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Theorem 1lt2nq 8597
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 8529 . . . . . 6  |-  1o  <N  ( 1o  +N  1o )
2 1pi 8507 . . . . . . 7  |-  1o  e.  N.
3 mulidpi 8510 . . . . . . 7  |-  ( 1o  e.  N.  ->  ( 1o  .N  1o )  =  1o )
42, 3ax-mp 8 . . . . . 6  |-  ( 1o 
.N  1o )  =  1o
5 addclpi 8516 . . . . . . . 8  |-  ( ( 1o  e.  N.  /\  1o  e.  N. )  -> 
( 1o  +N  1o )  e.  N. )
62, 2, 5mp2an 653 . . . . . . 7  |-  ( 1o 
+N  1o )  e. 
N.
7 mulidpi 8510 . . . . . . 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 4051 . . . . 5  |-  ( 1o 
.N  1o )  <N 
( ( 1o  +N  1o )  .N  1o )
10 ordpipq 8566 . . . . 5  |-  ( <. 1o ,  1o >.  <pQ  <. ( 1o  +N  1o ) ,  1o >.  <->  ( 1o  .N  1o )  <N  ( ( 1o  +N  1o )  .N  1o ) )
119, 10mpbir 200 . . . 4  |-  <. 1o ,  1o >.  <pQ  <. ( 1o  +N  1o ) ,  1o >.
12 df-1nq 8540 . . . 4  |-  1Q  =  <. 1o ,  1o >.
1312, 12oveq12i 5870 . . . . 5  |-  ( 1Q 
+pQ  1Q )  =  (
<. 1o ,  1o >.  +pQ 
<. 1o ,  1o >. )
14 addpipq 8561 . . . . . 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 654 . . . . 5  |-  ( <. 1o ,  1o >.  +pQ  <. 1o ,  1o >. )  =  <. ( ( 1o  .N  1o )  +N  ( 1o  .N  1o ) ) ,  ( 1o  .N  1o )
>.
164, 4oveq12i 5870 . . . . . 6  |-  ( ( 1o  .N  1o )  +N  ( 1o  .N  1o ) )  =  ( 1o  +N  1o )
1716, 4opeq12i 3801 . . . . 5  |-  <. (
( 1o  .N  1o )  +N  ( 1o  .N  1o ) ) ,  ( 1o  .N  1o )
>.  =  <. ( 1o 
+N  1o ) ,  1o >.
1813, 15, 173eqtri 2307 . . . 4  |-  ( 1Q 
+pQ  1Q )  =  <. ( 1o  +N  1o ) ,  1o >.
1911, 12, 183brtr4i 4051 . . 3  |-  1Q  <pQ  ( 1Q  +pQ  1Q )
20 lterpq 8594 . . 3  |-  ( 1Q 
<pQ  ( 1Q  +pQ  1Q ) 
<->  ( /Q `  1Q )  <Q  ( /Q `  ( 1Q  +pQ  1Q ) ) )
2119, 20mpbi 199 . 2  |-  ( /Q
`  1Q )  <Q 
( /Q `  ( 1Q  +pQ  1Q ) )
22 1nq 8552 . . . 4  |-  1Q  e.  Q.
23 nqerid 8557 . . . 4  |-  ( 1Q  e.  Q.  ->  ( /Q `  1Q )  =  1Q )
2422, 23ax-mp 8 . . 3  |-  ( /Q
`  1Q )  =  1Q
2524eqcomi 2287 . 2  |-  1Q  =  ( /Q `  1Q )
26 addpqnq 8562 . . 3  |-  ( ( 1Q  e.  Q.  /\  1Q  e.  Q. )  -> 
( 1Q  +Q  1Q )  =  ( /Q `  ( 1Q  +pQ  1Q ) ) )
2722, 22, 26mp2an 653 . 2  |-  ( 1Q 
+Q  1Q )  =  ( /Q `  ( 1Q  +pQ  1Q ) )
2821, 25, 273brtr4i 4051 1  |-  1Q  <Q  ( 1Q  +Q  1Q )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   <.cop 3643   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   1oc1o 6472   N.cnpi 8466    +N cpli 8467    .N cmi 8468    <N clti 8469    +pQ cplpq 8470    <pQ cltpq 8472   Q.cnq 8474   1Qc1q 8475   /Qcerq 8476    +Q cplq 8477    <Q cltq 8480
This theorem is referenced by:  ltaddnq  8598
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-omul 6484  df-er 6660  df-ni 8496  df-pli 8497  df-mi 8498  df-lti 8499  df-plpq 8532  df-ltpq 8534  df-enq 8535  df-nq 8536  df-erq 8537  df-plq 8538  df-1nq 8540  df-ltnq 8542
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