Theorem 1lt2nq 10896

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 10828	. . . . . 6 ⊢ 1o <N (1o +N 1o)
2		1pi 10806	. . . . . . 7 ⊢ 1o ∈ N
3		mulidpi 10809	. . . . . . 7 ⊢ (1o ∈ N → (1o ·N 1o) = 1o)
4	2, 3	ax-mp 5	. . . . . 6 ⊢ (1o ·N 1o) = 1o
5		addclpi 10815	. . . . . . . 8 ⊢ ((1o ∈ N ∧ 1o ∈ N) → (1o +N 1o) ∈ N)
6	2, 2, 5	mp2an 693	. . . . . . 7 ⊢ (1o +N 1o) ∈ N
7		mulidpi 10809	. . . . . . 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 5130	. . . . 5 ⊢ (1o ·N 1o) <N ((1o +N 1o) ·N 1o)
10		ordpipq 10865	. . . . 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 10839	. . . 4 ⊢ 1Q = ⟨1o, 1o⟩
13	12, 12	oveq12i 7380	. . . . 5 ⊢ (1Q +pQ 1Q) = (⟨1o, 1o⟩ +pQ ⟨1o, 1o⟩)
14		addpipq 10860	. . . . . 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 694	. . . . 5 ⊢ (⟨1o, 1o⟩ +pQ ⟨1o, 1o⟩) = ⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩
16	4, 4	oveq12i 7380	. . . . . 6 ⊢ ((1o ·N 1o) +N (1o ·N 1o)) = (1o +N 1o)
17	16, 4	opeq12i 4836	. . . . 5 ⊢ ⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩ = ⟨(1o +N 1o), 1o⟩
18	13, 15, 17	3eqtri 2764	. . . 4 ⊢ (1Q +pQ 1Q) = ⟨(1o +N 1o), 1o⟩
19	11, 12, 18	3brtr4i 5130	. . 3 ⊢ 1Q <pQ (1Q +pQ 1Q)
20		lterpq 10893	. . 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 10851	. . . 4 ⊢ 1Q ∈ Q
23		nqerid 10856	. . . 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 10861	. . 3 ⊢ ((1Q ∈ Q ∧ 1Q ∈ Q) → (1Q +Q 1Q) = ([Q]‘(1Q +pQ 1Q)))
27	22, 22, 26	mp2an 693	. 2 ⊢ (1Q +Q 1Q) = ([Q]‘(1Q +pQ 1Q))
28	21, 25, 27	3brtr4i 5130	1 ⊢ 1Q <Q (1Q +Q 1Q)

Syntax hints:   = wceq 1542   ∈ wcel 2114  ⟨cop 4588   class class class wbr 5100  ‘cfv 6500  (class class class)co 7368  1_oc1o 8400  Ncnpi 10767   +_N cpli 10768   ·_N cmi 10769   <_N clti 10770   +_pQ cplpq 10771   <_pQ cltpq 10773  Qcnq 10775  1_Qc1q 10776  [Q]cerq 10777   +_Q cplq 10778   <_Q cltq 10781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-oadd 8411  df-omul 8412  df-er 8645  df-ni 10795  df-pli 10796  df-mi 10797  df-lti 10798  df-plpq 10831  df-ltpq 10833  df-enq 10834  df-nq 10835  df-erq 10836  df-plq 10837  df-1nq 10839  df-ltnq 10841

This theorem is referenced by:  ltaddnq  10897