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Theorem 1lt2nq 7182
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 7116 . . . . 5 1o <N (1o +N 1o)
2 1pi 7091 . . . . . 6 1oN
3 mulidpi 7094 . . . . . 6 (1oN → (1o ·N 1o) = 1o)
42, 3ax-mp 5 . . . . 5 (1o ·N 1o) = 1o
54, 4oveq12i 5754 . . . . 5 ((1o ·N 1o) +N (1o ·N 1o)) = (1o +N 1o)
61, 4, 53brtr4i 3928 . . . 4 (1o ·N 1o) <N ((1o ·N 1o) +N (1o ·N 1o))
7 mulclpi 7104 . . . . . 6 ((1oN ∧ 1oN) → (1o ·N 1o) ∈ N)
82, 2, 7mp2an 422 . . . . 5 (1o ·N 1o) ∈ N
9 addclpi 7103 . . . . . 6 (((1o ·N 1o) ∈ N ∧ (1o ·N 1o) ∈ N) → ((1o ·N 1o) +N (1o ·N 1o)) ∈ N)
108, 8, 9mp2an 422 . . . . 5 ((1o ·N 1o) +N (1o ·N 1o)) ∈ N
11 ltmpig 7115 . . . . 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 1300 . . . 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 144 . . 3 (1o ·N (1o ·N 1o)) <N (1o ·N ((1o ·N 1o) +N (1o ·N 1o)))
14 ordpipqqs 7150 . . . 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 423 . . 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 145 . 2 [⟨1o, 1o⟩] ~Q <Q [⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q
17 df-1nqqs 7127 . 2 1Q = [⟨1o, 1o⟩] ~Q
1817, 17oveq12i 5754 . . 3 (1Q +Q 1Q) = ([⟨1o, 1o⟩] ~Q +Q [⟨1o, 1o⟩] ~Q )
19 addpipqqs 7146 . . . 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 423 . . 3 ([⟨1o, 1o⟩] ~Q +Q [⟨1o, 1o⟩] ~Q ) = [⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q
2118, 20eqtri 2138 . 2 (1Q +Q 1Q) = [⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q
2216, 17, 213brtr4i 3928 1 1Q <Q (1Q +Q 1Q)
Colors of variables: wff set class
Syntax hints:  wb 104   = wceq 1316  wcel 1465  cop 3500   class class class wbr 3899  (class class class)co 5742  1oc1o 6274  [cec 6395  Ncnpi 7048   +N cpli 7049   ·N cmi 7050   <N clti 7051   ~Q ceq 7055  1Qc1q 7057   +Q cplq 7058   <Q cltq 7061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-1nqqs 7127  df-ltnqqs 7129
This theorem is referenced by:  ltaddnq  7183
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