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Theorem rdivmuldivd 13387
Description: Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by Thierry Arnoux, 30-Oct-2017.)
Hypotheses
Ref Expression
dvrdir.b  |-  B  =  ( Base `  R
)
dvrdir.u  |-  U  =  (Unit `  R )
dvrdir.p  |-  .+  =  ( +g  `  R )
dvrdir.t  |-  ./  =  (/r
`  R )
rdivmuldivd.p  |-  .x.  =  ( .r `  R )
rdivmuldivd.r  |-  ( ph  ->  R  e.  CRing )
rdivmuldivd.a  |-  ( ph  ->  X  e.  B )
rdivmuldivd.b  |-  ( ph  ->  Y  e.  U )
rdivmuldivd.c  |-  ( ph  ->  Z  e.  B )
rdivmuldivd.d  |-  ( ph  ->  W  e.  U )
Assertion
Ref Expression
rdivmuldivd  |-  ( ph  ->  ( ( X  ./  Y )  .x.  ( Z  ./  W ) )  =  ( ( X 
.x.  Z )  ./  ( Y  .x.  W ) ) )

Proof of Theorem rdivmuldivd
StepHypRef Expression
1 dvrdir.b . . . . . 6  |-  B  =  ( Base `  R
)
21a1i 9 . . . . 5  |-  ( ph  ->  B  =  ( Base `  R ) )
3 rdivmuldivd.p . . . . . 6  |-  .x.  =  ( .r `  R )
43a1i 9 . . . . 5  |-  ( ph  ->  .x.  =  ( .r
`  R ) )
5 dvrdir.u . . . . . 6  |-  U  =  (Unit `  R )
65a1i 9 . . . . 5  |-  ( ph  ->  U  =  (Unit `  R ) )
7 eqidd 2188 . . . . 5  |-  ( ph  ->  ( invr `  R
)  =  ( invr `  R ) )
8 dvrdir.t . . . . . 6  |-  ./  =  (/r
`  R )
98a1i 9 . . . . 5  |-  ( ph  -> 
./  =  (/r `  R
) )
10 rdivmuldivd.r . . . . . 6  |-  ( ph  ->  R  e.  CRing )
1110crngringd 13256 . . . . 5  |-  ( ph  ->  R  e.  Ring )
12 rdivmuldivd.a . . . . 5  |-  ( ph  ->  X  e.  B )
13 rdivmuldivd.b . . . . 5  |-  ( ph  ->  Y  e.  U )
142, 4, 6, 7, 9, 11, 12, 13dvrvald 13377 . . . 4  |-  ( ph  ->  ( X  ./  Y
)  =  ( X 
.x.  ( ( invr `  R ) `  Y
) ) )
1514oveq1d 5903 . . 3  |-  ( ph  ->  ( ( X  ./  Y )  .x.  ( Z  ./  W ) )  =  ( ( X 
.x.  ( ( invr `  R ) `  Y
) )  .x.  ( Z  ./  W ) ) )
16 ringsrg 13292 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. SRing
)
1711, 16syl 14 . . . . . 6  |-  ( ph  ->  R  e. SRing )
182, 6, 17unitssd 13352 . . . . 5  |-  ( ph  ->  U  C_  B )
19 eqid 2187 . . . . . . 7  |-  ( invr `  R )  =  (
invr `  R )
205, 19unitinvcl 13366 . . . . . 6  |-  ( ( R  e.  Ring  /\  Y  e.  U )  ->  (
( invr `  R ) `  Y )  e.  U
)
2111, 13, 20syl2anc 411 . . . . 5  |-  ( ph  ->  ( ( invr `  R
) `  Y )  e.  U )
2218, 21sseldd 3168 . . . 4  |-  ( ph  ->  ( ( invr `  R
) `  Y )  e.  B )
23 rdivmuldivd.c . . . . 5  |-  ( ph  ->  Z  e.  B )
24 rdivmuldivd.d . . . . 5  |-  ( ph  ->  W  e.  U )
251, 5, 8dvrcl 13378 . . . . 5  |-  ( ( R  e.  Ring  /\  Z  e.  B  /\  W  e.  U )  ->  ( Z  ./  W )  e.  B )
2611, 23, 24, 25syl3anc 1248 . . . 4  |-  ( ph  ->  ( Z  ./  W
)  e.  B )
271, 3ringass 13263 . . . 4  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  ( ( invr `  R
