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Theorem 1lt2nq 8850
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 8782 . . . . . 6  |-  1o  <N  ( 1o  +N  1o )
2 1pi 8760 . . . . . . 7  |-  1o  e.  N.
3 mulidpi 8763 . . . . . . 7  |-  ( 1o  e.  N.  ->  ( 1o  .N  1o )  =  1o )
42, 3ax-mp 8 . . . . . 6  |-  ( 1o 
.N  1o )  =  1o
5 addclpi 8769 . . . . . . . 8  |-  ( ( 1o  e.  N.  /\  1o  e.  N. )  -> 
( 1o  +N  1o )  e.  N. )
62, 2, 5mp2an 654 . . . . . . 7  |-  ( 1o 
+N  1o )  e. 
N.
7 mulidpi 8763 . . . . . . 7  |-  ( ( 1o  +N  1o )  e.  N.  ->  (
( 1o  +N  1o )  .N  1o )  =  ( 1o  +N  1o ) )
86, 7ax-mp 8 . . . . . 6  |-  ( ( 1o  +N  1o )  .N  1o )  =  ( 1o  +N  1o )
91, 4, 83brtr4i 4240 . . . . 5  |-  ( 1o 
.N  1o )  <N 
( ( 1o  +N  1o )  .N  1o )
10 ordpipq 8819 . . . . 5  |-  ( <. 1o ,  1o >.  <pQ  <. ( 1o  +N  1o ) ,  1o >.  <->  ( 1o  .N  1o )  <N  ( ( 1o  +N  1o )  .N  1o ) )
119, 10mpbir 201 . . . 4  |-  <. 1o ,  1o >.  <pQ  <. ( 1o  +N  1o ) ,  1o >.
12 df-1nq 8793 . . . 4  |-  1Q  =  <. 1o ,  1o >.
1312, 12oveq12i 6093 . . . . 5  |-  ( 1Q 
+pQ  1Q )  =  (
<. 1o ,  1o >.  +pQ 
<. 1o ,  1o >. )
14 addpipq 8814 . . . . . 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 655 . . . . 5  |-  ( <. 1o ,  1o >.  +pQ  <. 1o ,  1o >. )  =  <. ( ( 1o  .N  1o )  +N  ( 1o  .N  1o ) ) ,  ( 1o  .N  1o )
>.
164, 4oveq12i 6093 . . . . . 6  |-  ( ( 1o  .N  1o )  +N  ( 1o  .N  1o ) )  =  ( 1o  +N  1o )
1716, 4opeq12i 3989 . . . . 5  |-  <. (
( 1o  .N  1o )  +N  ( 1o  .N  1o ) ) ,  ( 1o  .N  1o )
>.  =  <. ( 1o 
+N  1o ) ,  1o >.
1813, 15, 173eqtri 2460 . . . 4  |-  ( 1Q 
+pQ  1Q )  =  <. ( 1o  +N  1o ) ,  1o >.
1911, 12, 183brtr4i 4240 . . 3  |-  1Q  <pQ  ( 1Q  +pQ  1Q )
20 lterpq 8847 . . 3  |-  ( 1Q 
<pQ  ( 1Q  +pQ  1Q ) 
<->  ( /Q `  1Q )  <Q  ( /Q `  ( 1Q  +pQ  1Q ) ) )
2119, 20mpbi 200 . 2  |-  ( /Q
`  1Q )  <Q 
( /Q `  ( 1Q  +pQ  1Q ) )
22 1nq 8805 . . . 4  |-  1Q  e.  Q.
23 nqerid 8810 . . . 4  |-  ( 1Q  e.  Q.  ->  ( /Q `  1Q )  =  1Q )
2422, 23ax-mp 8 . . 3  |-  ( /Q
`  1Q )  =  1Q
2524eqcomi 2440 . 2  |-  1Q  =  ( /Q `  1Q )
26 addpqnq 8815 . . 3  |-  ( ( 1Q  e.  Q.  /\  1Q  e.  Q. )  -> 
( 1Q  +Q  1Q )  =  ( /Q `  ( 1Q  +pQ  1Q ) ) )
2722, 22, 26mp2an 654 . 2  |-  ( 1Q 
+Q  1Q )  =  ( /Q `  ( 1Q  +pQ  1Q ) )
2821, 25, 273brtr4i 4240 1  |-  1Q  <Q  ( 1Q  +Q  1Q )
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   <.cop 3817   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   1oc1o 6717   N.cnpi 8719    +N cpli 8720    .N cmi 8721    <N clti 8722    +pQ cplpq 8723    <pQ cltpq 8725   Q.cnq 8727   1Qc1q 8728   /Qcerq 8729    +Q cplq 8730    <Q cltq 8733
This theorem is referenced by:  ltaddnq  8851
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-omul 6729  df-er 6905  df-ni 8749  df-pli 8750  df-mi 8751  df-lti 8752  df-plpq 8785  df-ltpq 8787  df-enq 8788  df-nq 8789  df-erq 8790  df-plq 8791  df-1nq 8793  df-ltnq 8795
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