Theorem 1lt2nq 11013

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 10945	. . . . . 6 ⊢ 1o <N (1o +N 1o)
2		1pi 10923	. . . . . . 7 ⊢ 1o ∈ N
3		mulidpi 10926	. . . . . . 7 ⊢ (1o ∈ N → (1o ·N 1o) = 1o)
4	2, 3	ax-mp 5	. . . . . 6 ⊢ (1o ·N 1o) = 1o
5		addclpi 10932	. . . . . . . 8 ⊢ ((1o ∈ N ∧ 1o ∈ N) → (1o +N 1o) ∈ N)
6	2, 2, 5	mp2an 692	. . . . . . 7 ⊢ (1o +N 1o) ∈ N
7		mulidpi 10926	. . . . . . 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 5173	. . . . 5 ⊢ (1o ·N 1o) <N ((1o +N 1o) ·N 1o)
10		ordpipq 10982	. . . . 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 10956	. . . 4 ⊢ 1Q = ⟨1o, 1o⟩
13	12, 12	oveq12i 7443	. . . . 5 ⊢ (1Q +pQ 1Q) = (⟨1o, 1o⟩ +pQ ⟨1o, 1o⟩)
14		addpipq 10977	. . . . . 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 7443	. . . . . 6 ⊢ ((1o ·N 1o) +N (1o ·N 1o)) = (1o +N 1o)
17	16, 4	opeq12i 4878	. . . . 5 ⊢ ⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩ = ⟨(1o +N 1o), 1o⟩
18	13, 15, 17	3eqtri 2769	. . . 4 ⊢ (1Q +pQ 1Q) = ⟨(1o +N 1o), 1o⟩
19	11, 12, 18	3brtr4i 5173	. . 3 ⊢ 1Q <pQ (1Q +pQ 1Q)
20		lterpq 11010	. . 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 10968	. . . 4 ⊢ 1Q ∈ Q
23		nqerid 10973	. . . 4 ⊢ (1Q ∈ Q → ([Q]‘1Q) = 1Q)
24	22, 23	ax-mp 5	. . 3 ⊢ ([Q]‘1Q) = 1Q
25	24	eqcomi 2746	. 2 ⊢ 1Q = ([Q]‘1Q)
26		addpqnq 10978	. . 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 5173	1 ⊢ 1Q <Q (1Q +Q 1Q)

Syntax hints:   = wceq 1540   ∈ wcel 2108  ⟨cop 4632   class class class wbr 5143  ‘cfv 6561  (class class class)co 7431  1_oc1o 8499  Ncnpi 10884   +_N cpli 10885   ·_N cmi 10886   <_N clti 10887   +_pQ cplpq 10888   <_pQ cltpq 10890  Qcnq 10892  1_Qc1q 10893  [Q]cerq 10894   +_Q cplq 10895   <_Q cltq 10898

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-omul 8511  df-er 8745  df-ni 10912  df-pli 10913  df-mi 10914  df-lti 10915  df-plpq 10948  df-ltpq 10950  df-enq 10951  df-nq 10952  df-erq 10953  df-plq 10954  df-1nq 10956  df-ltnq 10958

This theorem is referenced by:  ltaddnq  11014