) `  Y )  e.  B  /\  ( Z  ./  W )  e.  B ) )  -> 
( ( X  .x.  ( ( invr `  R
) `  Y )
)  .x.  ( Z  ./  W ) )  =  ( X  .x.  (
( ( invr `  R
) `  Y )  .x.  ( Z  ./  W
) ) ) )
2811, 12, 22, 26, 27syl13anc 1250 . . 3  |-  ( ph  ->  ( ( X  .x.  ( ( invr `  R
) `  Y )
)  .x.  ( Z  ./  W ) )  =  ( X  .x.  (
( ( invr `  R
) `  Y )  .x.  ( Z  ./  W
) ) ) )
291, 3crngcom 13261 . . . . 5  |-  ( ( R  e.  CRing  /\  (
( invr `  R ) `  Y )  e.  B  /\  ( Z  ./  W
)  e.  B )  ->  ( ( (
invr `  R ) `  Y )  .x.  ( Z  ./  W ) )  =  ( ( Z 
./  W )  .x.  ( ( invr `  R
) `  Y )
) )
3010, 22, 26, 29syl3anc 1248 . . . 4  |-  ( ph  ->  ( ( ( invr `  R ) `  Y
)  .x.  ( Z  ./  W ) )  =  ( ( Z  ./  W )  .x.  (
( invr `  R ) `  Y ) ) )
3130oveq2d 5904 . . 3  |-  ( ph  ->  ( X  .x.  (
( ( invr `  R
) `  Y )  .x.  ( Z  ./  W
) ) )  =  ( X  .x.  (
( Z  ./  W
)  .x.  ( ( invr `  R ) `  Y ) ) ) )
3215, 28, 313eqtrd 2224 . 2  |-  ( ph  ->  ( ( X  ./  Y )  .x.  ( Z  ./  W ) )  =  ( X  .x.  ( ( Z  ./  W )  .x.  (
( invr `  R ) `  Y ) ) ) )
33 eqid 2187 . . . . . . . . 9  |-  ( (mulGrp `  R )s  U )  =  ( (mulGrp `  R )s  U
)
345, 33unitgrp 13359 . . . . . . . 8  |-  ( R  e.  Ring  ->  ( (mulGrp `  R )s  U )  e.  Grp )
3511, 34syl 14 . . . . . . 7  |-  ( ph  ->  ( (mulGrp `  R
)s 
U )  e.  Grp )
36 eqidd 2188 . . . . . . . . 9  |-  ( ph  ->  ( (mulGrp `  R
)s 
U )  =  ( (mulGrp `  R )s  U
) )
376, 36, 17unitgrpbasd 13358 . . . . . . . 8  |-  ( ph  ->  U  =  ( Base `  ( (mulGrp `  R
)s 
U ) ) )
3813, 37eleqtrd 2266 . . . . . . 7  |-  ( ph  ->  Y  e.  ( Base `  ( (mulGrp `  R
)s 
U ) ) )
3924, 37eleqtrd 2266 . . . . . . 7  |-  ( ph  ->  W  e.  ( Base `  ( (mulGrp `  R
)s 
U ) ) )
40 eqid 2187 . . . . . . . 8  |-  ( Base `  ( (mulGrp `  R
)s 
U ) )  =  ( Base `  (
(mulGrp `  R )s  U
) )
41 eqid 2187 . . . . . . . 8  |-  ( +g  `  ( (mulGrp `  R
)s 
U ) )  =  ( +g  `  (
(mulGrp `  R )s  U
) )
42 eqid 2187 . . . . . . . 8  |-  ( invg `  ( (mulGrp `  R )s  U ) )  =  ( invg `  ( (mulGrp `  R )s  U
) )
4340, 41, 42grpinvadd 12974 . . . . . . 7  |-  ( ( ( (mulGrp `  R
)s 
U )  e.  Grp  /\  Y  e.  ( Base `  ( (mulGrp `  R
)s 
U ) )  /\  W  e.  ( Base `  ( (mulGrp `  R
)s 
U ) ) )  ->  ( ( invg `  ( (mulGrp `  R )s  U ) ) `  ( Y ( +g  `  (
(mulGrp `  R )s  U
) ) W ) )  =  ( ( ( invg `  ( (mulGrp `  R )s  U
) ) `  W
) ( +g  `  (
(mulGrp `  R )s  U
) ) ( ( invg `  (
(mulGrp `  R )s  U
) ) `  Y
) ) )
4435, 38, 39, 43syl3anc 1248 . . . . . 6  |-  ( ph  ->  ( ( invg `  ( (mulGrp `  R
)s 
U ) ) `  ( Y ( +g  `  (
(mulGrp `  R )s  U
) ) W ) )  =  ( ( ( invg `  ( (mulGrp `  R )s  U
) ) `  W
