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Theorem nn0opthlem2d 10354
Description: Lemma for nn0opth2 10357. (Contributed by Jim Kingdon, 31-Oct-2021.)
Hypotheses
Ref Expression
nn0opthd.1  |-  ( ph  ->  A  e.  NN0 )
nn0opthd.2  |-  ( ph  ->  B  e.  NN0 )
nn0opthd.3  |-  ( ph  ->  C  e.  NN0 )
nn0opthd.4  |-  ( ph  ->  D  e.  NN0 )
Assertion
Ref Expression
nn0opthlem2d  |-  ( ph  ->  ( ( A  +  B )  <  C  ->  ( ( C  x.  C )  +  D
)  =/=  ( ( ( A  +  B
)  x.  ( A  +  B ) )  +  B ) ) )

Proof of Theorem nn0opthlem2d
StepHypRef Expression
1 nn0opthd.1 . . . . . . . 8  |-  ( ph  ->  A  e.  NN0 )
2 nn0opthd.2 . . . . . . . 8  |-  ( ph  ->  B  e.  NN0 )
31, 2nn0addcld 8932 . . . . . . 7  |-  ( ph  ->  ( A  +  B
)  e.  NN0 )
43nn0red 8929 . . . . . 6  |-  ( ph  ->  ( A  +  B
)  e.  RR )
54, 4remulcld 7714 . . . . 5  |-  ( ph  ->  ( ( A  +  B )  x.  ( A  +  B )
)  e.  RR )
62nn0red 8929 . . . . 5  |-  ( ph  ->  B  e.  RR )
75, 6readdcld 7713 . . . 4  |-  ( ph  ->  ( ( ( A  +  B )  x.  ( A  +  B
) )  +  B
)  e.  RR )
87adantr 272 . . 3  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B )  e.  RR )
9 nn0opthd.3 . . . . . . 7  |-  ( ph  ->  C  e.  NN0 )
109nn0red 8929 . . . . . 6  |-  ( ph  ->  C  e.  RR )
1110, 10remulcld 7714 . . . . 5  |-  ( ph  ->  ( C  x.  C
)  e.  RR )
1211adantr 272 . . . 4  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( C  x.  C )  e.  RR )
13 nn0opthd.4 . . . . . . 7  |-  ( ph  ->  D  e.  NN0 )
1413nn0red 8929 . . . . . 6  |-  ( ph  ->  D  e.  RR )
1511, 14readdcld 7713 . . . . 5  |-  ( ph  ->  ( ( C  x.  C )  +  D
)  e.  RR )
1615adantr 272 . . . 4  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( ( C  x.  C )  +  D )  e.  RR )
17 2re 8694 . . . . . . . . 9  |-  2  e.  RR
1817a1i 9 . . . . . . . 8  |-  ( ph  ->  2  e.  RR )
1918, 4remulcld 7714 . . . . . . 7  |-  ( ph  ->  ( 2  x.  ( A  +  B )
)  e.  RR )
205, 19readdcld 7713 . . . . . 6  |-  ( ph  ->  ( ( ( A  +  B )  x.  ( A  +  B
) )  +  ( 2  x.  ( A  +  B ) ) )  e.  RR )
2120adantr 272 . . . . 5  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  ( 2  x.  ( A  +  B
) ) )  e.  RR )
22 nn0addge2 8922 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  A  e.  NN0 )  ->  B  <_  ( A  +  B ) )
236, 1, 22syl2anc 406 . . . . . . . 8  |-  ( ph  ->  B  <_  ( A  +  B ) )
24 nn0addge1 8921 . . . . . . . . . 10  |-  ( ( ( A  +  B
)  e.  RR  /\  ( A  +  B
)  e.  NN0 )  ->  ( A  +  B
)  <_  ( ( A  +  B )  +  ( A  +  B ) ) )
254, 3, 24syl2anc 406 . . . . . . . . 9  |-  ( ph  ->  ( A  +  B
)  <_  ( ( A  +  B )  +  ( A  +  B ) ) )
264recnd 7712 . . . . . . . . . 10  |-  ( ph  ->  ( A  +  B
)  e.  CC )
27262timesd 8860 . . . . . . . . 9  |-  ( ph  ->  ( 2  x.  ( A  +  B )
)  =  ( ( A  +  B )  +  ( A  +  B ) ) )
2825, 27breqtrrd 3919 . . . . . . . 8  |-  ( ph  ->  ( A  +  B
)  <_  ( 2  x.  ( A  +  B ) ) )
296, 4, 19, 23, 28letrd 7803 . . . . . . 7  |-  ( ph  ->  B  <_  ( 2  x.  ( A  +  B ) ) )
