MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  1lt2nq Structured version   Visualization version   GIF version

Theorem 1lt2nq 11042
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 10974 . . . . . 6 1o <N (1o +N 1o)
2 1pi 10952 . . . . . . 7 1oN
3 mulidpi 10955 . . . . . . 7 (1oN → (1o ·N 1o) = 1o)
42, 3ax-mp 5 . . . . . 6 (1o ·N 1o) = 1o
5 addclpi 10961 . . . . . . . 8 ((1oN ∧ 1oN) → (1o +N 1o) ∈ N)
62, 2, 5mp2an 691 . . . . . . 7 (1o +N 1o) ∈ N
7 mulidpi 10955 . . . . . . 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 5196 . . . . 5 (1o ·N 1o) <N ((1o +N 1o) ·N 1o)
10 ordpipq 11011 . . . . 5 (⟨1o, 1o⟩ <pQ ⟨(1o +N 1o), 1o⟩ ↔ (1o ·N 1o) <N ((1o +N 1o) ·N 1o))
119, 10mpbir 231 . . . 4 ⟨1o, 1o⟩ <pQ ⟨(1o +N 1o), 1o
12 df-1nq 10985 . . . 4 1Q = ⟨1o, 1o
1312, 12oveq12i 7460 . . . . 5 (1Q +pQ 1Q) = (⟨1o, 1o⟩ +pQ ⟨1o, 1o⟩)
14 addpipq 11006 . . . . . 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 7460 . . . . . 6 ((1o ·N 1o) +N (1o ·N 1o)) = (1o +N 1o)
1716, 4opeq12i 4902 . . . . 5 ⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩ = ⟨(1o +N 1o), 1o
1813, 15, 173eqtri 2772 . . . 4 (1Q +pQ 1Q) = ⟨(1o +N 1o), 1o
1911, 12, 183brtr4i 5196 . . 3 1Q <pQ (1Q +pQ 1Q)
20 lterpq 11039 . . 3 (1Q <pQ (1Q +pQ 1Q) ↔ ([Q]‘1Q) <Q ([Q]‘(1Q +pQ 1Q)))
2119, 20mpbi 230 . 2 ([Q]‘1Q) <Q ([Q]‘(1Q +pQ 1Q))
22 1nq 10997 . . . 4 1QQ
23 nqerid 11002 . . . 4 (1QQ → ([Q]‘1Q) = 1Q)
2422, 23ax-mp 5 . . 3 ([Q]‘1Q) = 1Q
2524eqcomi 2749 . 2 1Q = ([Q]‘1Q)
26 addpqnq 11007 . . 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 5196 1 1Q <Q (1Q +Q 1Q)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2108  cop 4654   class class class wbr 5166  cfv 6573  (class class class)co 7448  1oc1o 8515  Ncnpi 10913   +N cpli 10914   ·N cmi 10915   <N clti 10916   +pQ cplpq 10917   <pQ cltpq 10919  Qcnq 10921  1Qc1q 10922  [Q]cerq 10923   +Q cplq 10924   <Q cltq 10927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-oadd 8526  df-omul 8527  df-er 8763  df-ni 10941  df-pli 10942  df-mi 10943  df-lti 10944  df-plpq 10977  df-ltpq 10979  df-enq 10980  df-nq 10981  df-erq 10982  df-plq 10983  df-1nq 10985  df-ltnq 10987
This theorem is referenced by:  ltaddnq  11043
  Copyright terms: Public domain W3C validator