) ( +g  `  (
(mulGrp `  R )s  U
) ) ( ( invg `  (
(mulGrp `  R )s  U
) ) `  Y
) ) )
456, 36, 7, 11invrfvald 13365 . . . . . . 7  |-  ( ph  ->  ( invr `  R
)  =  ( invg `  ( (mulGrp `  R )s  U ) ) )
4645fveq1d 5529 . . . . . 6  |-  ( ph  ->  ( ( invr `  R
) `  ( Y
( +g  `  ( (mulGrp `  R )s  U ) ) W ) )  =  ( ( invg `  ( (mulGrp `  R )s  U
) ) `  ( Y ( +g  `  (
(mulGrp `  R )s  U
) ) W ) ) )
4745fveq1d 5529 . . . . . . 7  |-  ( ph  ->  ( ( invr `  R
) `  W )  =  ( ( invg `  ( (mulGrp `  R )s  U ) ) `  W ) )
4845fveq1d 5529 . . . . . . 7  |-  ( ph  ->  ( ( invr `  R
) `  Y )  =  ( ( invg `  ( (mulGrp `  R )s  U ) ) `  Y ) )
4947, 48oveq12d 5906 . . . . . 6  |-  ( ph  ->  ( ( ( invr `  R ) `  W
) ( +g  `  (
(mulGrp `  R )s  U
) ) ( (
invr `  R ) `  Y ) )  =  ( ( ( invg `  ( (mulGrp `  R )s  U ) ) `  W ) ( +g  `  ( (mulGrp `  R
)s 
U ) ) ( ( invg `  ( (mulGrp `  R )s  U
) ) `  Y
) ) )
5044, 46, 493eqtr4d 2230 . . . . 5  |-  ( ph  ->  ( ( invr `  R
) `  ( Y
( +g  `  ( (mulGrp `  R )s  U ) ) W ) )  =  ( ( ( invr `  R
) `  W )
( +g  `  ( (mulGrp `  R )s  U ) ) ( ( invr `  R
) `  Y )
) )
51 basfn 12533 . . . . . . . . . . . 12  |-  Base  Fn  _V
5210elexd 2762 . . . . . . . . . . . 12  |-  ( ph  ->  R  e.  _V )
53 funfvex 5544 . . . . . . . . . . . . 13  |-  ( ( Fun  Base  /\  R  e. 
dom  Base )  ->  ( Base `  R )  e. 
_V )
5453funfni 5328 . . . . . . . . . . . 12  |-  ( (
Base  Fn  _V  /\  R  e.  _V )  ->  ( Base `  R )  e. 
_V )
5551, 52, 54sylancr 414 . . . . . . . . . . 11  |-  ( ph  ->  ( Base `  R
)  e.  _V )
561, 55eqeltrid 2274 . . . . . . . . . 10  |-  ( ph  ->  B  e.  _V )
5756, 18ssexd 4155 . . . . . . . . 9  |-  ( ph  ->  U  e.  _V )
58 ressex 12538 . . . . . . . . . 10  |-  ( ( R  e.  CRing  /\  U  e.  _V )  ->  ( Rs  U )  e.  _V )
59 eqid 2187 . . . . . . . . . . 11  |-  (mulGrp `  ( Rs  U ) )  =  (mulGrp `  ( Rs  U
) )
60 eqid 2187 . . . . . . . . . . 11  |-  ( .r
`  ( Rs  U ) )  =  ( .r
`  ( Rs  U ) )
6159, 60mgpplusgg 13174 . . . . . . . . . 10  |-  ( ( Rs  U )  e.  _V  ->  ( .r `  ( Rs  U ) )  =  ( +g  `  (mulGrp `  ( Rs  U ) ) ) )
6258, 61syl 14 . . . . . . . . 9  |-  ( ( R  e.  CRing  /\  U  e.  _V )  ->  ( .r `  ( Rs  U ) )  =  ( +g  `  (mulGrp `  ( Rs  U
) ) ) )
6310, 57, 62syl2anc 411 . . . . . . . 8  |-  ( ph  ->  ( .r `  ( Rs  U ) )  =  ( +g  `  (mulGrp `  ( Rs  U ) ) ) )
64 eqid 2187 . . . . . . . . . 10  |-  ( Rs  U )  =  ( Rs  U )
6564, 3ressmulrg 12617 . . . . . . . . 9  |-  ( ( U  e.  _V  /\  R  e.  CRing )  ->  .x.  =  ( .r `  ( Rs  U ) ) )
6657, 10, 65syl2anc 411 . . . . . . . 8  |-  ( ph  ->  .x.  =  ( .r
`  ( Rs  U ) ) )
