Theorem 1lt2nq 10779

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 10711	. . . . . 6 ⊢ 1o <N (1o +N 1o)
2		1pi 10689	. . . . . . 7 ⊢ 1o ∈ N
3		mulidpi 10692	. . . . . . 7 ⊢ (1o ∈ N → (1o ·N 1o) = 1o)
4	2, 3	ax-mp 5	. . . . . 6 ⊢ (1o ·N 1o) = 1o
5		addclpi 10698	. . . . . . . 8 ⊢ ((1o ∈ N ∧ 1o ∈ N) → (1o +N 1o) ∈ N)
6	2, 2, 5	mp2an 690	. . . . . . 7 ⊢ (1o +N 1o) ∈ N
7		mulidpi 10692	. . . . . . 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 5111	. . . . 5 ⊢ (1o ·N 1o) <N ((1o +N 1o) ·N 1o)
10		ordpipq 10748	. . . . 5 ⊢ (⟨1o, 1o⟩ <pQ ⟨(1o +N 1o), 1o⟩ ↔ (1o ·N 1o) <N ((1o +N 1o) ·N 1o))
11	9, 10	mpbir 230	. . . 4 ⊢ ⟨1o, 1o⟩ <pQ ⟨(1o +N 1o), 1o⟩
12		df-1nq 10722	. . . 4 ⊢ 1Q = ⟨1o, 1o⟩
13	12, 12	oveq12i 7319	. . . . 5 ⊢ (1Q +pQ 1Q) = (⟨1o, 1o⟩ +pQ ⟨1o, 1o⟩)
14		addpipq 10743	. . . . . 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 691	. . . . 5 ⊢ (⟨1o, 1o⟩ +pQ ⟨1o, 1o⟩) = ⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩
16	4, 4	oveq12i 7319	. . . . . 6 ⊢ ((1o ·N 1o) +N (1o ·N 1o)) = (1o +N 1o)
17	16, 4	opeq12i 4814	. . . . 5 ⊢ ⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩ = ⟨(1o +N 1o), 1o⟩
18	13, 15, 17	3eqtri 2768	. . . 4 ⊢ (1Q +pQ 1Q) = ⟨(1o +N 1o), 1o⟩
19	11, 12, 18	3brtr4i 5111	. . 3 ⊢ 1Q <pQ (1Q +pQ 1Q)
20		lterpq 10776	. . 3 ⊢ (1Q <pQ (1Q +pQ 1Q) ↔ ([Q]‘1Q) <Q ([Q]‘(1Q +pQ 1Q)))
21	19, 20	mpbi 229	. 2 ⊢ ([Q]‘1Q) <Q ([Q]‘(1Q +pQ 1Q))
22		1nq 10734	. . . 4 ⊢ 1Q ∈ Q
23		nqerid 10739	. . . 4 ⊢ (1Q ∈ Q → ([Q]‘1Q) = 1Q)
24	22, 23	ax-mp 5	. . 3 ⊢ ([Q]‘1Q) = 1Q
25	24	eqcomi 2745	. 2 ⊢ 1Q = ([Q]‘1Q)
26		addpqnq 10744	. . 3 ⊢ ((1Q ∈ Q ∧ 1Q ∈ Q) → (1Q +Q 1Q) = ([Q]‘(1Q +pQ 1Q)))
27	22, 22, 26	mp2an 690	. 2 ⊢ (1Q +Q 1Q) = ([Q]‘(1Q +pQ 1Q))
28	21, 25, 27	3brtr4i 5111	1 ⊢ 1Q <Q (1Q +Q 1Q)

Syntax hints:   = wceq 1539   ∈ wcel 2104  ⟨cop 4571   class class class wbr 5081  ‘cfv 6458  (class class class)co 7307  1_oc1o 8321  Ncnpi 10650   +_N cpli 10651   ·_N cmi 10652   <_N clti 10653   +_pQ cplpq 10654   <_pQ cltpq 10656  Qcnq 10658  1_Qc1q 10659  [Q]cerq 10660   +_Q cplq 10661   <_Q cltq 10664

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361  ax-un 7620

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3304  df-reu 3305  df-rab 3306  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5500  df-eprel 5506  df-po 5514  df-so 5515  df-fr 5555  df-we 5557  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-pred 6217  df-ord 6284  df-on 6285  df-lim 6286  df-suc 6287  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-ov 7310  df-oprab 7311  df-mpo 7312  df-om 7745  df-1st 7863  df-2nd 7864  df-frecs 8128  df-wrecs 8159  df-recs 8233  df-rdg 8272  df-1o 8328  df-oadd 8332  df-omul 8333  df-er 8529  df-ni 10678  df-pli 10679  df-mi 10680  df-lti 10681  df-plpq 10714  df-ltpq 10716  df-enq 10717  df-nq 10718  df-erq 10719  df-plq 10720  df-1nq 10722  df-ltnq 10724

This theorem is referenced by:  ltaddnq  10780