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Theorem 1lt2nq 10965
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 10897 . . . . . 6 1o <N (1o +N 1o)
2 1pi 10875 . . . . . . 7 1oN
3 mulidpi 10878 . . . . . . 7 (1oN → (1o ·N 1o) = 1o)
42, 3ax-mp 5 . . . . . 6 (1o ·N 1o) = 1o
5 addclpi 10884 . . . . . . . 8 ((1oN ∧ 1oN) → (1o +N 1o) ∈ N)
62, 2, 5mp2an 691 . . . . . . 7 (1o +N 1o) ∈ N
7 mulidpi 10878 . . . . . . 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 5178 . . . . 5 (1o ·N 1o) <N ((1o +N 1o) ·N 1o)
10 ordpipq 10934 . . . . 5 (⟨1o, 1o⟩ <pQ ⟨(1o +N 1o), 1o⟩ ↔ (1o ·N 1o) <N ((1o +N 1o) ·N 1o))
119, 10mpbir 230 . . . 4 ⟨1o, 1o⟩ <pQ ⟨(1o +N 1o), 1o
12 df-1nq 10908 . . . 4 1Q = ⟨1o, 1o
1312, 12oveq12i 7418 . . . . 5 (1Q +pQ 1Q) = (⟨1o, 1o⟩ +pQ ⟨1o, 1o⟩)
14 addpipq 10929 . . . . . 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 692 . . . . 5 (⟨1o, 1o⟩ +pQ ⟨1o, 1o⟩) = ⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩
164, 4oveq12i 7418 . . . . . 6 ((1o ·N 1o) +N (1o ·N 1o)) = (1o +N 1o)
1716, 4opeq12i 4878 . . . . 5 ⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩ = ⟨(1o +N 1o), 1o
1813, 15, 173eqtri 2765 . . . 4 (1Q +pQ 1Q) = ⟨(1o +N 1o), 1o
1911, 12, 183brtr4i 5178 . . 3 1Q <pQ (1Q +pQ 1Q)
20 lterpq 10962 . . 3 (1Q <pQ (1Q +pQ 1Q) ↔ ([Q]‘1Q) <Q ([Q]‘(1Q +pQ 1Q)))
2119, 20mpbi 229 . 2 ([Q]‘1Q) <Q ([Q]‘(1Q +pQ 1Q))
22 1nq 10920 . . . 4 1QQ
23 nqerid 10925 . . . 4 (1QQ → ([Q]‘1Q) = 1Q)
2422, 23ax-mp 5 . . 3 ([Q]‘1Q) = 1Q
2524eqcomi 2742 . 2 1Q = ([Q]‘1Q)
26 addpqnq 10930 . . 3 ((1QQ ∧ 1QQ) → (1Q +Q 1Q) = ([Q]‘(1Q +pQ 1Q)))
2722, 22, 26mp2an 691 . 2 (1Q +Q 1Q) = ([Q]‘(1Q +pQ 1Q))
2821, 25, 273brtr4i 5178 1 1Q <Q (1Q +Q 1Q)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  cop 4634   class class class wbr 5148  cfv 6541  (class class class)co 7406  1oc1o 8456  Ncnpi 10836   +N cpli 10837   ·N cmi 10838   <N clti 10839   +pQ cplpq 10840   <pQ cltpq 10842  Qcnq 10844  1Qc1q 10845  [Q]cerq 10846   +Q cplq 10847   <Q cltq 10850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-oadd 8467  df-omul 8468  df-er 8700  df-ni 10864  df-pli 10865  df-mi 10866  df-lti 10867  df-plpq 10900  df-ltpq 10902  df-enq 10903  df-nq 10904  df-erq 10905  df-plq 10906  df-1nq 10908  df-ltnq 10910
This theorem is referenced by:  ltaddnq  10966
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