67 eqid 2187 . . . . . . . . . . 11  |-  (mulGrp `  R )  =  (mulGrp `  R )
6864, 67mgpress 13181 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  U  e.  _V )  ->  (
(mulGrp `  R )s  U
)  =  (mulGrp `  ( Rs  U ) ) )
6911, 57, 68syl2anc 411 . . . . . . . . 9  |-  ( ph  ->  ( (mulGrp `  R
)s 
U )  =  (mulGrp `  ( Rs  U ) ) )
7069fveq2d 5531 . . . . . . . 8  |-  ( ph  ->  ( +g  `  (
(mulGrp `  R )s  U
) )  =  ( +g  `  (mulGrp `  ( Rs  U ) ) ) )
7163, 66, 703eqtr4d 2230 . . . . . . 7  |-  ( ph  ->  .x.  =  ( +g  `  ( (mulGrp `  R
)s 
U ) ) )
7271oveqd 5905 . . . . . 6  |-  ( ph  ->  ( Y  .x.  W
)  =  ( Y ( +g  `  (
(mulGrp `  R )s  U
) ) W ) )
7372fveq2d 5531 . . . . 5  |-  ( ph  ->  ( ( invr `  R
) `  ( Y  .x.  W ) )  =  ( ( invr `  R
) `  ( Y
( +g  `  ( (mulGrp `  R )s  U ) ) W ) ) )
7471oveqd 5905 . . . . 5  |-  ( ph  ->  ( ( ( invr `  R ) `  W
)  .x.  ( ( invr `  R ) `  Y ) )  =  ( ( ( invr `  R ) `  W
) ( +g  `  (
(mulGrp `  R )s  U
) ) ( (
invr `  R ) `  Y ) ) )
7550, 73, 743eqtr4d 2230 . . . 4  |-  ( ph  ->  ( ( invr `  R
) `  ( Y  .x.  W ) )  =  ( ( ( invr `  R ) `  W
)  .x.  ( ( invr `  R ) `  Y ) ) )
7675oveq2d 5904 . . 3  |-  ( ph  ->  ( ( X  .x.  Z )  .x.  (
( invr `  R ) `  ( Y  .x.  W
) ) )  =  ( ( X  .x.  Z )  .x.  (
( ( invr `  R
) `  W )  .x.  ( ( invr `  R
) `  Y )
) ) )
771, 3ringcl 13260 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .x.  Z )  e.  B )
7811, 12, 23, 77syl3anc 1248 . . . 4  |-  ( ph  ->  ( X  .x.  Z
)  e.  B )
795, 3unitmulcl 13356 . . . . 5  |-  ( ( R  e.  Ring  /\  Y  e.  U  /\  W  e.  U )  ->  ( Y  .x.  W )  e.  U )
8011, 13, 24, 79syl3anc 1248 . . . 4  |-  ( ph  ->  ( Y  .x.  W
)  e.  U )
812, 4, 6, 7, 9, 11, 78, 80dvrvald 13377 . . 3  |-  ( ph  ->  ( ( X  .x.  Z )  ./  ( Y  .x.  W ) )  =  ( ( X 
.x.  Z )  .x.  ( ( invr `  R
) `  ( Y  .x.  W ) ) ) )
825, 19unitinvcl 13366 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  W  e.  U )  ->  (
( invr `  R ) `  W )  e.  U
)
8311, 24, 82syl2anc 411 . . . . . . . 8  |-  ( ph  ->  ( ( invr `  R
) `  W )  e.  U )
8418, 83sseldd 3168 . . . . . . 7  |-  ( ph  ->  ( ( invr `  R
) `  W )  e.  B )
851, 3ringass 13263 . . . . . . 7  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Z  e.  B  /\  ( ( invr `  R
) `  W )  e.  B ) )  -> 
( ( X  .x.  Z )  .x.  (
( invr `  R ) `  W ) )  =  ( X  .x.  ( Z  .x.  ( ( invr `  R ) `  W
) ) ) )
8611, 12, 23, 84, 85syl13anc 1250 . . . . . 6  |-  ( ph  ->  ( ( X  .x.  Z )  .x.  (
( invr `  R ) `  W ) )  =  ( X  .x.  ( Z  .x.  ( ( invr `  R ) `  W
) ) ) )
872, 4, 6, 7, 9, 11, 23, 24dvrvald 13377 . . . . . . 7  |-  ( ph  ->  ( Z  ./  W
)  =  ( Z 
