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Theorem divalglemnqt 10527
Description: Lemma for divalg 10531. The  Q  <  T case involved in showing uniqueness. (Contributed by Jim Kingdon, 4-Dec-2021.)
Hypotheses
Ref Expression
divalglemnqt.d  |-  ( ph  ->  D  e.  NN )
divalglemnqt.r  |-  ( ph  ->  R  e.  ZZ )
divalglemnqt.s  |-  ( ph  ->  S  e.  ZZ )
divalglemnqt.q  |-  ( ph  ->  Q  e.  ZZ )
divalglemnqt.t  |-  ( ph  ->  T  e.  ZZ )
divalglemnqt.0s  |-  ( ph  ->  0  <_  S )
divalglemnqt.rd  |-  ( ph  ->  R  <  D )
divalglemnqt.eq  |-  ( ph  ->  ( ( Q  x.  D )  +  R
)  =  ( ( T  x.  D )  +  S ) )
Assertion
Ref Expression
divalglemnqt  |-  ( ph  ->  -.  Q  <  T
)

Proof of Theorem divalglemnqt
StepHypRef Expression
1 divalglemnqt.rd . . 3  |-  ( ph  ->  R  <  D )
21adantr 270 . 2  |-  ( (
ph  /\  Q  <  T )  ->  R  <  D )
3 divalglemnqt.d . . . . 5  |-  ( ph  ->  D  e.  NN )
43adantr 270 . . . 4  |-  ( (
ph  /\  Q  <  T )  ->  D  e.  NN )
54nnred 8171 . . 3  |-  ( (
ph  /\  Q  <  T )  ->  D  e.  RR )
6 divalglemnqt.r . . . . 5  |-  ( ph  ->  R  e.  ZZ )
76adantr 270 . . . 4  |-  ( (
ph  /\  Q  <  T )  ->  R  e.  ZZ )
87zred 8602 . . 3  |-  ( (
ph  /\  Q  <  T )  ->  R  e.  RR )
9 divalglemnqt.s . . . . . . 7  |-  ( ph  ->  S  e.  ZZ )
109adantr 270 . . . . . 6  |-  ( (
ph  /\  Q  <  T )  ->  S  e.  ZZ )
1110zred 8602 . . . . 5  |-  ( (
ph  /\  Q  <  T )  ->  S  e.  RR )
125, 11readdcld 7262 . . . 4  |-  ( (
ph  /\  Q  <  T )  ->  ( D  +  S )  e.  RR )
13 divalglemnqt.0s . . . . . 6  |-  ( ph  ->  0  <_  S )
1413adantr 270 . . . . 5  |-  ( (
ph  /\  Q  <  T )  ->  0  <_  S )
155, 11addge01d 7752 . . . . 5  |-  ( (
ph  /\  Q  <  T )  ->  ( 0  <_  S  <->  D  <_  ( D  +  S ) ) )
1614, 15mpbid 145 . . . 4  |-  ( (
ph  /\  Q  <  T )  ->  D  <_  ( D  +  S ) )
17 divalglemnqt.q . . . . . . . . . . 11  |-  ( ph  ->  Q  e.  ZZ )
1817adantr 270 . . . . . . . . . 10  |-  ( (
ph  /\  Q  <  T )  ->  Q  e.  ZZ )
1918zred 8602 . . . . . . . . 9  |-  ( (
ph  /\  Q  <  T )  ->  Q  e.  RR )
2019recnd 7261 . . . . . . . 8  |-  ( (
ph  /\  Q  <  T )  ->  Q  e.  CC )
215recnd 7261 . . . . . . . 8  |-  ( (
ph  /\  Q  <  T )  ->  D  e.  CC )
2220, 21mulcld 7253 . . . . . . 7  |-  ( (
ph  /\  Q  <  T )  ->  ( Q  x.  D )  e.  CC )
2311recnd 7261 . . . . . . 7  |-  ( (
ph  /\  Q  <  T )  ->  S  e.  CC )
2422, 21, 23addassd 7255 . . . . . 6  |-  ( (
ph  /\  Q  <  T )  ->  ( (
( Q  x.  D
)  +  D )  +  S )  =  ( ( Q  x.  D )  +  ( D  +  S ) ) )
2519, 5remulcld 7263 . . . . . . . . 9  |-  ( (
ph  /\  Q  <  T )  ->  ( Q  x.  D )  e.  RR )
2625, 5readdcld 7262 . . . . . . . 8  |-  ( (
ph  /\  Q  <  T )  ->  ( ( Q  x.  D )  +  D )  e.  RR )
27 divalglemnqt.t . . . . . . . . . . 11  |-  ( ph  ->  T  e.  ZZ )
2827adantr 270 . . . . . . . . . 10  |-  ( (
ph  /\  Q  <  T )  ->  T  e.  ZZ )
2928zred 8602 . . . . . . . . 9  |-  ( (
ph  /\  Q  <  T )  ->  T  e.  RR )
3029, 5remulcld 7263 . . . . . . . 8  |-  ( (
ph  /\  Q  <  T )  ->  ( T  x.  D )  e.  RR )
3120, 21adddirp1d 7259 . . . . . . . . 9  |-  ( (
