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Theorem nn0opthlem1 12451
Description: A rather pretty lemma for nn0opthi 12453. (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 10885 . . . 4  |-  1  e.  NN0
31, 2nn0addcli 10907 . . 3  |-  ( A  +  1 )  e. 
NN0
4 nn0opthlem1.2 . . 3  |-  C  e. 
NN0
53, 4nn0le2msqi 12450 . 2  |-  ( ( A  +  1 )  <_  C  <->  ( ( A  +  1 )  x.  ( A  + 
1 ) )  <_ 
( C  x.  C
) )
6 nn0ltp1le 10994 . . 3  |-  ( ( A  e.  NN0  /\  C  e.  NN0 )  -> 
( A  <  C  <->  ( A  +  1 )  <_  C ) )
71, 4, 6mp2an 676 . 2  |-  ( A  <  C  <->  ( A  +  1 )  <_  C )
81, 1nn0mulcli 10908 . . . . 5  |-  ( A  x.  A )  e. 
NN0
9 2nn0 10886 . . . . . 6  |-  2  e.  NN0
109, 1nn0mulcli 10908 . . . . 5  |-  ( 2  x.  A )  e. 
NN0
118, 10nn0addcli 10907 . . . 4  |-  ( ( A  x.  A )  +  ( 2  x.  A ) )  e. 
NN0
124, 4nn0mulcli 10908 . . . 4  |-  ( C  x.  C )  e. 
NN0
13 nn0ltp1le 10994 . . . 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 676 . . 3  |-  ( ( ( A  x.  A
)  +  ( 2  x.  A ) )  <  ( C  x.  C )  <->  ( (
( A  x.  A
)  +  ( 2  x.  A ) )  +  1 )  <_ 
( C  x.  C
) )
151nn0cni 10881 . . . . . . 7  |-  A  e.  CC
16 ax-1cn 9596 . . . . . . 7  |-  1  e.  CC
1715, 16binom2i 12381 . . . . . 6  |-  ( ( A  +  1 ) ^ 2 )  =  ( ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  1 ) ) )  +  ( 1 ^ 2 ) )
1815, 16addcli 9646 . . . . . . 7  |-  ( A  +  1 )  e.  CC
1918sqvali 12351 . . . . . 6  |-  ( ( A  +  1 ) ^ 2 )  =  ( ( A  + 
1 )  x.  ( A  +  1 ) )
2015sqvali 12351 . . . . . . . 8  |-  ( A ^ 2 )  =  ( A  x.  A
)
2120oveq1i 6315 . . . . . . 7  |-  ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  1 ) ) )  =  ( ( A  x.  A )  +  ( 2  x.  ( A  x.  1 ) ) )
2216sqvali 12351 . . . . . . 7  |-  ( 1 ^ 2 )  =  ( 1  x.  1 )
2321, 22oveq12i 6317 . . . . . 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 9644 . . . . . . . 8  |-  ( A  x.  1 )  =  A
2625oveq2i 6316 . . . . . . 7  |-  ( 2  x.  ( A  x.  1 ) )  =  ( 2  x.  A
)
2726oveq2i 6316 . . . . . 6  |-  ( ( A  x.  A )  +  ( 2  x.  ( A  x.  1 ) ) )  =  ( ( A  x.  A )  +  ( 2  x.  A ) )
2816mulid1i 9644 . . . . . 6  |-  ( 1  x.  1 )  =  1
2927, 28oveq12i 6317 . . . . 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 4433 . . 3  |-  ( ( ( A  +  1 )  x.  ( A  +  1 ) )  <_  ( C  x.  C )  <->  ( (
( A  x.  A
)  +  ( 2  x.  A ) )  +  1 )  <_ 
( C  x.  C
) )
3214, 31bitr4i 255 . 2  |-  ( ( ( A  x.  A
)  +  ( 2  x.  A ) )  <  ( C  x.  C )  <->  ( ( A  +  1 )  x.  ( A  + 
1 ) )  <_ 
( C  x.  C
) )
335, 7, 323bitr4i 280 1  |-  ( A  <  C  <->  ( ( A  x.  A )  +  ( 2  x.  A ) )  < 
( C  x.  C
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    e. wcel 1870   class class class wbr 4426  (class class class)co 6305   1c1 9539    + caddc 9541    x. cmul 9543    < clt 9674    <_ cle 9675   2c2 10659   NN0cn0 10869   ^cexp 12269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-seq 12211  df-exp 12270
This theorem is referenced by:  nn0opthlem2  12452
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