Theorem 1lt2nq 6504

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

Step	Hyp	Ref	Expression
1		1lt2pi 6438	. . . . 5 ⊢ 1𝑜 <N (1𝑜 +N 1𝑜)
2		1pi 6413	. . . . . 6 ⊢ 1𝑜 ∈ N
3		mulidpi 6416	. . . . . 6 ⊢ (1𝑜 ∈ N → (1𝑜 ·N 1𝑜) = 1𝑜)
4	2, 3	ax-mp 7	. . . . 5 ⊢ (1𝑜 ·N 1𝑜) = 1𝑜
5	4, 4	oveq12i 5524	. . . . 5 ⊢ ((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)) = (1𝑜 +N 1𝑜)
6	1, 4, 5	3brtr4i 3792	. . . 4 ⊢ (1𝑜 ·N 1𝑜) <N ((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜))
7		mulclpi 6426	. . . . . 6 ⊢ ((1𝑜 ∈ N ∧ 1𝑜 ∈ N) → (1𝑜 ·N 1𝑜) ∈ N)
8	2, 2, 7	mp2an 402	. . . . 5 ⊢ (1𝑜 ·N 1𝑜) ∈ N
9		addclpi 6425	. . . . . 6 ⊢ (((1𝑜 ·N 1𝑜) ∈ N ∧ (1𝑜 ·N 1𝑜) ∈ N) → ((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)) ∈ N)
10	8, 8, 9	mp2an 402	. . . . 5 ⊢ ((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)) ∈ N
11		ltmpig 6437	. . . . 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)))))
12	8, 10, 2, 11	mp3an 1232	. . . 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))))
13	6, 12	mpbi 133	. . 3 ⊢ (1𝑜 ·N (1𝑜 ·N 1𝑜)) <N (1𝑜 ·N ((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)))
14		ordpipqqs 6472	. . . 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)))))
15	2, 2, 10, 8, 14	mp4an 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))))
16	13, 15	mpbir 134	. 2 ⊢ [⟨1𝑜, 1𝑜⟩] ~Q <Q [⟨((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)), (1𝑜 ·N 1𝑜)⟩] ~Q
17		df-1nqqs 6449	. 2 ⊢ 1Q = [⟨1𝑜, 1𝑜⟩] ~Q
18	17, 17	oveq12i 5524	. . 3 ⊢ (1Q +Q 1Q) = ([⟨1𝑜, 1𝑜⟩] ~Q +Q [⟨1𝑜, 1𝑜⟩] ~Q )
19		addpipqqs 6468	. . . 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 )
20	2, 2, 2, 2, 19	mp4an 403	. . 3 ⊢ ([⟨1𝑜, 1𝑜⟩] ~Q +Q [⟨1𝑜, 1𝑜⟩] ~Q ) = [⟨((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)), (1𝑜 ·N 1𝑜)⟩] ~Q
21	18, 20	eqtri 2060	. 2 ⊢ (1Q +Q 1Q) = [⟨((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)), (1𝑜 ·N 1𝑜)⟩] ~Q
22	16, 17, 21	3brtr4i 3792	1 ⊢ 1Q <Q (1Q +Q 1Q)

Syntax hints:   ↔ wb 98   = wceq 1243   ∈ wcel 1393  ⟨cop 3378   class class class wbr 3764  (class class class)co 5512  1_𝑜c1o 5994  [cec 6104  Ncnpi 6370   +_N cpli 6371   ·_N cmi 6372   <_N clti 6373   ~_Q ceq 6377  1_Qc1q 6379   +_Q cplq 6380   <_Q cltq 6383

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-1nqqs 6449  df-ltnqqs 6451

This theorem is referenced by:  ltaddnq  6505