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Theorem divalglemnqt 11606
Description: Lemma for divalg 11610. 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 274 . 2  |-  ( (
ph  /\  Q  <  T )  ->  R  <  D )
3 divalglemnqt.d . . . . 5  |-  ( ph  ->  D  e.  NN )
43adantr 274 . . . 4  |-  ( (
ph  /\  Q  <  T )  ->  D  e.  NN )
54nnred 8726 . . 3  |-  ( (
ph  /\  Q  <  T )  ->  D  e.  RR )
6 divalglemnqt.r . . . . 5  |-  ( ph  ->  R  e.  ZZ )
76adantr 274 . . . 4  |-  ( (
ph  /\  Q  <  T )  ->  R  e.  ZZ )
87zred 9166 . . 3  |-  ( (
ph  /\  Q  <  T )  ->  R  e.  RR )
9 divalglemnqt.s . . . . . . 7  |-  ( ph  ->  S  e.  ZZ )
109adantr 274 . . . . . 6  |-  ( (
ph  /\  Q  <  T )  ->  S  e.  ZZ )
1110zred 9166 . . . . 5  |-  ( (
ph  /\  Q  <  T )  ->  S  e.  RR )
125, 11readdcld 7788 . . . 4  |-  ( (
ph  /\  Q  <  T )  ->  ( D  +  S )  e.  RR )
13 divalglemnqt.0s . . . . . 6  |-  ( ph  ->  0  <_  S )
1413adantr 274 . . . . 5  |-  ( (
ph  /\  Q  <  T )  ->  0  <_  S )
155, 11addge01d 8288 . . . . 5  |-  ( (
ph  /\  Q  <  T )  ->  ( 0  <_  S  <->  D  <_  ( D  +  S ) ) )
1614, 15mpbid 146 . . . 4  |-  ( (
ph  /\  Q  <  T )  ->  D  <_  ( D  +  S ) )
17 divalglemnqt.q . . . . . . . . . . 11  |-  ( ph  ->  Q  e.  ZZ )
1817adantr 274 . . . . . . . . . 10  |-  ( (
ph  /\  Q  <  T )  ->  Q  e.  ZZ )
1918zred 9166 . . . . . . . . 9  |-  ( (
ph  /\  Q  <  T )  ->  Q  e.  RR )
2019recnd 7787 . . . . . . . 8  |-  ( (
ph  /\  Q  <  T )  ->  Q  e.  CC )
215recnd 7787 . . . . . . . 8  |-  ( (
ph  /\  Q  <  T )  ->  D  e.  CC )
2220, 21mulcld 7779 . . . . . . 7  |-  ( (
ph  /\  Q  <  T )  ->  ( Q  x.  D )  e.  CC )
2311recnd 7787 . . . . . . 7  |-  ( (
ph  /\  Q  <  T )  ->  S  e.  CC )
2422, 21, 23addassd 7781 . . . . . 6  |-  ( (
ph  /\  Q  <  T )  ->  ( (
( Q  x.  D
)  +  D )  +  S )  =  ( ( Q  x.  D )  +  ( D  +  S ) ) )
2519, 5remulcld 7789 . . . . . . . . 9  |-  ( (
ph  /\  Q  <  T )  ->  ( Q  x.  D )  e.  RR )
2625, 5readdcld 7788 . . . . . . . 8  |-  ( (
ph  /\  Q  <  T )  ->  ( ( Q  x.  D )  +  D )  e.  RR )
27 divalglemnqt.t . . . . . . . . . . 11  |-  ( ph  ->  T  e.  ZZ )
2827adantr 274 . . . . . . . . . 10  |-  ( (
ph  /\  Q  <  T )  ->  T  e.  ZZ )
2928zred 9166 . . . . . . . . 9  |-  ( (
ph  /\  Q  <  T )  ->  T  e.  RR )
3029, 5remulcld 7789 . . . . . . . 8  |-  ( (
ph  /\  Q  <  T )  ->  ( T  x.  D )  e.  RR )
3120, 21adddirp1d 7785 . . . . . . . . 9  |-  ( (
ph  /\  Q  <  T )  ->  ( ( Q  +  1 )  x.  D )  =  ( ( Q  x.  D )  +  D
