Theorem 1lt2nq 10241

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 10173	. . . . . 6 ⊢ 1o <N (1o +N 1o)
2		1pi 10151	. . . . . . 7 ⊢ 1o ∈ N
3		mulidpi 10154	. . . . . . 7 ⊢ (1o ∈ N → (1o ·N 1o) = 1o)
4	2, 3	ax-mp 5	. . . . . 6 ⊢ (1o ·N 1o) = 1o
5		addclpi 10160	. . . . . . . 8 ⊢ ((1o ∈ N ∧ 1o ∈ N) → (1o +N 1o) ∈ N)
6	2, 2, 5	mp2an 688	. . . . . . 7 ⊢ (1o +N 1o) ∈ N
7		mulidpi 10154	. . . . . . 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 4992	. . . . 5 ⊢ (1o ·N 1o) <N ((1o +N 1o) ·N 1o)
10		ordpipq 10210	. . . . 5 ⊢ (⟨1o, 1o⟩ <pQ ⟨(1o +N 1o), 1o⟩ ↔ (1o ·N 1o) <N ((1o +N 1o) ·N 1o))
11	9, 10	mpbir 232	. . . 4 ⊢ ⟨1o, 1o⟩ <pQ ⟨(1o +N 1o), 1o⟩
12		df-1nq 10184	. . . 4 ⊢ 1Q = ⟨1o, 1o⟩
13	12, 12	oveq12i 7028	. . . . 5 ⊢ (1Q +pQ 1Q) = (⟨1o, 1o⟩ +pQ ⟨1o, 1o⟩)
14		addpipq 10205	. . . . . 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 689	. . . . 5 ⊢ (⟨1o, 1o⟩ +pQ ⟨1o, 1o⟩) = ⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩
16	4, 4	oveq12i 7028	. . . . . 6 ⊢ ((1o ·N 1o) +N (1o ·N 1o)) = (1o +N 1o)
17	16, 4	opeq12i 4715	. . . . 5 ⊢ ⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩ = ⟨(1o +N 1o), 1o⟩
18	13, 15, 17	3eqtri 2823	. . . 4 ⊢ (1Q +pQ 1Q) = ⟨(1o +N 1o), 1o⟩
19	11, 12, 18	3brtr4i 4992	. . 3 ⊢ 1Q <pQ (1Q +pQ 1Q)
20		lterpq 10238	. . 3 ⊢ (1Q <pQ (1Q +pQ 1Q) ↔ ([Q]‘1Q) <Q ([Q]‘(1Q +pQ 1Q)))
21	19, 20	mpbi 231	. 2 ⊢ ([Q]‘1Q) <Q ([Q]‘(1Q +pQ 1Q))
22		1nq 10196	. . . 4 ⊢ 1Q ∈ Q
23		nqerid 10201	. . . 4 ⊢ (1Q ∈ Q → ([Q]‘1Q) = 1Q)
24	22, 23	ax-mp 5	. . 3 ⊢ ([Q]‘1Q) = 1Q
25	24	eqcomi 2804	. 2 ⊢ 1Q = ([Q]‘1Q)
26		addpqnq 10206	. . 3 ⊢ ((1Q ∈ Q ∧ 1Q ∈ Q) → (1Q +Q 1Q) = ([Q]‘(1Q +pQ 1Q)))
27	22, 22, 26	mp2an 688	. 2 ⊢ (1Q +Q 1Q) = ([Q]‘(1Q +pQ 1Q))
28	21, 25, 27	3brtr4i 4992	1 ⊢ 1Q <Q (1Q +Q 1Q)

Syntax hints:   = wceq 1522   ∈ wcel 2081  ⟨cop 4478   class class class wbr 4962  ‘cfv 6225  (class class class)co 7016  1_oc1o 7946  Ncnpi 10112   +_N cpli 10113   ·_N cmi 10114   <_N clti 10115   +_pQ cplpq 10116   <_pQ cltpq 10118  Qcnq 10120  1_Qc1q 10121  [Q]cerq 10122   +_Q cplq 10123   <_Q cltq 10126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-oadd 7957  df-omul 7958  df-er 8139  df-ni 10140  df-pli 10141  df-mi 10142  df-lti 10143  df-plpq 10176  df-ltpq 10178  df-enq 10179  df-nq 10180  df-erq 10181  df-plq 10182  df-1nq 10184  df-ltnq 10186

This theorem is referenced by:  ltaddnq  10242