Theorem 1lt2nq 7609

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 7543	. . . . 5 ⊢ 1o <N (1o +N 1o)
2		1pi 7518	. . . . . 6 ⊢ 1o ∈ N
3		mulidpi 7521	. . . . . 6 ⊢ (1o ∈ N → (1o ·N 1o) = 1o)
4	2, 3	ax-mp 5	. . . . 5 ⊢ (1o ·N 1o) = 1o
5	4, 4	oveq12i 6022	. . . . 5 ⊢ ((1o ·N 1o) +N (1o ·N 1o)) = (1o +N 1o)
6	1, 4, 5	3brtr4i 4113	. . . 4 ⊢ (1o ·N 1o) <N ((1o ·N 1o) +N (1o ·N 1o))
7		mulclpi 7531	. . . . . 6 ⊢ ((1o ∈ N ∧ 1o ∈ N) → (1o ·N 1o) ∈ N)
8	2, 2, 7	mp2an 426	. . . . 5 ⊢ (1o ·N 1o) ∈ N
9		addclpi 7530	. . . . . 6 ⊢ (((1o ·N 1o) ∈ N ∧ (1o ·N 1o) ∈ N) → ((1o ·N 1o) +N (1o ·N 1o)) ∈ N)
10	8, 8, 9	mp2an 426	. . . . 5 ⊢ ((1o ·N 1o) +N (1o ·N 1o)) ∈ N
11		ltmpig 7542	. . . . 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 1371	. . . 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 145	. . 3 ⊢ (1o ·N (1o ·N 1o)) <N (1o ·N ((1o ·N 1o) +N (1o ·N 1o)))
14		ordpipqqs 7577	. . . 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 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))))
16	13, 15	mpbir 146	. 2 ⊢ [⟨1o, 1o⟩] ~Q <Q [⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q
17		df-1nqqs 7554	. 2 ⊢ 1Q = [⟨1o, 1o⟩] ~Q
18	17, 17	oveq12i 6022	. . 3 ⊢ (1Q +Q 1Q) = ([⟨1o, 1o⟩] ~Q +Q [⟨1o, 1o⟩] ~Q )
19		addpipqqs 7573	. . . 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 427	. . 3 ⊢ ([⟨1o, 1o⟩] ~Q +Q [⟨1o, 1o⟩] ~Q ) = [⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q
21	18, 20	eqtri 2250	. 2 ⊢ (1Q +Q 1Q) = [⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q
22	16, 17, 21	3brtr4i 4113	1 ⊢ 1Q <Q (1Q +Q 1Q)

Syntax hints:   ↔ wb 105   = wceq 1395   ∈ wcel 2200  ⟨cop 3669   class class class wbr 4083  (class class class)co 6010  1_oc1o 6566  [cec 6691  Ncnpi 7475   +_N cpli 7476   ·_N cmi 7477   <_N clti 7478   ~_Q ceq 7482  1_Qc1q 7484   +_Q cplq 7485   <_Q cltq 7488

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4381  df-id 4385  df-iord 4458  df-on 4460  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-irdg 6527  df-1o 6573  df-oadd 6577  df-omul 6578  df-er 6693  df-ec 6695  df-qs 6699  df-ni 7507  df-pli 7508  df-mi 7509  df-lti 7510  df-plpq 7547  df-enq 7550  df-nqqs 7551  df-plqqs 7552  df-1nqqs 7554  df-ltnqqs 7556

This theorem is referenced by:  ltaddnq  7610