Theorem 1lt2nq 9996

Description: One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
1lt2nq	⊢ 1Q <Q (1Q +Q 1Q)

Proof of Theorem 1lt2nq

Step	Hyp	Ref	Expression
1		1lt2pi 9928	. . . . . 6 ⊢ 1𝑜 <N (1𝑜 +N 1𝑜)
2		1pi 9906	. . . . . . 7 ⊢ 1𝑜 ∈ N
3		mulidpi 9909	. . . . . . 7 ⊢ (1𝑜 ∈ N → (1𝑜 ·N 1𝑜) = 1𝑜)
4	2, 3	ax-mp 5	. . . . . 6 ⊢ (1𝑜 ·N 1𝑜) = 1𝑜
5		addclpi 9915	. . . . . . . 8 ⊢ ((1𝑜 ∈ N ∧ 1𝑜 ∈ N) → (1𝑜 +N 1𝑜) ∈ N)
6	2, 2, 5	mp2an 664	. . . . . . 7 ⊢ (1𝑜 +N 1𝑜) ∈ N
7		mulidpi 9909	. . . . . . 7 ⊢ ((1𝑜 +N 1𝑜) ∈ N → ((1𝑜 +N 1𝑜) ·N 1𝑜) = (1𝑜 +N 1𝑜))
8	6, 7	ax-mp 5	. . . . . 6 ⊢ ((1𝑜 +N 1𝑜) ·N 1𝑜) = (1𝑜 +N 1𝑜)
9	1, 4, 8	3brtr4i 4814	. . . . 5 ⊢ (1𝑜 ·N 1𝑜) <N ((1𝑜 +N 1𝑜) ·N 1𝑜)
10		ordpipq 9965	. . . . 5 ⊢ (⟨1𝑜, 1𝑜⟩ <pQ ⟨(1𝑜 +N 1𝑜), 1𝑜⟩ ↔ (1𝑜 ·N 1𝑜) <N ((1𝑜 +N 1𝑜) ·N 1𝑜))
11	9, 10	mpbir 221	. . . 4 ⊢ ⟨1𝑜, 1𝑜⟩ <pQ ⟨(1𝑜 +N 1𝑜), 1𝑜⟩
12		df-1nq 9939	. . . 4 ⊢ 1Q = ⟨1𝑜, 1𝑜⟩
13	12, 12	oveq12i 6804	. . . . 5 ⊢ (1Q +pQ 1Q) = (⟨1𝑜, 1𝑜⟩ +pQ ⟨1𝑜, 1𝑜⟩)
14		addpipq 9960	. . . . . 6 ⊢ (((1𝑜 ∈ N ∧ 1𝑜 ∈ N) ∧ (1𝑜 ∈ N ∧ 1𝑜 ∈ N)) → (⟨1𝑜, 1𝑜⟩ +pQ ⟨1𝑜, 1𝑜⟩) = ⟨((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)), (1𝑜 ·N 1𝑜)⟩)
15	2, 2, 2, 2, 14	mp4an 665	. . . . 5 ⊢ (⟨1𝑜, 1𝑜⟩ +pQ ⟨1𝑜, 1𝑜⟩) = ⟨((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)), (1𝑜 ·N 1𝑜)⟩
16	4, 4	oveq12i 6804	. . . . . 6 ⊢ ((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)) = (1𝑜 +N 1𝑜)
17	16, 4	opeq12i 4542	. . . . 5 ⊢ ⟨((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)), (1𝑜 ·N 1𝑜)⟩ = ⟨(1𝑜 +N 1𝑜), 1𝑜⟩
18	13, 15, 17	3eqtri 2796	. . . 4 ⊢ (1Q +pQ 1Q) = ⟨(1𝑜 +N 1𝑜), 1𝑜⟩
19	11, 12, 18	3brtr4i 4814	. . 3 ⊢ 1Q <pQ (1Q +pQ 1Q)
20		lterpq 9993	. . 3 ⊢ (1Q <pQ (1Q +pQ 1Q) ↔ ([Q]‘1Q) <Q ([Q]‘(1Q +pQ 1Q)))
21	19, 20	mpbi 220	. 2 ⊢ ([Q]‘1Q) <Q ([Q]‘(1Q +pQ 1Q))
22		1nq 9951	. . . 4 ⊢ 1Q ∈ Q
23		nqerid 9956	. . . 4 ⊢ (1Q ∈ Q → ([Q]‘1Q) = 1Q)
24	22, 23	ax-mp 5	. . 3 ⊢ ([Q]‘1Q) = 1Q
25	24	eqcomi 2779	. 2 ⊢ 1Q = ([Q]‘1Q)
26		addpqnq 9961	. . 3 ⊢ ((1Q ∈ Q ∧ 1Q ∈ Q) → (1Q +Q 1Q) = ([Q]‘(1Q +pQ 1Q)))
27	22, 22, 26	mp2an 664	. 2 ⊢ (1Q +Q 1Q) = ([Q]‘(1Q +pQ 1Q))
28	21, 25, 27	3brtr4i 4814	1 ⊢ 1Q <Q (1Q +Q 1Q)

Syntax hints:   = wceq 1630   ∈ wcel 2144  ⟨cop 4320   class class class wbr 4784  ‘cfv 6031  (class class class)co 6792  1_𝑜c1o 7705  Ncnpi 9867   +_N cpli 9868   ·_N cmi 9869   <_N clti 9870   +_pQ cplpq 9871   <_pQ cltpq 9873  Qcnq 9875  1_Qc1q 9876  [Q]cerq 9877   +_Q cplq 9878   <_Q cltq 9881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-omul 7717  df-er 7895  df-ni 9895  df-pli 9896  df-mi 9897  df-lti 9898  df-plpq 9931  df-ltpq 9933  df-enq 9934  df-nq 9935  df-erq 9936  df-plq 9937  df-1nq 9939  df-ltnq 9941

This theorem is referenced by:  ltaddnq  9997