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Theorem 1lt2nq 9352
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 9284 . . . . . 6  |-  1o  <N  ( 1o  +N  1o )
2 1pi 9262 . . . . . . 7  |-  1o  e.  N.
3 mulidpi 9265 . . . . . . 7  |-  ( 1o  e.  N.  ->  ( 1o  .N  1o )  =  1o )
42, 3ax-mp 5 . . . . . 6  |-  ( 1o 
.N  1o )  =  1o
5 addclpi 9271 . . . . . . . 8  |-  ( ( 1o  e.  N.  /\  1o  e.  N. )  -> 
( 1o  +N  1o )  e.  N. )
62, 2, 5mp2an 672 . . . . . . 7  |-  ( 1o 
+N  1o )  e. 
N.
7 mulidpi 9265 . . . . . . 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 4475 . . . . 5  |-  ( 1o 
.N  1o )  <N 
( ( 1o  +N  1o )  .N  1o )
10 ordpipq 9321 . . . . 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 9295 . . . 4  |-  1Q  =  <. 1o ,  1o >.
1312, 12oveq12i 6297 . . . . 5  |-  ( 1Q 
+pQ  1Q )  =  (
<. 1o ,  1o >.  +pQ 
<. 1o ,  1o >. )
14 addpipq 9316 . . . . . 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 673 . . . . 5  |-  ( <. 1o ,  1o >.  +pQ  <. 1o ,  1o >. )  =  <. ( ( 1o  .N  1o )  +N  ( 1o  .N  1o ) ) ,  ( 1o  .N  1o )
>.
164, 4oveq12i 6297 . . . . . 6  |-  ( ( 1o  .N  1o )  +N  ( 1o  .N  1o ) )  =  ( 1o  +N  1o )
1716, 4opeq12i 4218 . . . . 5  |-  <. (
( 1o  .N  1o )  +N  ( 1o  .N  1o ) ) ,  ( 1o  .N  1o )
>.  =  <. ( 1o 
+N  1o ) ,  1o >.
1813, 15, 173eqtri 2500 . . . 4  |-  ( 1Q 
+pQ  1Q )  =  <. ( 1o  +N  1o ) ,  1o >.
1911, 12, 183brtr4i 4475 . . 3  |-  1Q  <pQ  ( 1Q  +pQ  1Q )
20 lterpq 9349 . . 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 9307 . . . 4  |-  1Q  e.  Q.
23 nqerid 9312 . . . 4  |-  ( 1Q  e.  Q.  ->  ( /Q `  1Q )  =  1Q )
2422, 23ax-mp 5 . . 3  |-  ( /Q
`  1Q )  =  1Q
2524eqcomi 2480 . 2  |-  1Q  =  ( /Q `  1Q )
26 addpqnq 9317 . . 3  |-  ( ( 1Q  e.  Q.  /\  1Q  e.  Q. )  -> 
( 1Q  +Q  1Q )  =  ( /Q `  ( 1Q  +pQ  1Q ) ) )
2722, 22, 26mp2an 672 . 2  |-  ( 1Q 
+Q  1Q )  =  ( /Q `  ( 1Q  +pQ  1Q ) )
2821, 25, 273brtr4i 4475 1  |-  1Q  <Q  ( 1Q  +Q  1Q )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    e. wcel 1767   <.cop 4033   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   1oc1o 7124   N.cnpi 9223    +N cpli 9224    .N cmi 9225    <N clti 9226    +pQ cplpq 9227    <pQ cltpq 9229   Q.cnq 9231   1Qc1q 9232   /Qcerq 9233    +Q cplq 9234    <Q cltq 9237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-omul 7136  df-er 7312  df-ni 9251  df-pli 9252  df-mi 9253  df-lti 9254  df-plpq 9287  df-ltpq 9289  df-enq 9290  df-nq 9291  df-erq 9292  df-plq 9293  df-1nq 9295  df-ltnq 9297
This theorem is referenced by:  ltaddnq  9353
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