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Theorem divalglemnqt 11927
Description: Lemma for divalg 11931. 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 276 . 2  |-  ( (
ph  /\  Q  <  T )  ->  R  <  D )
3 divalglemnqt.d . . . . 5  |-  ( ph  ->  D  e.  NN )
43adantr 276 . . . 4  |-  ( (
ph  /\  Q  <  T )  ->  D  e.  NN )
54nnred 8934 . . 3  |-  ( (
ph  /\  Q  <  T )  ->  D  e.  RR )
6 divalglemnqt.r . . . . 5  |-  ( ph  ->  R  e.  ZZ )
76adantr 276 . . . 4  |-  ( (
ph  /\  Q  <  T )  ->  R  e.  ZZ )
87zred 9377 . . 3  |-  ( (
ph  /\  Q  <  T )  ->  R  e.  RR )
9 divalglemnqt.s . . . . . . 7  |-  ( ph  ->  S  e.  ZZ )
109adantr 276 . . . . . 6  |-  ( (
ph  /\  Q  <  T )  ->  S  e.  ZZ )
1110zred 9377 . . . . 5  |-  ( (
ph  /\  Q  <  T )  ->  S  e.  RR )
125, 11readdcld 7989 . . . 4  |-  ( (
ph  /\  Q  <  T )  ->  ( D  +  S )  e.  RR )
13 divalglemnqt.0s . . . . . 6  |-  ( ph  ->  0  <_  S )
1413adantr 276 . . . . 5  |-  ( (
ph  /\  Q  <  T )  ->  0  <_  S )
155, 11addge01d 8492 . . . . 5  |-  ( (
ph  /\  Q  <  T )  ->  ( 0  <_  S  <->  D  <_  ( D  +  S ) ) )
1614, 15mpbid 147 . . . 4  |-  ( (
ph  /\  Q  <  T )  ->  D  <_  ( D  +  S ) )
17 divalglemnqt.q . . . . . . . . . . 11  |-  ( ph  ->  Q  e.  ZZ )
1817adantr 276 . . . . . . . . . 10  |-  ( (
ph  /\  Q  <  T )  ->  Q  e.  ZZ )
1918zred 9377 . . . . . . . . 9  |-  ( (
ph  /\  Q  <  T )  ->  Q  e.  RR )
2019recnd 7988 . . . . . . . 8  |-  ( (
ph  /\  Q  <  T )  ->  Q  e.  CC )
215recnd 7988 . . . . . . . 8  |-  ( (
ph  /\  Q  <  T )  ->  D  e.  CC )
2220, 21mulcld 7980 . . . . . . 7  |-  ( (
ph  /\  Q  <  T )  ->  ( Q  x.  D )  e.  CC )
2311recnd 7988 . . . . . . 7  |-  ( (
ph  /\  Q  <  T )  ->  S  e.  CC )
2422, 21, 23addassd 7982 . . . . . 6  |-  ( (
ph  /\  Q  <  T )  ->  ( (
( Q  x.  D
)  +  D )  +  S )  =  ( ( Q  x.  D )  +  ( D  +  S ) ) )
2519, 5remulcld 7990 . . . . . . . . 9  |-  ( (
ph  /\  Q  <  T )  ->  ( Q  x.  D )  e.  RR )
2625, 5readdcld 7989 . . . . . . . 8  |-  ( (
ph  /\  Q  <  T )  ->  ( ( Q  x.  D )  +  D )  e.  RR )
27 divalglemnqt.t . . . . . . . . . . 11  |-  ( ph  ->  T  e.  ZZ )
2827adantr 276 . . . . . . . . . 10  |-  ( (
ph  /\  Q  <  T )  ->  T  e.  ZZ )
2928zred 9377 . . . . . . . . 9  |-  ( (
ph  /\  Q  <  T )  ->  T  e.  RR )
3029, 5remulcld 7990 . . . . . . . 8  |-  ( (
ph  /\  Q  <  T )  ->  ( T  x.  D )  e.  RR )
3120, 21adddirp1d 7986 . . . . . . . . 9  |-  ( (
ph  /\  Q  <  T )  ->  ( ( Q  +  1 )  x.  D )  =  ( ( Q  x.  D )  +  D
