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Theorem 1lt2nq 7404
Description: One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.)
Assertion
Ref Expression
1lt2nq 1Q <Q (1Q +Q 1Q)

Proof of Theorem 1lt2nq
StepHypRef Expression
1 1lt2pi 7338 . . . . 5 1o <N (1o +N 1o)
2 1pi 7313 . . . . . 6 1oN
3 mulidpi 7316 . . . . . 6 (1oN → (1o ·N 1o) = 1o)
42, 3ax-mp 5 . . . . 5 (1o ·N 1o) = 1o
54, 4oveq12i 5886 . . . . 5 ((1o ·N 1o) +N (1o ·N 1o)) = (1o +N 1o)
61, 4, 53brtr4i 4033 . . . 4 (1o ·N 1o) <N ((1o ·N 1o) +N (1o ·N 1o))
7 mulclpi 7326 . . . . . 6 ((1oN ∧ 1oN) → (1o ·N 1o) ∈ N)
82, 2, 7mp2an 426 . . . . 5 (1o ·N 1o) ∈ N
9 addclpi 7325 . . . . . 6 (((1o ·N 1o) ∈ N ∧ (1o ·N 1o) ∈ N) → ((1o ·N 1o) +N (1o ·N 1o)) ∈ N)
108, 8, 9mp2an 426 . . . . 5 ((1o ·N 1o) +N (1o ·N 1o)) ∈ N
11 ltmpig 7337 . . . . 5 (((1o ·N 1o) ∈ N ∧ ((1o ·N 1o) +N (1o ·N 1o)) ∈ N ∧ 1oN) → ((1o ·N 1o) <N ((1o ·N 1o) +N (1o ·N 1o)) ↔ (1o ·N (1o ·N 1o)) <N (1o ·N ((1o ·N 1o) +N (1o ·N 1o)))))
128, 10, 2, 11mp3an 1337 . . . 4 ((1o ·N 1o) <N ((1o ·N 1o) +N (1o ·N 1o)) ↔ (1o ·N (1o ·N 1o)) <N (1o ·N ((1o ·N 1o) +N (1o ·N 1o))))
136, 12mpbi 145 . . 3 (1o ·N (1o ·N 1o)) <N (1o ·N ((1o ·N 1o) +N (1o ·N 1o)))
14 ordpipqqs 7372 . . . 4 (((1oN ∧ 1oN) ∧ (((1o ·N 1o) +N (1o ·N 1o)) ∈ N ∧ (1o ·N 1o) ∈ N)) → ([⟨1o, 1o⟩] ~Q <Q [⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q ↔ (1o ·N (1o ·N 1o)) <N (1o ·N ((1o ·N 1o) +N (1o ·N 1o)))))
152, 2, 10, 8, 14mp4an 427 . . 3 ([⟨1o, 1o⟩] ~Q <Q [⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q ↔ (1o ·N (1o ·N 1o)) <N (1o ·N ((1o ·N 1o) +N (1o ·N 1o))))
1613, 15mpbir 146 . 2 [⟨1o, 1o⟩] ~Q <Q [⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q
17 df-1nqqs 7349 . 2 1Q = [⟨1o, 1o⟩] ~Q
1817, 17oveq12i 5886 . . 3 (1Q +Q 1Q) = ([⟨1o, 1o⟩] ~Q +Q [⟨1o, 1o⟩] ~Q )
19 addpipqqs 7368 . . . 4 (((1oN ∧ 1oN) ∧ (1oN ∧ 1oN)) → ([⟨1o, 1o⟩] ~Q +Q [⟨1o, 1o⟩] ~Q ) = [⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q )
202, 2, 2, 2, 19mp4an 427 . . 3 ([⟨1o, 1o⟩] ~Q +Q [⟨1o, 1o⟩] ~Q ) = [⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q
2118, 20eqtri 2198 . 2 (1Q +Q 1Q) = [⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q
2216, 17, 213brtr4i 4033 1 1Q <Q (1Q +Q 1Q)
Colors of variables: wff set class
Syntax hints:  wb 105   = wceq 1353  wcel 2148  cop 3595   class class class wbr 4003  (class class class)co 5874  1oc1o 6409  [cec 6532  Ncnpi 7270   +N cpli 7271   ·N cmi 7272   <N clti 7273   ~Q ceq 7277  1Qc1q 7279   +Q cplq 7280   <Q cltq 7283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-1nqqs 7349  df-ltnqqs 7351
This theorem is referenced by:  ltaddnq  7405
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