Theorem 1lt2nq 7561

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 7495	. . . . 5 ⊢ 1o <N (1o +N 1o)
2		1pi 7470	. . . . . 6 ⊢ 1o ∈ N
3		mulidpi 7473	. . . . . 6 ⊢ (1o ∈ N → (1o ·N 1o) = 1o)
4	2, 3	ax-mp 5	. . . . 5 ⊢ (1o ·N 1o) = 1o
5	4, 4	oveq12i 5986	. . . . 5 ⊢ ((1o ·N 1o) +N (1o ·N 1o)) = (1o +N 1o)
6	1, 4, 5	3brtr4i 4092	. . . 4 ⊢ (1o ·N 1o) <N ((1o ·N 1o) +N (1o ·N 1o))
7		mulclpi 7483	. . . . . 6 ⊢ ((1o ∈ N ∧ 1o ∈ N) → (1o ·N 1o) ∈ N)
8	2, 2, 7	mp2an 426	. . . . 5 ⊢ (1o ·N 1o) ∈ N
9		addclpi 7482	. . . . . 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 7494	. . . . 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 1352	. . . 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 7529	. . . 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 7506	. 2 ⊢ 1Q = [⟨1o, 1o⟩] ~Q
18	17, 17	oveq12i 5986	. . 3 ⊢ (1Q +Q 1Q) = ([⟨1o, 1o⟩] ~Q +Q [⟨1o, 1o⟩] ~Q )
19		addpipqqs 7525	. . . 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 2230	. 2 ⊢ (1Q +Q 1Q) = [⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q
22	16, 17, 21	3brtr4i 4092	1 ⊢ 1Q <Q (1Q +Q 1Q)

Syntax hints:   ↔ wb 105   = wceq 1375   ∈ wcel 2180  ⟨cop 3649   class class class wbr 4062  (class class class)co 5974  1_oc1o 6525  [cec 6648  Ncnpi 7427   +_N cpli 7428   ·_N cmi 7429   <_N clti 7430   ~_Q ceq 7434  1_Qc1q 7436   +_Q cplq 7437   <_Q cltq 7440

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-iinf 4657

This theorem depends on definitions:  df-bi 117  df-dc 839  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-eprel 4357  df-id 4361  df-iord 4434  df-on 4436  df-suc 4439  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-recs 6421  df-irdg 6486  df-1o 6532  df-oadd 6536  df-omul 6537  df-er 6650  df-ec 6652  df-qs 6656  df-ni 7459  df-pli 7460  df-mi 7461  df-lti 7462  df-plpq 7499  df-enq 7502  df-nqqs 7503  df-plqqs 7504  df-1nqqs 7506  df-ltnqqs 7508

This theorem is referenced by:  ltaddnq  7562