . 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 4613 . . . . 5 (1𝑜 ·N 1𝑜) <N ((1𝑜 +N 1𝑜) ·N 1𝑜)
10 ordpipq 9643 . . . . 5 (⟨1𝑜, 1𝑜⟩ <pQ ⟨(1𝑜 +N 1𝑜), 1𝑜⟩ ↔ (1𝑜 ·N 1𝑜) <N ((1𝑜 +N 1𝑜) ·N 1𝑜))
119, 10mpbir 220 . . . 4 ⟨1𝑜, 1𝑜⟩ <pQ ⟨(1𝑜 +N 1𝑜), 1𝑜
12 df-1nq 9617 . . . 4 1Q = ⟨1𝑜, 1𝑜
1312, 12oveq12i 6561 . . . . 5 (1Q +pQ 1Q) = (⟨1𝑜, 1𝑜⟩ +pQ ⟨1𝑜, 1𝑜⟩)
14 addpipq 9638 . . . . . 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 705 . . . . 5 (⟨1𝑜, 1𝑜⟩ +pQ ⟨1𝑜, 1𝑜⟩) = ⟨((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)), (1𝑜 ·N 1𝑜)⟩
164, 4oveq12i 6561 . . . . . 6 ((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)) = (1𝑜 +N 1𝑜)
1716, 4opeq12i 4345 . . . . 5 ⟨((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)), (1𝑜 ·N 1𝑜)⟩ = ⟨(1𝑜 +N 1𝑜), 1𝑜
1813, 15, 173eqtri 2636 . . . 4 (1Q +pQ 1Q) = ⟨(1𝑜 +N 1𝑜), 1𝑜
1911, 12, 183brtr4i 4613 . . 3 1Q <pQ (1Q +pQ 1Q)
20 lterpq 9671 . . 3 (1Q <pQ (1Q +pQ 1Q) ↔ ([Q]‘1Q) <Q ([Q]‘(1Q +pQ 1Q)))
2119, 20mpbi 219 . 2 ([Q]‘1Q) <Q ([Q]‘(1Q +pQ 1Q))
22 1nq 9629 . . . 4 1QQ
23 nqerid 9634 . . . 4 (1QQ → ([Q]‘1Q) = 1Q)
2422, 23ax-mp 5 . . 3 ([Q]‘1Q) = 1Q
2524eqcomi 2619 . 2 1Q = ([Q]‘1Q)
26 addpqnq 9639 . . 3 ((1QQ ∧ 1QQ) → (1Q +Q 1Q) = ([Q]‘(1Q +pQ 1Q)))
2722, 22, 26mp2an 704 . 2 (1Q +Q 1Q) = ([Q]‘(1Q +pQ 1Q))
2821, 25, 273brtr4i 4613 1 1Q <Q (1Q +Q 1Q)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1475  wcel 1977  cop 4131   class class class wbr 4583  cfv 5804  (class class class)co 6549  1𝑜c1o 7440  Ncnpi 9545   +N cpli 9546   ·N cmi 9547   <N clti 9548   +pQ cplpq 9549   <pQ cltpq 9551  Qcnq 9553  1Qc1q 9554  [Q]cerq 9555   +Q cplq 9556   <Q cltq 9559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452  df-er 7629  df-ni 9573  df-pli 9574  df-mi 9575  df-lti 9576  df-plpq 9609  df-ltpq 9611  df-enq 9612  df-nq 9613  df-erq 9614  df-plq 9615  df-1nq 9617  df-ltnq 9619
This theorem is referenced by:  ltaddnq  9675
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