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Theorem 1lt2nq 9381
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 9313 . . . . . 6  |-  1o  <N  ( 1o  +N  1o )
2 1pi 9291 . . . . . . 7  |-  1o  e.  N.
3 mulidpi 9294 . . . . . . 7  |-  ( 1o  e.  N.  ->  ( 1o  .N  1o )  =  1o )
42, 3ax-mp 5 . . . . . 6  |-  ( 1o 
.N  1o )  =  1o
5 addclpi 9300 . . . . . . . 8  |-  ( ( 1o  e.  N.  /\  1o  e.  N. )  -> 
( 1o  +N  1o )  e.  N. )
62, 2, 5mp2an 670 . . . . . . 7  |-  ( 1o 
+N  1o )  e. 
N.
7 mulidpi 9294 . . . . . . 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 4423 . . . . 5  |-  ( 1o 
.N  1o )  <N 
( ( 1o  +N  1o )  .N  1o )
10 ordpipq 9350 . . . . 5  |-  ( <. 1o ,  1o >.  <pQ  <. ( 1o  +N  1o ) ,  1o >.  <->  ( 1o  .N  1o )  <N  ( ( 1o  +N  1o )  .N  1o ) )
119, 10mpbir 209 . . . 4  |-  <. 1o ,  1o >.  <pQ  <. ( 1o  +N  1o ) ,  1o >.
12 df-1nq 9324 . . . 4  |-  1Q  =  <. 1o ,  1o >.
1312, 12oveq12i 6290 . . . . 5  |-  ( 1Q 
+pQ  1Q )  =  (
<. 1o ,  1o >.  +pQ 
<. 1o ,  1o >. )
14 addpipq 9345 . . . . . 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 671 . . . . 5  |-  ( <. 1o ,  1o >.  +pQ  <. 1o ,  1o >. )  =  <. ( ( 1o  .N  1o )  +N  ( 1o  .N  1o ) ) ,  ( 1o  .N  1o )
>.
164, 4oveq12i 6290 . . . . . 6  |-  ( ( 1o  .N  1o )  +N  ( 1o  .N  1o ) )  =  ( 1o  +N  1o )
1716, 4opeq12i 4164 . . . . 5  |-  <. (
( 1o  .N  1o )  +N  ( 1o  .N  1o ) ) ,  ( 1o  .N  1o )
>.  =  <. ( 1o 
+N  1o ) ,  1o >.
1813, 15, 173eqtri 2435 . . . 4  |-  ( 1Q 
+pQ  1Q )  =  <. ( 1o  +N  1o ) ,  1o >.
1911, 12, 183brtr4i 4423 . . 3  |-  1Q  <pQ  ( 1Q  +pQ  1Q )
20 lterpq 9378 . . 3  |-  ( 1Q 
<pQ  ( 1Q  +pQ  1Q ) 
<->  ( /Q `  1Q )  <Q  ( /Q `  ( 1Q  +pQ  1Q ) ) )
2119, 20mpbi 208 . 2  |-  ( /Q
`  1Q )  <Q 
( /Q `  ( 1Q  +pQ  1Q ) )
22 1nq 9336 . . . 4  |-  1Q  e.  Q.
23 nqerid 9341 . . . 4  |-  ( 1Q  e.  Q.  ->  ( /Q `  1Q )  =  1Q )
2422, 23ax-mp 5 . . 3  |-  ( /Q
`  1Q )  =  1Q
2524eqcomi 2415 . 2  |-  1Q  =  ( /Q `  1Q )
26 addpqnq 9346 . . 3  |-  ( ( 1Q  e.  Q.  /\  1Q  e.  Q. )  -> 
( 1Q  +Q  1Q )  =  ( /Q `  ( 1Q  +pQ  1Q ) ) )
2722, 22, 26mp2an 670 . 2  |-  ( 1Q 
+Q  1Q )  =  ( /Q `  ( 1Q  +pQ  1Q ) )
2821, 25, 273brtr4i 4423 1  |-  1Q  <Q  ( 1Q  +Q  1Q )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405    e. wcel 1842   <.cop 3978   class class class wbr 4395   ` cfv 5569  (class class class)co 6278   1oc1o 7160   N.cnpi 9252    +N cpli 9253    .N cmi 9254    <N clti 9255    +pQ cplpq 9256    <pQ cltpq 9258   Q.cnq 9260   1Qc1q 9261   /Qcerq 9262    +Q cplq 9263    <Q cltq 9266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-omul 7172  df-er 7348  df-ni 9280  df-pli 9281  df-mi 9282  df-lti 9283  df-plpq 9316  df-ltpq 9318  df-enq 9319  df-nq 9320  df-erq 9321  df-plq 9322  df-1nq 9324  df-ltnq 9326
This theorem is referenced by:  ltaddnq  9382
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