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Theorem nn0opthlem1 11376
Description: A rather pretty lemma for nn0opthi 11378. (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 10073 . . . 4  |-  1  e.  NN0
31, 2nn0addcli 10093 . . 3  |-  ( A  +  1 )  e. 
NN0
4 nn0opthlem1.2 . . 3  |-  C  e. 
NN0
53, 4nn0le2msqi 11375 . 2  |-  ( ( A  +  1 )  <_  C  <->  ( ( A  +  1 )  x.  ( A  + 
1 ) )  <_ 
( C  x.  C
) )
6 nn0ltp1le 10166 . . 3  |-  ( ( A  e.  NN0  /\  C  e.  NN0 )  -> 
( A  <  C  <->  ( A  +  1 )  <_  C ) )
71, 4, 6mp2an 653 . 2  |-  ( A  <  C  <->  ( A  +  1 )  <_  C )
81, 1nn0mulcli 10094 . . . . 5  |-  ( A  x.  A )  e. 
NN0
9 2nn0 10074 . . . . . 6  |-  2  e.  NN0
109, 1nn0mulcli 10094 . . . . 5  |-  ( 2  x.  A )  e. 
NN0
118, 10nn0addcli 10093 . . . 4  |-  ( ( A  x.  A )  +  ( 2  x.  A ) )  e. 
NN0
124, 4nn0mulcli 10094 . . . 4  |-  ( C  x.  C )  e. 
NN0
13 nn0ltp1le 10166 . . . 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 653 . . 3  |-  ( ( ( A  x.  A
)  +  ( 2  x.  A ) )  <  ( C  x.  C )  <->  ( (
( A  x.  A
)  +  ( 2  x.  A ) )  +  1 )  <_ 
( C  x.  C
) )
151nn0cni 10069 . . . . . . 7  |-  A  e.  CC
16 ax-1cn 8885 . . . . . . 7  |-  1  e.  CC
1715, 16binom2i 11305 . . . . . 6  |-  ( ( A  +  1 ) ^ 2 )  =  ( ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  1 ) ) )  +  ( 1 ^ 2 ) )
1815, 16addcli 8931 . . . . . . 7  |-  ( A  +  1 )  e.  CC
1918sqvali 11276 . . . . . 6  |-  ( ( A  +  1 ) ^ 2 )  =  ( ( A  + 
1 )  x.  ( A  +  1 ) )
2015sqvali 11276 . . . . . . . 8  |-  ( A ^ 2 )  =  ( A  x.  A
)
2120oveq1i 5955 . . . . . . 7  |-  ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  1 ) ) )  =  ( ( A  x.  A )  +  ( 2  x.  ( A  x.  1 ) ) )
2216sqvali 11276 . . . . . . 7  |-  ( 1 ^ 2 )  =  ( 1  x.  1 )
2321, 22oveq12i 5957 . . . . . 6  |-  ( ( ( A ^ 2 )  +  ( 2  x.  ( A  x.  1 ) ) )  +  ( 1 ^ 2 ) )  =  ( ( ( A  x.  A )  +  ( 2  x.  ( A  x.  1 ) ) )  +  ( 1  x.  1 ) )
2417, 19, 233eqtr3i 2386 . . . . 5  |-  ( ( A  +  1 )  x.  ( A  + 
1 ) )  =  ( ( ( A  x.  A )  +  ( 2  x.  ( A  x.  1 ) ) )  +  ( 1  x.  1 ) )
2515mulid1i 8929 . . . . . . . 8  |-  ( A  x.  1 )  =  A
2625oveq2i 5956 . . . . . . 7  |-  ( 2  x.  ( A  x.  1 ) )  =  ( 2  x.  A
)
2726oveq2i 5956 . . . . . 6  |-  ( ( A  x.  A )  +  ( 2  x.  ( A  x.  1 ) ) )  =  ( ( A  x.  A )  +  ( 2  x.  A ) )
2816mulid1i 8929 . . . . . 6  |-  ( 1  x.  1 )  =  1
2927, 28oveq12i 5957 . . . . 5  |-  ( ( ( A  x.  A
)  +  ( 2  x.  ( A  x.  1 ) ) )  +  ( 1  x.  1 ) )  =  ( ( ( A  x.  A )  +  ( 2  x.  A
) )  +  1 )
3024, 29eqtri 2378 . . . 4  |-  ( ( A  +  1 )  x.  ( A  + 
1 ) )  =  ( ( ( A  x.  A )  +  ( 2  x.  A
) )  +  1 )
3130breq1i 4111 . . 3  |-  ( ( ( A  +  1 )  x.  ( A  +  1 ) )  <_  ( C  x.  C )  <->  ( (
( A  x.  A
)  +  ( 2  x.  A ) )  +  1 )  <_ 
( C  x.  C
) )
3214, 31bitr4i 243 . 2  |-  ( ( ( A  x.  A
)  +  ( 2  x.  A ) )  <  ( C  x.  C )  <->  ( ( A  +  1 )  x.  ( A  + 
1 ) )  <_ 
( C  x.  C
) )
335, 7, 323bitr4i 268 1  |-  ( A  <  C  <->  ( ( A  x.  A )  +  ( 2  x.  A ) )  < 
( C  x.  C
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    e. wcel 1710   class class class wbr 4104  (class class class)co 5945   1c1 8828    + caddc 8830    x. cmul 8832    < clt 8957    <_ cle 8958   2c2 9885   NN0cn0 10057   ^cexp 11197
This theorem is referenced by:  nn0opthlem2  11377
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-nn 9837  df-2 9894  df-n0 10058  df-z 10117  df-uz 10323  df-seq 11139  df-exp 11198
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