ph  /\  Q  <  T )  ->  ( ( Q  +  1 )  x.  D )  =  ( ( Q  x.  D )  +  D
) )
32 peano2re 7363 . . . . . . . . . . 11  |-  ( Q  e.  RR  ->  ( Q  +  1 )  e.  RR )
3319, 32syl 14 . . . . . . . . . 10  |-  ( (
ph  /\  Q  <  T )  ->  ( Q  +  1 )  e.  RR )
344nnnn0d 8460 . . . . . . . . . . 11  |-  ( (
ph  /\  Q  <  T )  ->  D  e.  NN0 )
3534nn0ge0d 8463 . . . . . . . . . 10  |-  ( (
ph  /\  Q  <  T )  ->  0  <_  D )
36 simpr 108 . . . . . . . . . . 11  |-  ( (
ph  /\  Q  <  T )  ->  Q  <  T )
37 zltp1le 8538 . . . . . . . . . . . 12  |-  ( ( Q  e.  ZZ  /\  T  e.  ZZ )  ->  ( Q  <  T  <->  ( Q  +  1 )  <_  T ) )
3817, 28, 37syl2an2r 560 . . . . . . . . . . 11  |-  ( (
ph  /\  Q  <  T )  ->  ( Q  <  T  <->  ( Q  + 
1 )  <_  T
) )
3936, 38mpbid 145 . . . . . . . . . 10  |-  ( (
ph  /\  Q  <  T )  ->  ( Q  +  1 )  <_  T )
4033, 29, 5, 35, 39lemul1ad 8136 . . . . . . . . 9  |-  ( (
ph  /\  Q  <  T )  ->  ( ( Q  +  1 )  x.  D )  <_ 
( T  x.  D
) )
4131, 40eqbrtrrd 3827 . . . . . . . 8  |-  ( (
ph  /\  Q  <  T )  ->  ( ( Q  x.  D )  +  D )  <_  ( T  x.  D )
)
4226, 30, 11, 41leadd1dd 7778 . . . . . . 7  |-  ( (
ph  /\  Q  <  T )  ->  ( (
( Q  x.  D
)  +  D )  +  S )  <_ 
( ( T  x.  D )  +  S
) )
43 divalglemnqt.eq . . . . . . . 8  |-  ( ph  ->  ( ( Q  x.  D )  +  R
)  =  ( ( T  x.  D )  +  S ) )
4443adantr 270 . . . . . . 7  |-  ( (
ph  /\  Q  <  T )  ->  ( ( Q  x.  D )  +  R )  =  ( ( T  x.  D
)  +  S ) )
4542, 44breqtrrd 3831 . . . . . 6  |-  ( (
ph  /\  Q  <  T )  ->  ( (
( Q  x.  D
)  +  D )  +  S )  <_ 
( ( Q  x.  D )  +  R
) )
4624, 45eqbrtrrd 3827 . . . . 5  |-  ( (
ph  /\  Q  <  T )  ->  ( ( Q  x.  D )  +  ( D  +  S ) )  <_ 
( ( Q  x.  D )  +  R
) )
4712, 8, 25leadd2d 7759 . . . . 5  |-  ( (
ph  /\  Q  <  T )  ->  ( ( D  +  S )  <_  R  <->  ( ( Q  x.  D )  +  ( D  +  S
) )  <_  (
( Q  x.  D
)  +  R ) ) )
4846, 47mpbird 165 . . . 4  |-  ( (
ph  /\  Q  <  T )  ->  ( D  +  S )  <_  R
)
495, 12, 8, 16, 48letrd 7352 . . 3  |-  ( (
ph  /\  Q  <  T )  ->  D  <_  R )
505, 8, 49lensymd 7350 . 2  |-  ( (
ph  /\  Q  <  T )  ->  -.  R  <  D )
512, 50pm2.65da 620 1  |-  ( ph  ->  -.  Q  <  T
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   class class class wbr 3805  (class class class)co 5563   RRcr 7094   0cc0 7095   1c1 7096    + caddc 7098    x. cmul 7100    < clt 7267    <_ cle 7268   NNcn 8158   ZZcz 8484
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 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-mulrcl 7189  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-precex 7200  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-apti 7205  ax-pre-ltadd 7206  ax-pre-mulgt0 7207  ax-pre-mulext 7208
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-br 3806  df-opab 3860  df-id 4076  df-po 4079  df-iso 4080  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-iota 4917  df-fun 4954  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-reap 7794  df-ap 7801  df-inn 8159  df-n0 8408  df-z 8485
This theorem is referenced by:  divalglemeunn  10528  divalglemeuneg  10530
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