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Theorem nn0opthlem1 11566
Description: A rather pretty lemma for nn0opthi 11568. (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 10242 . . . 4  |-  1  e.  NN0
31, 2nn0addcli 10262 . . 3  |-  ( A  +  1 )  e. 
NN0
4 nn0opthlem1.2 . . 3  |-  C  e. 
NN0
53, 4nn0le2msqi 11565 . 2  |-  ( ( A  +  1 )  <_  C  <->  ( ( A  +  1 )  x.  ( A  + 
1 ) )  <_ 
( C  x.  C
) )
6 nn0ltp1le 10337 . . 3  |-  ( ( A  e.  NN0  /\  C  e.  NN0 )  -> 
( A  <  C  <->  ( A  +  1 )  <_  C ) )
71, 4, 6mp2an 655 . 2  |-  ( A  <  C  <->  ( A  +  1 )  <_  C )
81, 1nn0mulcli 10263 . . . . 5  |-  ( A  x.  A )  e. 
NN0
9 2nn0 10243 . . . . . 6  |-  2  e.  NN0
109, 1nn0mulcli 10263 . . . . 5  |-  ( 2  x.  A )  e. 
NN0
118, 10nn0addcli 10262 . . . 4  |-  ( ( A  x.  A )  +  ( 2  x.  A ) )  e. 
NN0
124, 4nn0mulcli 10263 . . . 4  |-  ( C  x.  C )  e. 
NN0
13 nn0ltp1le 10337 . . . 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 655 . . 3  |-  ( ( ( A  x.  A
)  +  ( 2  x.  A ) )  <  ( C  x.  C )  <->  ( (
( A  x.  A
)  +  ( 2  x.  A ) )  +  1 )  <_ 
( C  x.  C
) )
151nn0cni 10238 . . . . . . 7  |-  A  e.  CC
16 ax-1cn 9053 . . . . . . 7  |-  1  e.  CC
1715, 16binom2i 11495 . . . . . 6  |-  ( ( A  +  1 ) ^ 2 )  =  ( ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  1 ) ) )  +  ( 1 ^ 2 ) )
1815, 16addcli 9099 . . . . . . 7  |-  ( A  +  1 )  e.  CC
1918sqvali 11466 . . . . . 6  |-  ( ( A  +  1 ) ^ 2 )  =  ( ( A  + 
1 )  x.  ( A  +  1 ) )
2015sqvali 11466 . . . . . . . 8  |-  ( A ^ 2 )  =  ( A  x.  A
)
2120oveq1i 6094 . . . . . . 7  |-  ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  1 ) ) )  =  ( ( A  x.  A )  +  ( 2  x.  ( A  x.  1 ) ) )
2216sqvali 11466 . . . . . . 7  |-  ( 1 ^ 2 )  =  ( 1  x.  1 )
2321, 22oveq12i 6096 . . . . . 6  |-  ( ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  1 ) ) )  +  ( 1 ^ 2 ) )  =  ( ( ( A  x.  A )  +  ( 2  x.  ( A  x.  1 ) ) )  +  ( 1  x.  1 ) )
2417, 19, 233eqtr3i 2466 . . . . 5  |-  ( ( A  +  1 )  x.  ( A  + 
1 ) )  =  ( ( ( A  x.  A )  +  ( 2  x.  ( A  x.  1 ) ) )  +  ( 1  x.  1 ) )
2515mulid1i 9097 . . . . . . . 8  |-  ( A  x.  1 )  =  A
2625oveq2i 6095 . . . . . . 7  |-  ( 2  x.  ( A  x.  1 ) )  =  ( 2  x.  A
)
2726oveq2i 6095 . . . . . 6  |-  ( ( A  x.  A )  +  ( 2  x.  ( A  x.  1 ) ) )  =  ( ( A  x.  A )  +  ( 2  x.  A ) )
2816mulid1i 9097 . . . . . 6  |-  ( 1  x.  1 )  =  1
2927, 28oveq12i 6096 . . . . 5  |-  ( ( ( A  x.  A
)  +  ( 2  x.  ( A  x.  1 ) ) )  +  ( 1  x.  1 ) )  =  ( ( ( A  x.  A )  +  ( 2  x.  A
) )  +  1 )
3024, 29eqtri 2458 . . . 4  |-  ( ( A  +  1 )  x.  ( A  + 
1 ) )  =  ( ( ( A  x.  A )  +  ( 2  x.  A
) )  +  1 )
3130breq1i 4222 . . 3  |-  ( ( ( A  +  1 )  x.  ( A  +  1 ) )  <_  ( C  x.  C )  <->  ( (
( A  x.  A
)  +  ( 2  x.  A ) )  +  1 )  <_ 
( C  x.  C
) )
3214, 31bitr4i 245 . 2  |-  ( ( ( A  x.  A
)  +  ( 2  x.  A ) )  <  ( C  x.  C )  <->  ( ( A  +  1 )  x.  ( A  + 
1 ) )  <_ 
( C  x.  C
) )
335, 7, 323bitr4i 270 1  |-  ( A  <  C  <->  ( ( A  x.  A )  +  ( 2  x.  A ) )  < 
( C  x.  C
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    e. wcel 1726   class class class wbr 4215  (class class class)co 6084   1c1 8996    + caddc 8998    x. cmul 9000    < clt 9125    <_ cle 9126   2c2 10054   NN0cn0 10226   ^cexp 11387
This theorem is referenced by:  nn0opthlem2  11567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-n0 10227  df-z 10288  df-uz 10494  df-seq 11329  df-exp 11388
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