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Theorem nn0opthlem2d 10499
Description: Lemma for nn0opth2 10502. (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 9058 . . . . . . 7  |-  ( ph  ->  ( A  +  B
)  e.  NN0 )
43nn0red 9055 . . . . . 6  |-  ( ph  ->  ( A  +  B
)  e.  RR )
54, 4remulcld 7820 . . . . 5  |-  ( ph  ->  ( ( A  +  B )  x.  ( A  +  B )
)  e.  RR )
62nn0red 9055 . . . . 5  |-  ( ph  ->  B  e.  RR )
75, 6readdcld 7819 . . . 4  |-  ( ph  ->  ( ( ( A  +  B )  x.  ( A  +  B
) )  +  B
)  e.  RR )
87adantr 274 . . 3  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B )  e.  RR )
9 nn0opthd.3 . . . . . . 7  |-  ( ph  ->  C  e.  NN0 )
109nn0red 9055 . . . . . 6  |-  ( ph  ->  C  e.  RR )
1110, 10remulcld 7820 . . . . 5  |-  ( ph  ->  ( C  x.  C
)  e.  RR )
1211adantr 274 . . . 4  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( C  x.  C )  e.  RR )
13 nn0opthd.4 . . . . . . 7  |-  ( ph  ->  D  e.  NN0 )
1413nn0red 9055 . . . . . 6  |-  ( ph  ->  D  e.  RR )
1511, 14readdcld 7819 . . . . 5  |-  ( ph  ->  ( ( C  x.  C )  +  D
)  e.  RR )
1615adantr 274 . . . 4  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( ( C  x.  C )  +  D )  e.  RR )
17 2re 8814 . . . . . . . . 9  |-  2  e.  RR
1817a1i 9 . . . . . . . 8  |-  ( ph  ->  2  e.  RR )
1918, 4remulcld 7820 . . . . . . 7  |-  ( ph  ->  ( 2  x.  ( A  +  B )
)  e.  RR )
205, 19readdcld 7819 . . . . . 6  |-  ( ph  ->  ( ( ( A  +  B )  x.  ( A  +  B
) )  +  ( 2  x.  ( A  +  B ) ) )  e.  RR )
2120adantr 274 . . . . 5  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  ( 2  x.  ( A  +  B
) ) )  e.  RR )
22 nn0addge2 9048 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  A  e.  NN0 )  ->  B  <_  ( A  +  B ) )
236, 1, 22syl2anc 409 . . . . . . . 8  |-  ( ph  ->  B  <_  ( A  +  B ) )
24 nn0addge1 9047 . . . . . . . . . 10  |-  ( ( ( A  +  B
)  e.  RR  /\  ( A  +  B
)  e.  NN0 )  ->  ( A  +  B
)  <_  ( ( A  +  B )  +  ( A  +  B ) ) )
254, 3, 24syl2anc 409 . . . . . . . . 9  |-  ( ph  ->  ( A  +  B
)  <_  ( ( A  +  B )  +  ( A  +  B ) ) )
264recnd 7818 . . . . . . . . . 10  |-  ( ph  ->  ( A  +  B
)  e.  CC )
27262timesd 8986 . . . . . . . . 9  |-  ( ph  ->  ( 2  x.  ( A  +  B )
)  =  ( ( A  +  B )  +  ( A  +  B ) ) )
2825, 27breqtrrd 3964 . . . . . . . 8  |-  ( ph  ->  ( A  +  B
)  <_  ( 2  x.  ( A  +  B ) ) )
296, 4, 19, 23, 28letrd 7910 . . . . . . 7  |-  ( ph  ->  B  <_  ( 2  x.  ( A  +  B ) ) )
306, 19, 5, 29leadd2dd 8346 . . . . . 6  |-  ( ph  ->  ( ( ( A  +  B )  x.  ( A  +  B
) )  +  B
)  <_  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  ( 2  x.  ( A  +  B
) ) ) )
3130adantr 274 . . . . 5  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B )  <_ 
( ( ( A  +  B )  x.  ( A  +  B
) )  +  ( 2  x.  ( A  +  B ) ) ) )
323, 9nn0opthlem1d 10498 . . . . . 6  |-  ( ph  ->  ( ( A  +  B )  <  C  <->  ( ( ( A  +  B )  x.  ( A  +  B )
)  +  ( 2  x.  ( A  +  B ) ) )  <  ( C  x.  C ) ) )
3332biimpa 294 . . . . 5  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  ( 2  x.  ( A  +  B
) ) )  < 
( C  x.  C
) )
348, 21, 12, 31, 33lelttrd 7911 . . . 4  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B )  < 
( C  x.  C
) )
35 nn0addge1 9047 . . . . . 6  |-  ( ( ( C  x.  C
)  e.  RR  /\  D  e.  NN0 )  -> 
( C  x.  C
)  <_  ( ( C  x.  C )  +  D ) )
3611, 13, 35syl2anc 409 . . . . 5  |-  ( ph  ->  ( C  x.  C
)  <_  ( ( C  x.  C )  +  D ) )
3736adantr 274 . . . 4  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( C  x.  C )  <_  (
( C  x.  C
)  +  D ) )
388, 12, 16, 34, 37ltletrd 8209 . . 3  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B )  < 
( ( C  x.  C )  +  D
) )
398, 38gtned 7900 . 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 1481    =/= wne 2309   class class class wbr 3937  (class class class)co 5782   RRcr 7643    + caddc 7647    x. cmul 7649    < clt 7824    <_ cle 7825   2c2 8795   NN0cn0 9001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-frec 6296  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-2 8803  df-n0 9002  df-z 9079  df-uz 9351  df-seqfrec 10250  df-exp 10324
This theorem is referenced by:  nn0opthd  10500
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