Theorem 1lt2nq 10926

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 10858	. . . . . 6 ⊢ 1o <N (1o +N 1o)
2		1pi 10836	. . . . . . 7 ⊢ 1o ∈ N
3		mulidpi 10839	. . . . . . 7 ⊢ (1o ∈ N → (1o ·N 1o) = 1o)
4	2, 3	ax-mp 5	. . . . . 6 ⊢ (1o ·N 1o) = 1o
5		addclpi 10845	. . . . . . . 8 ⊢ ((1o ∈ N ∧ 1o ∈ N) → (1o +N 1o) ∈ N)
6	2, 2, 5	mp2an 692	. . . . . . 7 ⊢ (1o +N 1o) ∈ N
7		mulidpi 10839	. . . . . . 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 5137	. . . . 5 ⊢ (1o ·N 1o) <N ((1o +N 1o) ·N 1o)
10		ordpipq 10895	. . . . 5 ⊢ (⟨1o, 1o⟩ <pQ ⟨(1o +N 1o), 1o⟩ ↔ (1o ·N 1o) <N ((1o +N 1o) ·N 1o))
11	9, 10	mpbir 231	. . . 4 ⊢ ⟨1o, 1o⟩ <pQ ⟨(1o +N 1o), 1o⟩
12		df-1nq 10869	. . . 4 ⊢ 1Q = ⟨1o, 1o⟩
13	12, 12	oveq12i 7399	. . . . 5 ⊢ (1Q +pQ 1Q) = (⟨1o, 1o⟩ +pQ ⟨1o, 1o⟩)
14		addpipq 10890	. . . . . 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 693	. . . . 5 ⊢ (⟨1o, 1o⟩ +pQ ⟨1o, 1o⟩) = ⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩
16	4, 4	oveq12i 7399	. . . . . 6 ⊢ ((1o ·N 1o) +N (1o ·N 1o)) = (1o +N 1o)
17	16, 4	opeq12i 4842	. . . . 5 ⊢ ⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩ = ⟨(1o +N 1o), 1o⟩
18	13, 15, 17	3eqtri 2756	. . . 4 ⊢ (1Q +pQ 1Q) = ⟨(1o +N 1o), 1o⟩
19	11, 12, 18	3brtr4i 5137	. . 3 ⊢ 1Q <pQ (1Q +pQ 1Q)
20		lterpq 10923	. . 3 ⊢ (1Q <pQ (1Q +pQ 1Q) ↔ ([Q]‘1Q) <Q ([Q]‘(1Q +pQ 1Q)))
21	19, 20	mpbi 230	. 2 ⊢ ([Q]‘1Q) <Q ([Q]‘(1Q +pQ 1Q))
22		1nq 10881	. . . 4 ⊢ 1Q ∈ Q
23		nqerid 10886	. . . 4 ⊢ (1Q ∈ Q → ([Q]‘1Q) = 1Q)
24	22, 23	ax-mp 5	. . 3 ⊢ ([Q]‘1Q) = 1Q
25	24	eqcomi 2738	. 2 ⊢ 1Q = ([Q]‘1Q)
26		addpqnq 10891	. . 3 ⊢ ((1Q ∈ Q ∧ 1Q ∈ Q) → (1Q +Q 1Q) = ([Q]‘(1Q +pQ 1Q)))
27	22, 22, 26	mp2an 692	. 2 ⊢ (1Q +Q 1Q) = ([Q]‘(1Q +pQ 1Q))
28	21, 25, 27	3brtr4i 5137	1 ⊢ 1Q <Q (1Q +Q 1Q)

Syntax hints:   = wceq 1540   ∈ wcel 2109  ⟨cop 4595   class class class wbr 5107  ‘cfv 6511  (class class class)co 7387  1_oc1o 8427  Ncnpi 10797   +_N cpli 10798   ·_N cmi 10799   <_N clti 10800   +_pQ cplpq 10801   <_pQ cltpq 10803  Qcnq 10805  1_Qc1q 10806  [Q]cerq 10807   +_Q cplq 10808   <_Q cltq 10811

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-oadd 8438  df-omul 8439  df-er 8671  df-ni 10825  df-pli 10826  df-mi 10827  df-lti 10828  df-plpq 10861  df-ltpq 10863  df-enq 10864  df-nq 10865  df-erq 10866  df-plq 10867  df-1nq 10869  df-ltnq 10871

This theorem is referenced by:  ltaddnq  10927