) )
32 peano2re 7891 . . . . . . . . . . 11  |-  ( Q  e.  RR  ->  ( Q  +  1 )  e.  RR )
3319, 32syl 14 . . . . . . . . . 10  |-  ( (
ph  /\  Q  <  T )  ->  ( Q  +  1 )  e.  RR )
344nnnn0d 9023 . . . . . . . . . . 11  |-  ( (
ph  /\  Q  <  T )  ->  D  e.  NN0 )
3534nn0ge0d 9026 . . . . . . . . . 10  |-  ( (
ph  /\  Q  <  T )  ->  0  <_  D )
36 simpr 109 . . . . . . . . . . 11  |-  ( (
ph  /\  Q  <  T )  ->  Q  <  T )
37 zltp1le 9101 . . . . . . . . . . . 12  |-  ( ( Q  e.  ZZ  /\  T  e.  ZZ )  ->  ( Q  <  T  <->  ( Q  +  1 )  <_  T ) )
3817, 28, 37syl2an2r 584 . . . . . . . . . . 11  |-  ( (
ph  /\  Q  <  T )  ->  ( Q  <  T  <->  ( Q  + 
1 )  <_  T
) )
3936, 38mpbid 146 . . . . . . . . . 10  |-  ( (
ph  /\  Q  <  T )  ->  ( Q  +  1 )  <_  T )
4033, 29, 5, 35, 39lemul1ad 8690 . . . . . . . . 9  |-  ( (
ph  /\  Q  <  T )  ->  ( ( Q  +  1 )  x.  D )  <_ 
( T  x.  D
) )
4131, 40eqbrtrrd 3947 . . . . . . . 8  |-  ( (
ph  /\  Q  <  T )  ->  ( ( Q  x.  D )  +  D )  <_  ( T  x.  D )
)
4226, 30, 11, 41leadd1dd 8314 . . . . . . 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 274 . . . . . . 7  |-  ( (
ph  /\  Q  <  T )  ->  ( ( Q  x.  D )  +  R )  =  ( ( T  x.  D
)  +  S ) )
4542, 44breqtrrd 3951 . . . . . 6  |-  ( (
ph  /\  Q  <  T )  ->  ( (
( Q  x.  D
)  +  D )  +  S )  <_ 
( ( Q  x.  D )  +  R
) )
4624, 45eqbrtrrd 3947 . . . . 5  |-  ( (
ph  /\  Q  <  T )  ->  ( ( Q  x.  D )  +  ( D  +  S ) )  <_ 
( ( Q  x.  D )  +  R
) )
4712, 8, 25leadd2d 8295 . . . . 5  |-  ( (
ph  /\  Q  <  T )  ->  ( ( D  +  S )  <_  R  <->  ( ( Q  x.  D )  +  ( D  +  S
) )  <_  (
( Q  x.  D
)  +  R ) ) )
4846, 47mpbird 166 . . . 4  |-  ( (
ph  /\  Q  <  T )  ->  ( D  +  S )  <_  R
)
495, 12, 8, 16, 48letrd 7879 . . 3  |-  ( (
ph  /\  Q  <  T )  ->  D  <_  R )
505, 8, 49lensymd 7877 . 2  |-  ( (
ph  /\  Q  <  T )  ->  -.  R  <  D )
512, 50pm2.65da 650 1  |-  ( ph  ->  -.  Q  <  T
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   class class class wbr 3924  (class class class)co 5767   RRcr 7612   0cc0 7613   1c1 7614    + caddc 7616    x. cmul 7618    < clt 7793    <_ cle 7794   NNcn 8713   ZZcz 9047
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-br 3925  df-opab 3985  df-id 4210  df-po 4213  df-iso 4214  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-inn 8714  df-n0 8971  df-z 9048
This theorem is referenced by:  divalglemeunn  11607  divalglemeuneg  11609
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