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Theorem 1lt2nq 9400
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
StepHypRef Expression
1 1lt2pi 9332 . . . . . 6  |-  1o  <N  ( 1o  +N  1o )
2 1pi 9310 . . . . . . 7  |-  1o  e.  N.
3 mulidpi 9313 . . . . . . 7  |-  ( 1o  e.  N.  ->  ( 1o  .N  1o )  =  1o )
42, 3ax-mp 5 . . . . . 6  |-  ( 1o 
.N  1o )  =  1o
5 addclpi 9319 . . . . . . . 8  |-  ( ( 1o  e.  N.  /\  1o  e.  N. )  -> 
( 1o  +N  1o )  e.  N. )
62, 2, 5mp2an 677 . . . . . . 7  |-  ( 1o 
+N  1o )  e. 
N.
7 mulidpi 9313 . . . . . . 7  |-  ( ( 1o  +N  1o )  e.  N.  ->  (
( 1o  +N  1o )  .N  1o )  =  ( 1o  +N  1o ) )
86, 7ax-mp 5 . . . . . 6  |-  ( ( 1o  +N  1o )  .N  1o )  =  ( 1o  +N  1o )
91, 4, 83brtr4i 4450 . . . . 5  |-  ( 1o 
.N  1o )  <N 
( ( 1o  +N  1o )  .N  1o )
10 ordpipq 9369 . . . . 5  |-  ( <. 1o ,  1o >.  <pQ  <. ( 1o  +N  1o ) ,  1o >.  <->  ( 1o  .N  1o )  <N  ( ( 1o  +N  1o )  .N  1o ) )
119, 10mpbir 213 . . . 4  |-  <. 1o ,  1o >.  <pQ  <. ( 1o  +N  1o ) ,  1o >.
12 df-1nq 9343 . . . 4  |-  1Q  =  <. 1o ,  1o >.
1312, 12oveq12i 6315 . . . . 5  |-  ( 1Q 
+pQ  1Q )  =  (
<. 1o ,  1o >.  +pQ 
<. 1o ,  1o >. )
14 addpipq 9364 . . . . . 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 )
>. )
152, 2, 2, 2, 14mp4an 678 . . . . 5  |-  ( <. 1o ,  1o >.  +pQ  <. 1o ,  1o >. )  =  <. ( ( 1o  .N  1o )  +N  ( 1o  .N  1o ) ) ,  ( 1o  .N  1o )
>.
164, 4oveq12i 6315 . . . . . 6  |-  ( ( 1o  .N  1o )  +N  ( 1o  .N  1o ) )  =  ( 1o  +N  1o )
1716, 4opeq12i 4190 . . . . 5  |-  <. (
( 1o  .N  1o )  +N  ( 1o  .N  1o ) ) ,  ( 1o  .N  1o )
>.  =  <. ( 1o 
+N  1o ) ,  1o >.
1813, 15, 173eqtri 2456 . . . 4  |-  ( 1Q 
+pQ  1Q )  =  <. ( 1o  +N  1o ) ,  1o >.
1911, 12, 183brtr4i 4450 . . 3  |-  1Q  <pQ  ( 1Q  +pQ  1Q )
20 lterpq 9397 . . 3  |-  ( 1Q 
<pQ  ( 1Q  +pQ  1Q ) 
<->  ( /Q `  1Q )  <Q  ( /Q `  ( 1Q  +pQ  1Q ) ) )
2119, 20mpbi 212 . 2  |-  ( /Q
`  1Q )  <Q 
( /Q `  ( 1Q  +pQ  1Q ) )
22 1nq 9355 . . . 4  |-  1Q  e.  Q.
23 nqerid 9360 . . . 4  |-  ( 1Q  e.  Q.  ->  ( /Q `  1Q )  =  1Q )
2422, 23ax-mp 5 . . 3  |-  ( /Q
`  1Q )  =  1Q
2524eqcomi 2436 . 2  |-  1Q  =  ( /Q `  1Q )
26 addpqnq 9365 . . 3  |-  ( ( 1Q  e.  Q.  /\  1Q  e.  Q. )  -> 
( 1Q  +Q  1Q )  =  ( /Q `  ( 1Q  +pQ  1Q ) ) )
2722, 22, 26mp2an 677 . 2  |-  ( 1Q 
+Q  1Q )  =  ( /Q `  ( 1Q  +pQ  1Q ) )
2821, 25, 273brtr4i 4450 1  |-  1Q  <Q  ( 1Q  +Q  1Q )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1438    e. wcel 1869   <.cop 4003   class class class wbr 4421   ` cfv 5599  (class class class)co 6303   1oc1o 7181   N.cnpi 9271    +N cpli 9272    .N cmi 9273    <N clti 9274    +pQ cplpq 9275    <pQ cltpq 9277   Q.cnq 9279   1Qc1q 9280   /Qcerq 9281    +Q cplq 9282    <Q cltq 9285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-1st 6805  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-oadd 7192  df-omul 7193  df-er 7369  df-ni 9299  df-pli 9300  df-mi 9301  df-lti 9302  df-plpq 9335  df-ltpq 9337  df-enq 9338  df-nq 9339  df-erq 9340  df-plq 9341  df-1nq 9343  df-ltnq 9345
This theorem is referenced by:  ltaddnq  9401
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