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Theorem divalglemqt 11894
Description: Lemma for divalg 11899. The  Q  =  T case involved in showing uniqueness. (Contributed by Jim Kingdon, 5-Dec-2021.)
Hypotheses
Ref Expression
divalglemqt.d  |-  ( ph  ->  D  e.  ZZ )
divalglemqt.r  |-  ( ph  ->  R  e.  ZZ )
divalglemqt.s  |-  ( ph  ->  S  e.  ZZ )
divalglemqt.q  |-  ( ph  ->  Q  e.  ZZ )
divalglemqt.t  |-  ( ph  ->  T  e.  ZZ )
divalglemqt.qt  |-  ( ph  ->  Q  =  T )
divalglemqt.eq  |-  ( ph  ->  ( ( Q  x.  D )  +  R
)  =  ( ( T  x.  D )  +  S ) )
Assertion
Ref Expression
divalglemqt  |-  ( ph  ->  R  =  S )

Proof of Theorem divalglemqt
StepHypRef Expression
1 divalglemqt.qt . . . 4  |-  ( ph  ->  Q  =  T )
21oveq1d 5883 . . 3  |-  ( ph  ->  ( Q  x.  D
)  =  ( T  x.  D ) )
3 divalglemqt.q . . . . 5  |-  ( ph  ->  Q  e.  ZZ )
4 divalglemqt.d . . . . 5  |-  ( ph  ->  D  e.  ZZ )
53, 4zmulcld 9357 . . . 4  |-  ( ph  ->  ( Q  x.  D
)  e.  ZZ )
65zcnd 9352 . . 3  |-  ( ph  ->  ( Q  x.  D
)  e.  CC )
72, 6eqeltrrd 2255 . 2  |-  ( ph  ->  ( T  x.  D
)  e.  CC )
8 divalglemqt.r . . 3  |-  ( ph  ->  R  e.  ZZ )
98zcnd 9352 . 2  |-  ( ph  ->  R  e.  CC )
10 divalglemqt.s . . 3  |-  ( ph  ->  S  e.  ZZ )
1110zcnd 9352 . 2  |-  ( ph  ->  S  e.  CC )
122oveq1d 5883 . . 3  |-  ( ph  ->  ( ( Q  x.  D )  +  R
)  =  ( ( T  x.  D )  +  R ) )
13 divalglemqt.eq . . 3  |-  ( ph  ->  ( ( Q  x.  D )  +  R
)  =  ( ( T  x.  D )  +  S ) )
1412, 13eqtr3d 2212 . 2  |-  ( ph  ->  ( ( T  x.  D )  +  R
)  =  ( ( T  x.  D )  +  S ) )
157, 9, 11, 14addcanad 8120 1  |-  ( ph  ->  R  =  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148  (class class class)co 5868   CCcc 7787    + caddc 7792    x. cmul 7794   ZZcz 9229
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-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-setind 4532  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-mulrcl 7888  ax-addcom 7889  ax-mulcom 7890  ax-addass 7891  ax-mulass 7892  ax-distr 7893  ax-i2m1 7894  ax-1rid 7896  ax-0id 7897  ax-rnegex 7898  ax-cnre 7900
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-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-iota 5173  df-fun 5213  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-sub 8107  df-neg 8108  df-inn 8896  df-n0 9153  df-z 9230
This theorem is referenced by:  divalglemeunn  11896  divalglemeuneg  11898
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