306, 19, 5, 29leadd2dd 8234 . . . . . 6  |-  ( ph  ->  ( ( ( A  +  B )  x.  ( A  +  B
) )  +  B
)  <_  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  ( 2  x.  ( A  +  B
) ) ) )
3130adantr 272 . . . . 5  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B )  <_ 
( ( ( A  +  B )  x.  ( A  +  B
) )  +  ( 2  x.  ( A  +  B ) ) ) )
323, 9nn0opthlem1d 10353 . . . . . 6  |-  ( ph  ->  ( ( A  +  B )  <  C  <->  ( ( ( A  +  B )  x.  ( A  +  B )
)  +  ( 2  x.  ( A  +  B ) ) )  <  ( C  x.  C ) ) )
3332biimpa 292 . . . . 5  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  ( 2  x.  ( A  +  B
) ) )  < 
( C  x.  C
) )
348, 21, 12, 31, 33lelttrd 7804 . . . 4  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B )  < 
( C  x.  C
) )
35 nn0addge1 8921 . . . . . 6  |-  ( ( ( C  x.  C
)  e.  RR  /\  D  e.  NN0 )  -> 
( C  x.  C
)  <_  ( ( C  x.  C )  +  D ) )
3611, 13, 35syl2anc 406 . . . . 5  |-  ( ph  ->  ( C  x.  C
)  <_  ( ( C  x.  C )  +  D ) )
3736adantr 272 . . . 4  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( C  x.  C )  <_  (
( C  x.  C
)  +  D ) )
388, 12, 16, 34, 37ltletrd 8098 . . 3  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B )  < 
( ( C  x.  C )  +  D
) )
398, 38gtned 7793 . 2  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( ( C  x.  C )  +  D )  =/=  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B ) )
4039ex 114 1  |-  ( ph  ->  ( ( A  +  B )  <  C  ->  ( ( C  x.  C )  +  D
)  =/=  ( ( ( A  +  B
)  x.  ( A  +  B ) )  +  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1461    =/= wne 2280   class class class wbr 3893  (class class class)co 5726   RRcr 7540    + caddc 7544    x. cmul 7546    < clt 7718    <_ cle 7719   2c2 8675   NN0cn0 8875
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-iinf 4460  ax-cnex 7630  ax-resscn 7631  ax-1cn 7632  ax-1re 7633  ax-icn 7634  ax-addcl 7635  ax-addrcl 7636  ax-mulcl 7637  ax-mulrcl 7638  ax-addcom 7639  ax-mulcom 7640  ax-addass 7641  ax-mulass 7642  ax-distr 7643  ax-i2m1 7644  ax-0lt1 7645  ax-1rid 7646  ax-0id 7647  ax-rnegex 7648  ax-precex 7649  ax-cnre 7650  ax-pre-ltirr 7651  ax-pre-ltwlin 7652  ax-pre-lttrn 7653  ax-pre-apti 7654  ax-pre-ltadd 7655  ax-pre-mulgt0 7656  ax-pre-mulext 7657
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-nel 2376  df-ral 2393  df-rex 2394  df-reu 2395  df-rmo 2396  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-if 3439  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-id 4173  df-po 4176  df-iso 4177  df-iord 4246  df-on 4248  df-ilim 4249  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-riota 5682  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-recs 6154  df-frec 6240  df-pnf 7720  df-mnf 7721  df-xr 7722  df-ltxr 7723  df-le 7724  df-sub 7852  df-neg 7853  df-reap 8249  df-ap 8256  df-div 8340  df-inn 8625  df-2 8683  df-n0 8876  df-z 8953  df-uz 9223  df-seqfrec 10106  df-exp 10180
This theorem is referenced by:  nn0opthd  10355
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