ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nn0opthlem2d Unicode version

Theorem nn0opthlem2d 9745
Description: Lemma for nn0opth2 9748. (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 8412 . . . . . . 7  |-  ( ph  ->  ( A  +  B
)  e.  NN0 )
43nn0red 8409 . . . . . 6  |-  ( ph  ->  ( A  +  B
)  e.  RR )
54, 4remulcld 7211 . . . . 5  |-  ( ph  ->  ( ( A  +  B )  x.  ( A  +  B )
)  e.  RR )
62nn0red 8409 . . . . 5  |-  ( ph  ->  B  e.  RR )
75, 6readdcld 7210 . . . 4  |-  ( ph  ->  ( ( ( A  +  B )  x.  ( A  +  B
) )  +  B
)  e.  RR )
87adantr 270 . . 3  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B )  e.  RR )
9 nn0opthd.3 . . . . . . 7  |-  ( ph  ->  C  e.  NN0 )
109nn0red 8409 . . . . . 6  |-  ( ph  ->  C  e.  RR )
1110, 10remulcld 7211 . . . . 5  |-  ( ph  ->  ( C  x.  C
)  e.  RR )
1211adantr 270 . . . 4  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( C  x.  C )  e.  RR )
13 nn0opthd.4 . . . . . . 7  |-  ( ph  ->  D  e.  NN0 )
1413nn0red 8409 . . . . . 6  |-  ( ph  ->  D  e.  RR )
1511, 14readdcld 7210 . . . . 5  |-  ( ph  ->  ( ( C  x.  C )  +  D
)  e.  RR )
1615adantr 270 . . . 4  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( ( C  x.  C )  +  D )  e.  RR )
17 2re 8176 . . . . . . . . 9  |-  2  e.  RR
1817a1i 9 . . . . . . . 8  |-  ( ph  ->  2  e.  RR )
1918, 4remulcld 7211 . . . . . . 7  |-  ( ph  ->  ( 2  x.  ( A  +  B )
)  e.  RR )
205, 19readdcld 7210 . . . . . 6  |-  ( ph  ->  ( ( ( A  +  B )  x.  ( A  +  B
) )  +  ( 2  x.  ( A  +  B ) ) )  e.  RR )
2120adantr 270 . . . . 5  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  ( 2  x.  ( A  +  B
) ) )  e.  RR )
22 nn0addge2 8402 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  A  e.  NN0 )  ->  B  <_  ( A  +  B ) )
236, 1, 22syl2anc 403 . . . . . . . 8  |-  ( ph  ->  B  <_  ( A  +  B ) )
24 nn0addge1 8401 . . . . . . . . . 10  |-  ( ( ( A  +  B
)  e.  RR  /\  ( A  +  B
)  e.  NN0 )  ->  ( A  +  B
)  <_  ( ( A  +  B )  +  ( A  +  B ) ) )
254, 3, 24syl2anc 403 . . . . . . . . 9  |-  ( ph  ->  ( A  +  B
)  <_  ( ( A  +  B )  +  ( A  +  B ) ) )
264recnd 7209 . . . . . . . . . 10  |-  ( ph  ->  ( A  +  B
)  e.  CC )
27262timesd 8340 . . . . . . . . 9  |-  ( ph  ->  ( 2  x.  ( A  +  B )
)  =  ( ( A  +  B )  +  ( A  +  B ) ) )
2825, 27breqtrrd 3819 . . . . . . . 8  |-  ( ph  ->  ( A  +  B
)  <_  ( 2  x.  ( A  +  B ) ) )
296, 4, 19, 23, 28letrd 7300 . . . . . . 7  |-  ( ph  ->  B  <_  ( 2  x.  ( A  +  B ) ) )
306, 19, 5, 29leadd2dd 7727 . . . . . 6  |-  ( ph  ->  ( ( ( A  +  B )  x.  ( A  +  B
) )  +  B
)  <_  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  ( 2  x.  ( A  +  B
) ) ) )
3130adantr 270 . . . . 5  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B )  <_ 
( ( ( A  +  B )  x.  ( A  +  B
) )  +  ( 2  x.  ( A  +  B ) ) ) )
323, 9nn0opthlem1d 9744 . . . . . 6  |-  ( ph  ->  ( ( A  +  B )  <  C  <->  ( ( ( A  +  B )  x.  ( A  +  B )
)  +  ( 2  x.  ( A  +  B ) ) )  <  ( C  x.  C ) ) )
3332biimpa 290 . . . . 5  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  ( 2  x.  ( A  +  B
) ) )  < 
( C  x.  C
) )
348, 21, 12, 31, 33lelttrd 7301 . . . 4  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B )  < 
( C  x.  C
) )
35 nn0addge1 8401 . . . . . 6  |-  ( ( ( C  x.  C
)  e.  RR  /\  D  e.  NN0 )  -> 
( C  x.  C
)  <_  ( ( C  x.  C )  +  D ) )
3611, 13, 35syl2anc 403 . . . . 5  |-  ( ph  ->  ( C  x.  C
)  <_  ( ( C  x.  C )  +  D ) )
3736adantr 270 . . . 4  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( C  x.  C )  <_  (
( C  x.  C
)  +  D ) )
388, 12, 16, 34, 37ltletrd 7594 . . 3  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B )  < 
( ( C  x.  C )  +  D
) )
398, 38gtned 7290 . 2  |-  ( (
ph  /\  ( A  +  B )  <  C
)  ->  ( ( C  x.  C )  +  D )  =/=  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B ) )
4039ex 113 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 102    e. wcel 1434    =/= wne 2246   class class class wbr 3793  (class class class)co 5543   RRcr 7042    + caddc 7046    x. cmul 7048    < clt 7215    <_ cle 7216   2c2 8156   NN0cn0 8355
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-mulrcl 7137  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-precex 7148  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-apti 7153  ax-pre-ltadd 7154  ax-pre-mulgt0 7155  ax-pre-mulext 7156
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-if 3360  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-ilim 4132  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-frec 6040  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-reap 7742  df-ap 7749  df-div 7828  df-inn 8107  df-2 8165  df-n0 8356  df-z 8433  df-uz 8701  df-iseq 9522  df-iexp 9573
This theorem is referenced by:  nn0opthd  9746
  Copyright terms: Public domain W3C validator