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Theorem nn0opthlem1 11524
Description: A rather pretty lemma for nn0opthi 11526. (Contributed by Raph Levien, 10-Dec-2002.)
Hypotheses
Ref Expression
nn0opthlem1.1  |-  A  e. 
NN0
nn0opthlem1.2  |-  C  e. 
NN0
Assertion
Ref Expression
nn0opthlem1  |-  ( A  <  C  <->  ( ( A  x.  A )  +  ( 2  x.  A ) )  < 
( C  x.  C
) )

Proof of Theorem nn0opthlem1
StepHypRef Expression
1 nn0opthlem1.1 . . . 4  |-  A  e. 
NN0
2 1nn0 10201 . . . 4  |-  1  e.  NN0
31, 2nn0addcli 10221 . . 3  |-  ( A  +  1 )  e. 
NN0
4 nn0opthlem1.2 . . 3  |-  C  e. 
NN0
53, 4nn0le2msqi 11523 . 2  |-  ( ( A  +  1 )  <_  C  <->  ( ( A  +  1 )  x.  ( A  + 
1 ) )  <_ 
( C  x.  C
) )
6 nn0ltp1le 10296 . . 3  |-  ( ( A  e.  NN0  /\  C  e.  NN0 )  -> 
( A  <  C  <->  ( A  +  1 )  <_  C ) )
71, 4, 6mp2an 654 . 2  |-  ( A  <  C  <->  ( A  +  1 )  <_  C )
81, 1nn0mulcli 10222 . . . . 5  |-  ( A  x.  A )  e. 
NN0
9 2nn0 10202 . . . . . 6  |-  2  e.  NN0
109, 1nn0mulcli 10222 . . . . 5  |-  ( 2  x.  A )  e. 
NN0
118, 10nn0addcli 10221 . . . 4  |-  ( ( A  x.  A )  +  ( 2  x.  A ) )  e. 
NN0
124, 4nn0mulcli 10222 . . . 4  |-  ( C  x.  C )  e. 
NN0
13 nn0ltp1le 10296 . . . 4  |-  ( ( ( ( A  x.  A )  +  ( 2  x.  A ) )  e.  NN0  /\  ( C  x.  C
)  e.  NN0 )  ->  ( ( ( A  x.  A )  +  ( 2  x.  A
) )  <  ( C  x.  C )  <->  ( ( ( A  x.  A )  +  ( 2  x.  A ) )  +  1 )  <_  ( C  x.  C ) ) )
1411, 12, 13mp2an 654 . . 3  |-  ( ( ( A  x.  A
)  +  ( 2  x.  A ) )  <  ( C  x.  C )  <->  ( (
( A  x.  A
)  +  ( 2  x.  A ) )  +  1 )  <_ 
( C  x.  C
) )
151nn0cni 10197 . . . . . . 7  |-  A  e.  CC
16 ax-1cn 9012 . . . . . . 7  |-  1  e.  CC
1715, 16binom2i 11453 . . . . . 6  |-  ( ( A  +  1 ) ^ 2 )  =  ( ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  1 ) ) )  +  ( 1 ^ 2 ) )
1815, 16addcli 9058 . . . . . . 7  |-  ( A  +  1 )  e.  CC
1918sqvali 11424 . . . . . 6  |-  ( ( A  +  1 ) ^ 2 )  =  ( ( A  + 
1 )  x.  ( A  +  1 ) )
2015sqvali 11424 . . . . . . . 8  |-  ( A ^ 2 )  =  ( A  x.  A
)
2120oveq1i 6058 . . . . . . 7  |-  ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  1 ) ) )  =  ( ( A  x.  A )  +  ( 2  x.  ( A  x.  1 ) ) )
2216sqvali 11424 . . . . . . 7  |-  ( 1 ^ 2 )  =  ( 1  x.  1 )
2321, 22oveq12i 6060 . . . . . 6  |-  ( ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  1 ) ) )  +  ( 1 ^ 2 ) )  =  ( ( ( A  x.  A )  +  ( 2  x.  ( A  x.  1 ) ) )  +  ( 1  x.  1 ) )
2417, 19, 233eqtr3i 2440 . . . . 5  |-  ( ( A  +  1 )  x.  ( A  + 
1 ) )  =  ( ( ( A  x.  A )  +  ( 2  x.  ( A  x.  1 ) ) )  +  ( 1  x.  1 ) )
2515mulid1i 9056 . . . . . . . 8  |-  ( A  x.  1 )  =  A
2625oveq2i 6059 . . . . . . 7  |-  ( 2  x.  ( A  x.  1 ) )  =  ( 2  x.  A
)
2726oveq2i 6059 . . . . . 6  |-  ( ( A  x.  A )  +  ( 2  x.  ( A  x.  1 ) ) )  =  ( ( A  x.  A )  +  ( 2  x.  A ) )
2816mulid1i 9056 . . . . . 6  |-  ( 1  x.  1 )  =  1
2927, 28oveq12i 6060 . . . . 5  |-  ( ( ( A  x.  A
)  +  ( 2  x.  ( A  x.  1 ) ) )  +  ( 1  x.  1 ) )  =  ( ( ( A  x.  A )  +  ( 2  x.  A
) )  +  1 )
3024, 29eqtri 2432 . . . 4  |-  ( ( A  +  1 )  x.  ( A  + 
1 ) )  =  ( ( ( A  x.  A )  +  ( 2  x.  A
) )  +  1 )
3130breq1i 4187 . . 3  |-  ( ( ( A  +  1 )  x.  ( A  +  1 ) )  <_  ( C  x.  C )  <->  ( (
( A  x.  A
)  +  ( 2  x.  A ) )  +  1 )  <_ 
( C  x.  C
) )
3214, 31bitr4i 244 . 2  |-  ( ( ( A  x.  A
)  +  ( 2  x.  A ) )  <  ( C  x.  C )  <->  ( ( A  +  1 )  x.  ( A  + 
1 ) )  <_ 
( C  x.  C
) )
335, 7, 323bitr4i 269 1  |-  ( A  <  C  <->  ( ( A  x.  A )  +  ( 2  x.  A ) )  < 
( C  x.  C
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    e. wcel 1721   class class class wbr 4180  (class class class)co 6048   1c1 8955    + caddc 8957    x. cmul 8959    < clt 9084    <_ cle 9085   2c2 10013   NN0cn0 10185   ^cexp 11345
This theorem is referenced by:  nn0opthlem2  11525
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-n0 10186  df-z 10247  df-uz 10453  df-seq 11287  df-exp 11346
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