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Theorem nn0opthlem2d 10733
Description: Lemma for nn0opth2 10736. (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 9263 . . . . . . 7  |-  ( ph  ->  ( A  +  B
)  e.  NN0 )
43nn0red 9260 . . . . . 6  |-  ( ph  ->  ( A  +  B
)  e.  RR )
54, 4remulcld 8018 . . . . 5  |-  ( ph  ->  ( ( A  +  B )  x.  ( A  +  B )
)  e.  RR )
62nn0red 9260 . . . . 5  |-  ( ph  ->  B  e.  RR )
75, 6readdcld 8017 . . . 4  |-  ( ph  ->  ( ( ( A  +  B )  x.  ( A  +  B
) )  +  B
)  e.  RR )
87adantr 276 . . 3  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B )  e.  RR )
9 nn0opthd.3 . . . . . . 7  |-  ( ph  ->  C  e.  NN0 )
109nn0red 9260 . . . . . 6  |-  ( ph  ->  C  e.  RR )
1110, 10remulcld 8018 . . . . 5  |-  ( ph  ->  ( C  x.  C
)  e.  RR )
1211adantr 276 . . . 4  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( C  x.  C )  e.  RR )
13 nn0opthd.4 . . . . . . 7  |-  ( ph  ->  D  e.  NN0 )
1413nn0red 9260 . . . . . 6  |-  ( ph  ->  D  e.  RR )
1511, 14readdcld 8017 . . . . 5  |-  ( ph  ->  ( ( C  x.  C )  +  D
)  e.  RR )
1615adantr 276 . . . 4  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( ( C  x.  C )  +  D )  e.  RR )
17 2re 9019 . . . . . . . . 9  |-  2  e.  RR
1817a1i 9 . . . . . . . 8  |-  ( ph  ->  2  e.  RR )
1918, 4remulcld 8018 . . . . . . 7  |-  ( ph  ->  ( 2  x.  ( A  +  B )
)  e.  RR )
205, 19readdcld 8017 . . . . . 6  |-  ( ph  ->  ( ( ( A  +  B )  x.  ( A  +  B
) )  +  ( 2  x.  ( A  +  B ) ) )  e.  RR )
2120adantr 276 . . . . 5  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  ( 2  x.  ( A  +  B
) ) )  e.  RR )
22 nn0addge2 9253 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  A  e.  NN0 )  ->  B  <_  ( A  +  B ) )
236, 1, 22syl2anc 411 . . . . . . . 8  |-  ( ph  ->  B  <_  ( A  +  B ) )
24 nn0addge1 9252 . . . . . . . . . 10  |-  ( ( ( A  +  B
)  e.  RR  /\  ( A  +  B
)  e.  NN0 )  ->  ( A  +  B
)  <_  ( ( A  +  B )  +  ( A  +  B ) ) )
254, 3, 24syl2anc 411 . . . . . . . . 9  |-  ( ph  ->  ( A  +  B
)  <_  ( ( A  +  B )  +  ( A  +  B ) ) )
264recnd 8016 . . . . . . . . . 10  |-  ( ph  ->  ( A  +  B
)  e.  CC )
27262timesd 9191 . . . . . . . . 9  |-  ( ph  ->  ( 2  x.  ( A  +  B )
)  =  ( ( A  +  B )  +  ( A  +  B ) ) )
2825, 27breqtrrd 4046 . . . . . . . 8  |-  ( ph  ->  ( A  +  B
)  <_  ( 2  x.  ( A  +  B ) ) )
296, 4, 19, 23, 28letrd 8111 . . . . . . 7  |-  ( ph  ->  B  <_  ( 2  x.  ( A  +  B ) ) )
306, 19, 5, 29leadd2dd 8547 . . . . . 6  |-  ( ph  ->  ( ( ( A  +  B )  x.  ( A  +  B
) )  +  B
)  <_  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  ( 2  x.  ( A  +  B
) ) ) )
3130adantr 276 . . . . 5  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B )  <_ 
( ( ( A  +  B )  x.  ( A  +  B
) )  +  ( 2  x.  ( A  +  B ) ) ) )
323, 9nn0opthlem1d 10732 . . . . . 6  |-  ( ph  ->  ( ( A  +  B )  <  C  <->  ( ( ( A  +  B )  x.  ( A  +  B )
)  +  ( 2  x.  ( A  +  B ) ) )  <  ( C  x.  C ) ) )
3332biimpa 296 . . . . 5  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  ( 2  x.  ( A  +  B
) ) )  < 
( C  x.  C
) )
348, 21, 12, 31, 33lelttrd 8112 . . . 4  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B )  < 
( C  x.  C
) )
35 nn0addge1 9252 . . . . . 6  |-  ( ( ( C  x.  C
)  e.  RR  /\  D  e.  NN0 )  -> 
( C  x.  C
)  <_  ( ( C  x.  C )  +  D ) )
3611, 13, 35syl2anc 411 . . . . 5  |-  ( ph  ->  ( C  x.  C
)  <_  ( ( C  x.  C )  +  D ) )
3736adantr 276 . . . 4  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( C  x.  C )  <_  (
( C  x.  C
)  +  D ) )
388, 12, 16, 34, 37ltletrd 8410 . . 3  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B )  < 
( ( C  x.  C )  +  D
) )
398, 38gtned 8100 . 2  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( ( C  x.  C )  +  D )  =/=  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B ) )
4039ex 115 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 104    e. wcel 2160    =/= wne 2360   class class class wbr 4018  (class class class)co 5896   RRcr 7840    + caddc 7844    x. cmul 7846    < clt 8022    <_ cle 8023   2c2 9000   NN0cn0 9206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-mulrcl 7940  ax-addcom 7941  ax-mulcom 7942  ax-addass 7943  ax-mulass 7944  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-1rid 7948  ax-0id 7949  ax-rnegex 7950  ax-precex 7951  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-apti 7956  ax-pre-ltadd 7957  ax-pre-mulgt0 7958  ax-pre-mulext 7959
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-frec 6416  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161  df-reap 8562  df-ap 8569  df-div 8660  df-inn 8950  df-2 9008  df-n0 9207  df-z 9284  df-uz 9559  df-seqfrec 10477  df-exp 10551
This theorem is referenced by:  nn0opthd  10734
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