) )
32 peano2re 8095 . . . . . . . . . . 11  |-  ( Q  e.  RR  ->  ( Q  +  1 )  e.  RR )
3319, 32syl 14 . . . . . . . . . 10  |-  ( (
ph  /\  Q  <  T )  ->  ( Q  +  1 )  e.  RR )
344nnnn0d 9231 . . . . . . . . . . 11  |-  ( (
ph  /\  Q  <  T )  ->  D  e.  NN0 )
3534nn0ge0d 9234 . . . . . . . . . 10  |-  ( (
ph  /\  Q  <  T )  ->  0  <_  D )
36 simpr 110 . . . . . . . . . . 11  |-  ( (
ph  /\  Q  <  T )  ->  Q  <  T )
37 zltp1le 9309 . . . . . . . . . . . 12  |-  ( ( Q  e.  ZZ  /\  T  e.  ZZ )  ->  ( Q  <  T  <->  ( Q  +  1 )  <_  T ) )
3817, 28, 37syl2an2r 595 . . . . . . . . . . 11  |-  ( (
ph  /\  Q  <  T )  ->  ( Q  <  T  <->  ( Q  + 
1 )  <_  T
) )
3936, 38mpbid 147 . . . . . . . . . 10  |-  ( (
ph  /\  Q  <  T )  ->  ( Q  +  1 )  <_  T )
4033, 29, 5, 35, 39lemul1ad 8898 . . . . . . . . 9  |-  ( (
ph  /\  Q  <  T )  ->  ( ( Q  +  1 )  x.  D )  <_ 
( T  x.  D
) )
4131, 40eqbrtrrd 4029 . . . . . . . 8  |-  ( (
ph  /\  Q  <  T )  ->  ( ( Q  x.  D )  +  D )  <_  ( T  x.  D )
)
4226, 30, 11, 41leadd1dd 8518 . . . . . . 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 276 . . . . . . 7  |-  ( (
ph  /\  Q  <  T )  ->  ( ( Q  x.  D )  +  R )  =  ( ( T  x.  D
)  +  S ) )
4542, 44breqtrrd 4033 . . . . . 6  |-  ( (
ph  /\  Q  <  T )  ->  ( (
( Q  x.  D
)  +  D )  +  S )  <_ 
( ( Q  x.  D )  +  R
) )
4624, 45eqbrtrrd 4029 . . . . 5  |-  ( (
ph  /\  Q  <  T )  ->  ( ( Q  x.  D )  +  ( D  +  S ) )  <_ 
( ( Q  x.  D )  +  R
) )
4712, 8, 25leadd2d 8499 . . . . 5  |-  ( (
ph  /\  Q  <  T )  ->  ( ( D  +  S )  <_  R  <->  ( ( Q  x.  D )  +  ( D  +  S
) )  <_  (
( Q  x.  D
)  +  R ) ) )
4846, 47mpbird 167 . . . 4  |-  ( (
ph  /\  Q  <  T )  ->  ( D  +  S )  <_  R
)
495, 12, 8, 16, 48letrd 8083 . . 3  |-  ( (
ph  /\  Q  <  T )  ->  D  <_  R )
505, 8, 49lensymd 8081 . 2  |-  ( (
ph  /\  Q  <  T )  ->  -.  R  <  D )
512, 50pm2.65da 661 1  |-  ( ph  ->  -.  Q  <  T
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   class class class wbr 4005  (class class class)co 5877   RRcr 7812   0cc0 7813   1c1 7814    + caddc 7816    x. cmul 7818    < clt 7994    <_ cle 7995   NNcn 8921   ZZcz 9255
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-id 4295  df-po 4298  df-iso 4299  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-inn 8922  df-n0 9179  df-z 9256
This theorem is referenced by:  divalglemeunn  11928  divalglemeuneg  11930
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