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Theorem nn0opthlem2 11564
Description: Lemma for nn0opthi 11565. (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 10259 . . . 4  |-  ( A  +  B )  e. 
NN0
4 nn0opth.3 . . . 4  |-  C  e. 
NN0
53, 4nn0opthlem1 11563 . . 3  |-  ( ( A  +  B )  <  C  <->  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  ( 2  x.  ( A  +  B
) ) )  < 
( C  x.  C
) )
62nn0rei 10234 . . . . . 6  |-  B  e.  RR
76, 1nn0addge2i 10271 . . . . 5  |-  B  <_ 
( A  +  B
)
83, 2nn0lele2xi 10274 . . . . . 6  |-  ( B  <_  ( A  +  B )  ->  B  <_  ( 2  x.  ( A  +  B )
) )
9 2re 10071 . . . . . . . 8  |-  2  e.  RR
103nn0rei 10234 . . . . . . . 8  |-  ( A  +  B )  e.  RR
119, 10remulcli 9106 . . . . . . 7  |-  ( 2  x.  ( A  +  B ) )  e.  RR
1210, 10remulcli 9106 . . . . . . 7  |-  ( ( A  +  B )  x.  ( A  +  B ) )  e.  RR
136, 11, 12leadd2i 9585 . . . . . 6  |-  ( B  <_  ( 2  x.  ( A  +  B
) )  <->  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B )  <_ 
( ( ( A  +  B )  x.  ( A  +  B
) )  +  ( 2  x.  ( A  +  B ) ) ) )
148, 13sylib 190 . . . . 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 9105 . . . . 5  |-  ( ( ( A  +  B
)  x.  ( A  +  B ) )  +  B )  e.  RR
1712, 11readdcli 9105 . . . . 5  |-  ( ( ( A  +  B
)  x.  ( A  +  B ) )  +  ( 2  x.  ( A  +  B
) ) )  e.  RR
184nn0rei 10234 . . . . . 6  |-  C  e.  RR
1918, 18remulcli 9106 . . . . 5  |-  ( C  x.  C )  e.  RR
2016, 17, 19lelttri 9202 . . . 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 653 . . 3  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  ( 2  x.  ( A  +  B ) ) )  <  ( C  x.  C )  ->  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B )  <  ( C  x.  C ) )
225, 21sylbi 189 . 2  |-  ( ( A  +  B )  <  C  ->  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B )  <  ( C  x.  C ) )
23 nn0opth.4 . . . 4  |-  D  e. 
NN0
2419, 23nn0addge1i 10270 . . 3  |-  ( C  x.  C )  <_ 
( ( C  x.  C )  +  D
)
2523nn0rei 10234 . . . . 5  |-  D  e.  RR
2619, 25readdcli 9105 . . . 4  |-  ( ( C  x.  C )  +  D )  e.  RR
2716, 19, 26ltletri 9203 . . 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 654 . 2  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  <  ( C  x.  C )  ->  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B )  <  ( ( C  x.  C )  +  D ) )
2916, 26ltnei 9199 . 2  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  <  ( ( C  x.  C )  +  D )  ->  (
( C  x.  C
)  +  D )  =/=  ( ( ( A  +  B )  x.  ( A  +  B ) )  +  B ) )
3022, 28, 293syl 19 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 1726    =/= wne 2601   class class class wbr 4214  (class class class)co 6083    + caddc 8995    x. cmul 8997    < clt 9122    <_ cle 9123   2c2 10051   NN0cn0 10223
This theorem is referenced by:  nn0opthi  11565
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
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 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-n0 10224  df-z 10285  df-uz 10491  df-seq 11326  df-exp 11385
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