Theorem 1lt2nq 9795

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 9727	. . . . . 6 |- 1o <N ( 1o +N 1o )
2		1pi 9705	. . . . . . 7 |- 1o e. N.
3		mulidpi 9708	. . . . . . 7 |- ( 1o e. N. -> ( 1o .N 1o ) = 1o )
4	2, 3	ax-mp 5	. . . . . 6 |- ( 1o .N 1o ) = 1o
5		addclpi 9714	. . . . . . . 8 |- ( ( 1o e. N. /\ 1o e. N. ) -> ( 1o +N 1o ) e. N. )
6	2, 2, 5	mp2an 708	. . . . . . 7 |- ( 1o +N 1o ) e. N.
7		mulidpi 9708	. . . . . . 7 |- ( ( 1o +N 1o ) e. 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 4683	. . . . 5 |- ( 1o .N 1o ) <N ( ( 1o +N 1o ) .N 1o )
10		ordpipq 9764	. . . . 5 |- ( <. 1o , 1o >. <pQ <. ( 1o +N 1o ) , 1o >. <-> ( 1o .N 1o ) <N ( ( 1o +N 1o ) .N 1o ) )
11	9, 10	mpbir 221	. . . 4 |- <. 1o , 1o >. <pQ <. ( 1o +N 1o ) , 1o >.
12		df-1nq 9738	. . . 4 |- 1Q = <. 1o , 1o >.
13	12, 12	oveq12i 6662	. . . . 5 |- ( 1Q +pQ 1Q ) = ( <. 1o , 1o >. +pQ <. 1o , 1o >. )
14		addpipq 9759	. . . . . 6 |- ( ( ( 1o e. N. /\ 1o e. N. ) /\ ( 1o e. N. /\ 1o e. N. ) ) -> ( <. 1o , 1o >. +pQ <. 1o , 1o >. ) = <. ( ( 1o .N 1o ) +N ( 1o .N 1o ) ) , ( 1o .N 1o ) >. )
15	2, 2, 2, 2, 14	mp4an 709	. . . . 5 |- ( <. 1o , 1o >. +pQ <. 1o , 1o >. ) = <. ( ( 1o .N 1o ) +N ( 1o .N 1o ) ) , ( 1o .N 1o ) >.
16	4, 4	oveq12i 6662	. . . . . 6 |- ( ( 1o .N 1o ) +N ( 1o .N 1o ) ) = ( 1o +N 1o )
17	16, 4	opeq12i 4407	. . . . 5 |- <. ( ( 1o .N 1o ) +N ( 1o .N 1o ) ) , ( 1o .N 1o ) >. = <. ( 1o +N 1o ) , 1o >.
18	13, 15, 17	3eqtri 2648	. . . 4 |- ( 1Q +pQ 1Q ) = <. ( 1o +N 1o ) , 1o >.
19	11, 12, 18	3brtr4i 4683	. . 3 |- 1Q <pQ ( 1Q +pQ 1Q )
20		lterpq 9792	. . 3 |- ( 1Q <pQ ( 1Q +pQ 1Q ) <-> ( /Q ` 1Q ) <Q ( /Q ` ( 1Q +pQ 1Q ) ) )
21	19, 20	mpbi 220	. 2 |- ( /Q ` 1Q ) <Q ( /Q ` ( 1Q +pQ 1Q ) )
22		1nq 9750	. . . 4 |- 1Q e. Q.
23		nqerid 9755	. . . 4 |- ( 1Q e. Q. -> ( /Q ` 1Q ) = 1Q )
24	22, 23	ax-mp 5	. . 3 |- ( /Q ` 1Q ) = 1Q
25	24	eqcomi 2631	. 2 |- 1Q = ( /Q ` 1Q )
26		addpqnq 9760	. . 3 |- ( ( 1Q e. Q. /\ 1Q e. Q. ) -> ( 1Q +Q 1Q ) = ( /Q ` ( 1Q +pQ 1Q ) ) )
27	22, 22, 26	mp2an 708	. 2 |- ( 1Q +Q 1Q ) = ( /Q ` ( 1Q +pQ 1Q ) )
28	21, 25, 27	3brtr4i 4683	1 |- 1Q <Q ( 1Q +Q 1Q )

Syntax hints:   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   1oc1o 7553   N.cnpi 9666   +N cpli 9667   .N cmi 9668   <N clti 9669   +pQ cplpq 9670   <pQ cltpq 9672   Q.cnq 9674   1Qc1q 9675   /Qcerq 9676   +Q cplq 9677   <Q cltq 9680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ni 9694  df-pli 9695  df-mi 9696  df-lti 9697  df-plpq 9730  df-ltpq 9732  df-enq 9733  df-nq 9734  df-erq 9735  df-plq 9736  df-1nq 9738  df-ltnq 9740

This theorem is referenced by:  ltaddnq  9796