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Theorem 1lt2nq 9742
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 9674 . . . . . 6 1𝑜 <N (1𝑜 +N 1𝑜)
2 1pi 9652 . . . . . . 7 1𝑜N
3 mulidpi 9655 . . . . . . 7 (1𝑜N → (1𝑜 ·N 1𝑜) = 1𝑜)
42, 3ax-mp 5 . . . . . 6 (1𝑜 ·N 1𝑜) = 1𝑜
5 addclpi 9661 . . . . . . . 8 ((1𝑜N ∧ 1𝑜N) → (1𝑜 +N 1𝑜) ∈ N)
62, 2, 5mp2an 707 . . . . . . 7 (1𝑜 +N 1𝑜) ∈ N
7 mulidpi 9655 . . . . . . 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 4645 . . . . 5 (1𝑜 ·N 1𝑜) <N ((1𝑜 +N 1𝑜) ·N 1𝑜)
10 ordpipq 9711 . . . . 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 9685 . . . 4 1Q = ⟨1𝑜, 1𝑜
1312, 12oveq12i 6619 . . . . 5 (1Q +pQ 1Q) = (⟨1𝑜, 1𝑜⟩ +pQ ⟨1𝑜, 1𝑜⟩)
14 addpipq 9706 . . . . . 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 708 . . . . 5 (⟨1𝑜, 1𝑜⟩ +pQ ⟨1𝑜, 1𝑜⟩) = ⟨((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)), (1𝑜 ·N 1𝑜)⟩
164, 4oveq12i 6619 . . . . . 6 ((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)) = (1𝑜 +N 1𝑜)
1716, 4opeq12i 4377 . . . . 5 ⟨((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)), (1𝑜 ·N 1𝑜)⟩ = ⟨(1𝑜 +N 1𝑜), 1𝑜
1813, 15, 173eqtri 2647 . . . 4 (1Q +pQ 1Q) = ⟨(1𝑜 +N 1𝑜), 1𝑜
1911, 12, 183brtr4i 4645 . . 3 1Q <pQ (1Q +pQ 1Q)
20 lterpq 9739 . . 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 9697 . . . 4 1QQ
23 nqerid 9702 . . . 4 (1QQ → ([Q]‘1Q) = 1Q)
2422, 23ax-mp 5 . . 3 ([Q]‘1Q) = 1Q
2524eqcomi 2630 . 2 1Q = ([Q]‘1Q)
26 addpqnq 9707 . . 3 ((1QQ ∧ 1QQ) → (1Q +Q 1Q) = ([Q]‘(1Q +pQ 1Q)))
2722, 22, 26mp2an 707 . 2 (1Q +Q 1Q) = ([Q]‘(1Q +pQ 1Q))
2821, 25, 273brtr4i 4645 1 1Q <Q (1Q +Q 1Q)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  cop 4156   class class class wbr 4615  cfv 5849  (class class class)co 6607  1𝑜c1o 7501  Ncnpi 9613   +N cpli 9614   ·N cmi 9615   <N clti 9616   +pQ cplpq 9617   <pQ cltpq 9619  Qcnq 9621  1Qc1q 9622  [Q]cerq 9623   +Q cplq 9624   <Q cltq 9627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-1o 7508  df-oadd 7512  df-omul 7513  df-er 7690  df-ni 9641  df-pli 9642  df-mi 9643  df-lti 9644  df-plpq 9677  df-ltpq 9679  df-enq 9680  df-nq 9681  df-erq 9682  df-plq 9683  df-1nq 9685  df-ltnq 9687
This theorem is referenced by:  ltaddnq  9743
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