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Theorem nn0opthlem2 11284
Description: Lemma for nn0opthi 11285. (Contributed by Raph Levien, 10-Dec-2002.) (Revised by Scott Fenton, 8-Sep-2010.)
Hypotheses
Ref Expression
nn0opth.1  |-  A  e. 
NN0
nn0opth.2  |-  B  e. 
NN0
nn0opth.3  |-  C  e. 
NN0
nn0opth.4  |-  D  e. 
NN0
Assertion
Ref Expression
nn0opthlem2  |-  ( ( A  +  B )  <  C  ->  (
( C  x.  C
)  +  D )  =/=  ( ( ( A  +  B )  x.  ( A  +  B ) )  +  B ) )

Proof of Theorem nn0opthlem2
StepHypRef Expression
1 nn0opth.1 . . . . 5  |-  A  e. 
NN0
2 nn0opth.2 . . . . 5  |-  B  e. 
NN0
31, 2nn0addcli 10001 . . . 4  |-  ( A  +  B )  e. 
NN0
4 nn0opth.3 . . . 4  |-  C  e. 
NN0
53, 4nn0opthlem1 11283 . . 3  |-  ( ( A  +  B )  <  C  <->  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  ( 2  x.  ( A  +  B
) ) )  < 
( C  x.  C
) )
62nn0rei 9976 . . . . . 6  |-  B  e.  RR
76, 1nn0addge2i 10013 . . . . 5  |-  B  <_ 
( A  +  B
)
83, 2nn0lele2xi 10016 . . . . . 6  |-  ( B  <_  ( A  +  B )  ->  B  <_  ( 2  x.  ( A  +  B )
) )
9 2re 9815 . . . . . . . 8  |-  2  e.  RR
103nn0rei 9976 . . . . . . . 8  |-  ( A  +  B )  e.  RR
119, 10remulcli 8851 . . . . . . 7  |-  ( 2  x.  ( A  +  B ) )  e.  RR
1210, 10remulcli 8851 . . . . . . 7  |-  ( ( A  +  B )  x.  ( A  +  B ) )  e.  RR
136, 11, 12leadd2i 9329 . . . . . 6  |-  ( B  <_  ( 2  x.  ( A  +  B
) )  <->  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B )  <_ 
( ( ( A  +  B )  x.  ( A  +  B
) )  +  ( 2  x.  ( A  +  B ) ) ) )
148, 13sylib 188 . . . . 5  |-  ( B  <_  ( A  +  B )  ->  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B )  <_  ( ( ( A  +  B )  x.  ( A  +  B ) )  +  ( 2  x.  ( A  +  B )
) ) )
157, 14ax-mp 8 . . . 4  |-  ( ( ( A  +  B
)  x.  ( A  +  B ) )  +  B )  <_ 
( ( ( A  +  B )  x.  ( A  +  B
) )  +  ( 2  x.  ( A  +  B ) ) )
1612, 6readdcli 8850 . . . . 5  |-  ( ( ( A  +  B
)  x.  ( A  +  B ) )  +  B )  e.  RR
1712, 11readdcli 8850 . . . . 5  |-  ( ( ( A  +  B
)  x.  ( A  +  B ) )  +  ( 2  x.  ( A  +  B
) ) )  e.  RR
184nn0rei 9976 . . . . . 6  |-  C  e.  RR
1918, 18remulcli 8851 . . . . 5  |-  ( C  x.  C )  e.  RR
2016, 17, 19lelttri 8946 . . . 4  |-  ( ( ( ( ( A  +  B )  x.  ( A  +  B
) )  +  B
)  <_  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  ( 2  x.  ( A  +  B
) ) )  /\  ( ( ( A  +  B )  x.  ( A  +  B
) )  +  ( 2  x.  ( A  +  B ) ) )  <  ( C  x.  C ) )  ->  ( ( ( A  +  B )  x.  ( A  +  B ) )  +  B )  <  ( C  x.  C )
)
2115, 20mpan 651 . . 3  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  ( 2  x.  ( A  +  B ) ) )  <  ( C  x.  C )  ->  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B )  <  ( C  x.  C ) )
225, 21sylbi 187 . 2  |-  ( ( A  +  B )  <  C  ->  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B )  <  ( C  x.  C ) )
23 nn0opth.4 . . . 4  |-  D  e. 
NN0
2419, 23nn0addge1i 10012 . . 3  |-  ( C  x.  C )  <_ 
( ( C  x.  C )  +  D
)
2523nn0rei 9976 . . . . 5  |-  D  e.  RR
2619, 25readdcli 8850 . . . 4  |-  ( ( C  x.  C )  +  D )  e.  RR
2716, 19, 26ltletri 8947 . . 3  |-  ( ( ( ( ( A  +  B )  x.  ( A  +  B
) )  +  B
)  <  ( C  x.  C )  /\  ( C  x.  C )  <_  ( ( C  x.  C )  +  D
) )  ->  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B )  <  ( ( C  x.  C )  +  D ) )
2824, 27mpan2 652 . 2  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  <  ( C  x.  C )  ->  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B )  <  ( ( C  x.  C )  +  D ) )
2916, 26ltnei 8943 . 2  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  <  ( ( C  x.  C )  +  D )  ->  (
( C  x.  C
)  +  D )  =/=  ( ( ( A  +  B )  x.  ( A  +  B ) )  +  B ) )
3022, 28, 293syl 18 1  |-  ( ( A  +  B )  <  C  ->  (
( C  x.  C
)  +  D )  =/=  ( ( ( A  +  B )  x.  ( A  +  B ) )  +  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684    =/= wne 2446   class class class wbr 4023  (class class class)co 5858    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868   2c2 9795   NN0cn0 9965
This theorem is referenced by:  nn0opthi  11285
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-seq 11047  df-exp 11105
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