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Theorem nn0opthlem1 12317
Description: A rather pretty lemma for nn0opthi 12319. (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 10812 . . . 4  |-  1  e.  NN0
31, 2nn0addcli 10834 . . 3  |-  ( A  +  1 )  e. 
NN0
4 nn0opthlem1.2 . . 3  |-  C  e. 
NN0
53, 4nn0le2msqi 12316 . 2  |-  ( ( A  +  1 )  <_  C  <->  ( ( A  +  1 )  x.  ( A  + 
1 ) )  <_ 
( C  x.  C
) )
6 nn0ltp1le 10921 . . 3  |-  ( ( A  e.  NN0  /\  C  e.  NN0 )  -> 
( A  <  C  <->  ( A  +  1 )  <_  C ) )
71, 4, 6mp2an 672 . 2  |-  ( A  <  C  <->  ( A  +  1 )  <_  C )
81, 1nn0mulcli 10835 . . . . 5  |-  ( A  x.  A )  e. 
NN0
9 2nn0 10813 . . . . . 6  |-  2  e.  NN0
109, 1nn0mulcli 10835 . . . . 5  |-  ( 2  x.  A )  e. 
NN0
118, 10nn0addcli 10834 . . . 4  |-  ( ( A  x.  A )  +  ( 2  x.  A ) )  e. 
NN0
124, 4nn0mulcli 10835 . . . 4  |-  ( C  x.  C )  e. 
NN0
13 nn0ltp1le 10921 . . . 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 672 . . 3  |-  ( ( ( A  x.  A
)  +  ( 2  x.  A ) )  <  ( C  x.  C )  <->  ( (
( A  x.  A
)  +  ( 2  x.  A ) )  +  1 )  <_ 
( C  x.  C
) )
151nn0cni 10808 . . . . . . 7  |-  A  e.  CC
16 ax-1cn 9551 . . . . . . 7  |-  1  e.  CC
1715, 16binom2i 12246 . . . . . 6  |-  ( ( A  +  1 ) ^ 2 )  =  ( ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  1 ) ) )  +  ( 1 ^ 2 ) )
1815, 16addcli 9601 . . . . . . 7  |-  ( A  +  1 )  e.  CC
1918sqvali 12216 . . . . . 6  |-  ( ( A  +  1 ) ^ 2 )  =  ( ( A  + 
1 )  x.  ( A  +  1 ) )
2015sqvali 12216 . . . . . . . 8  |-  ( A ^ 2 )  =  ( A  x.  A
)
2120oveq1i 6295 . . . . . . 7  |-  ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  1 ) ) )  =  ( ( A  x.  A )  +  ( 2  x.  ( A  x.  1 ) ) )
2216sqvali 12216 . . . . . . 7  |-  ( 1 ^ 2 )  =  ( 1  x.  1 )
2321, 22oveq12i 6297 . . . . . 6  |-  ( ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  1 ) ) )  +  ( 1 ^ 2 ) )  =  ( ( ( A  x.  A )  +  ( 2  x.  ( A  x.  1 ) ) )  +  ( 1  x.  1 ) )
2417, 19, 233eqtr3i 2504 . . . . 5  |-  ( ( A  +  1 )  x.  ( A  + 
1 ) )  =  ( ( ( A  x.  A )  +  ( 2  x.  ( A  x.  1 ) ) )  +  ( 1  x.  1 ) )
2515mulid1i 9599 . . . . . . . 8  |-  ( A  x.  1 )  =  A
2625oveq2i 6296 . . . . . . 7  |-  ( 2  x.  ( A  x.  1 ) )  =  ( 2  x.  A
)
2726oveq2i 6296 . . . . . 6  |-  ( ( A  x.  A )  +  ( 2  x.  ( A  x.  1 ) ) )  =  ( ( A  x.  A )  +  ( 2  x.  A ) )
2816mulid1i 9599 . . . . . 6  |-  ( 1  x.  1 )  =  1
2927, 28oveq12i 6297 . . . . 5  |-  ( ( ( A  x.  A
)  +  ( 2  x.  ( A  x.  1 ) ) )  +  ( 1  x.  1 ) )  =  ( ( ( A  x.  A )  +  ( 2  x.  A
) )  +  1 )
3024, 29eqtri 2496 . . . 4  |-  ( ( A  +  1 )  x.  ( A  + 
1 ) )  =  ( ( ( A  x.  A )  +  ( 2  x.  A
) )  +  1 )
3130breq1i 4454 . . 3  |-  ( ( ( A  +  1 )  x.  ( A  +  1 ) )  <_  ( C  x.  C )  <->  ( (
( A  x.  A
)  +  ( 2  x.  A ) )  +  1 )  <_ 
( C  x.  C
) )
3214, 31bitr4i 252 . 2  |-  ( ( ( A  x.  A
)  +  ( 2  x.  A ) )  <  ( C  x.  C )  <->  ( ( A  +  1 )  x.  ( A  + 
1 ) )  <_ 
( C  x.  C
) )
335, 7, 323bitr4i 277 1  |-  ( A  <  C  <->  ( ( A  x.  A )  +  ( 2  x.  A ) )  < 
( C  x.  C
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    e. wcel 1767   class class class wbr 4447  (class class class)co 6285   1c1 9494    + caddc 9496    x. cmul 9498    < clt 9629    <_ cle 9630   2c2 10586   NN0cn0 10796   ^cexp 12135
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  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
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-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  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-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-2nd 6786  df-recs 7043  df-rdg 7077  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-n0 10797  df-z 10866  df-uz 11084  df-seq 12077  df-exp 12136
This theorem is referenced by:  nn0opthlem2  12318
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