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Theorem 1lt2nq 6389
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 6324 . . . . 5 1𝑜 <N (1𝑜 +N 1𝑜)
2 1pi 6299 . . . . . 6 1𝑜 N
3 mulidpi 6302 . . . . . 6 (1𝑜 N → (1𝑜 ·N 1𝑜) = 1𝑜)
42, 3ax-mp 7 . . . . 5 (1𝑜 ·N 1𝑜) = 1𝑜
54, 4oveq12i 5467 . . . . 5 ((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)) = (1𝑜 +N 1𝑜)
61, 4, 53brtr4i 3783 . . . 4 (1𝑜 ·N 1𝑜) <N ((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜))
7 mulclpi 6312 . . . . . 6 ((1𝑜 N 1𝑜 N) → (1𝑜 ·N 1𝑜) N)
82, 2, 7mp2an 402 . . . . 5 (1𝑜 ·N 1𝑜) N
9 addclpi 6311 . . . . . 6 (((1𝑜 ·N 1𝑜) N (1𝑜 ·N 1𝑜) N) → ((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)) N)
108, 8, 9mp2an 402 . . . . 5 ((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)) N
11 ltmpig 6323 . . . . 5 (((1𝑜 ·N 1𝑜) N ((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)) N 1𝑜 N) → ((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 1231 . . . 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 133 . . 3 (1𝑜 ·N (1𝑜 ·N 1𝑜)) <N (1𝑜 ·N ((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)))
14 ordpipqqs 6358 . . . 4 (((1𝑜 N 1𝑜 N) (((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 403 . . 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 134 . 2 [⟨1𝑜, 1𝑜⟩] ~Q <Q [⟨((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)), (1𝑜 ·N 1𝑜)⟩] ~Q
17 df-1nqqs 6335 . 2 1Q = [⟨1𝑜, 1𝑜⟩] ~Q
1817, 17oveq12i 5467 . . 3 (1Q +Q 1Q) = ([⟨1𝑜, 1𝑜⟩] ~Q +Q [⟨1𝑜, 1𝑜⟩] ~Q )
19 addpipqqs 6354 . . . 4 (((1𝑜 N 1𝑜 N) (1𝑜 N 1𝑜 N)) → ([⟨1𝑜, 1𝑜⟩] ~Q +Q [⟨1𝑜, 1𝑜⟩] ~Q ) = [⟨((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)), (1𝑜 ·N 1𝑜)⟩] ~Q )
202, 2, 2, 2, 19mp4an 403 . . 3 ([⟨1𝑜, 1𝑜⟩] ~Q +Q [⟨1𝑜, 1𝑜⟩] ~Q ) = [⟨((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)), (1𝑜 ·N 1𝑜)⟩] ~Q
2118, 20eqtri 2057 . 2 (1Q +Q 1Q) = [⟨((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)), (1𝑜 ·N 1𝑜)⟩] ~Q
2216, 17, 213brtr4i 3783 1 1Q <Q (1Q +Q 1Q)
Colors of variables: wff set class
Syntax hints:  wb 98   = wceq 1242   wcel 1390  cop 3370   class class class wbr 3755  (class class class)co 5455  1𝑜c1o 5933  [cec 6040  Ncnpi 6256   +N cpli 6257   ·N cmi 6258   <N clti 6259   ~Q ceq 6263  1Qc1q 6265   +Q cplq 6266   <Q cltq 6269
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-1nqqs 6335  df-ltnqqs 6337
This theorem is referenced by:  ltaddnq  6390
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