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Theorem 1lt2nq 7407
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 7341 . . . . 5 1o <N (1o +N 1o)
2 1pi 7316 . . . . . 6 1oN
3 mulidpi 7319 . . . . . 6 (1oN → (1o ·N 1o) = 1o)
42, 3ax-mp 5 . . . . 5 (1o ·N 1o) = 1o
54, 4oveq12i 5889 . . . . 5 ((1o ·N 1o) +N (1o ·N 1o)) = (1o +N 1o)
61, 4, 53brtr4i 4035 . . . 4 (1o ·N 1o) <N ((1o ·N 1o) +N (1o ·N 1o))
7 mulclpi 7329 . . . . . 6 ((1oN ∧ 1oN) → (1o ·N 1o) ∈ N)
82, 2, 7mp2an 426 . . . . 5 (1o ·N 1o) ∈ N
9 addclpi 7328 . . . . . 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 7340 . . . . 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 7375 . . . 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 7352 . 2 1Q = [⟨1o, 1o⟩] ~Q
1817, 17oveq12i 5889 . . 3 (1Q +Q 1Q) = ([⟨1o, 1o⟩] ~Q +Q [⟨1o, 1o⟩] ~Q )
19 addpipqqs 7371 . . . 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 4035 1 1Q <Q (1Q +Q 1Q)
Colors of variables: wff set class
Syntax hints:  wb 105   = wceq 1353  wcel 2148  cop 3597   class class class wbr 4005  (class class class)co 5877  1oc1o 6412  [cec 6535  Ncnpi 7273   +N cpli 7274   ·N cmi 7275   <N clti 7276   ~Q ceq 7280  1Qc1q 7282   +Q cplq 7283   <Q cltq 7286
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 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589
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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-1nqqs 7352  df-ltnqqs 7354
This theorem is referenced by:  ltaddnq  7408
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