.x.  ( ( invr `  R ) `  W
) ) )
8887oveq2d 5904 . . . . . 6  |-  ( ph  ->  ( X  .x.  ( Z  ./  W ) )  =  ( X  .x.  ( Z  .x.  ( (
invr `  R ) `  W ) ) ) )
8986, 88eqtr4d 2223 . . . . 5  |-  ( ph  ->  ( ( X  .x.  Z )  .x.  (
( invr `  R ) `  W ) )  =  ( X  .x.  ( Z  ./  W ) ) )
9089oveq1d 5903 . . . 4  |-  ( ph  ->  ( ( ( X 
.x.  Z )  .x.  ( ( invr `  R
) `  W )
)  .x.  ( ( invr `  R ) `  Y ) )  =  ( ( X  .x.  ( Z  ./  W ) )  .x.  ( (
invr `  R ) `  Y ) ) )
911, 3ringass 13263 . . . . 5  |-  ( ( R  e.  Ring  /\  (
( X  .x.  Z
)  e.  B  /\  ( ( invr `  R
) `  W )  e.  B  /\  (
( invr `  R ) `  Y )  e.  B
) )  ->  (
( ( X  .x.  Z )  .x.  (
( invr `  R ) `  W ) )  .x.  ( ( invr `  R
) `  Y )
)  =  ( ( X  .x.  Z ) 
.x.  ( ( (
invr `  R ) `  W )  .x.  (
( invr `  R ) `  Y ) ) ) )
9211, 78, 84, 22, 91syl13anc 1250 . . . 4  |-  ( ph  ->  ( ( ( X 
.x.  Z )  .x.  ( ( invr `  R
) `  W )
)  .x.  ( ( invr `  R ) `  Y ) )  =  ( ( X  .x.  Z )  .x.  (
( ( invr `  R
) `  W )  .x.  ( ( invr `  R
) `  Y )
) ) )
931, 3ringass 13263 . . . . 5  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  ( Z  ./  W )  e.  B  /\  (
( invr `  R ) `  Y )  e.  B
) )  ->  (
( X  .x.  ( Z  ./  W ) ) 
.x.  ( ( invr `  R ) `  Y
) )  =  ( X  .x.  ( ( Z  ./  W )  .x.  ( ( invr `  R
) `  Y )
) ) )
9411, 12, 26, 22, 93syl13anc 1250 . . . 4  |-  ( ph  ->  ( ( X  .x.  ( Z  ./  W ) )  .x.  ( (
invr `  R ) `  Y ) )  =  ( X  .x.  (
( Z  ./  W
)  .x.  ( ( invr `  R ) `  Y ) ) ) )
9590, 92, 943eqtr3rd 2229 . . 3  |-  ( ph  ->  ( X  .x.  (
( Z  ./  W
)  .x.  ( ( invr `  R ) `  Y ) ) )  =  ( ( X 
.x.  Z )  .x.  ( ( ( invr `  R ) `  W
)  .x.  ( ( invr `  R ) `  Y ) ) ) )
9676, 81, 953eqtr4rd 2231 . 2  |-  ( ph  ->  ( X  .x.  (
( Z  ./  W
)  .x.  ( ( invr `  R ) `  Y ) ) )  =  ( ( X 
.x.  Z )  ./  ( Y  .x.  W ) ) )
9732, 96eqtrd 2220 1  |-  ( ph  ->  ( ( X  ./  Y )  .x.  ( Z  ./  W ) )  =  ( ( X 
.x.  Z )  ./  ( Y  .x.  W ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1363    e. wcel 2158   _Vcvv 2749    Fn wfn 5223   ` cfv 5228  (class class class)co 5888   Basecbs 12475   ↾s cress 12476   +g cplusg 12550   .rcmulr 12551   Grpcgrp 12898   invgcminusg 12899  mulGrpcmgp 13170  SRingcsrg 13210   Ringcrg 13243   CRingccrg 13244  Unitcui 13330   invrcinvr 13363  /rcdvr 13374
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-addcom 7924  ax-addass 7926  ax-i2m1 7929  ax-0lt1 7930  ax-0id 7932  ax-rnegex 7933  ax-pre-ltirr 7936  ax-pre-lttrn 7938  ax-pre-ltadd 7940
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-tpos 6259  df-pnf 8007  df-mnf 8008  df-ltxr 8010  df-inn 8933  df-2 8991  df-3 8992  df-ndx 12478  df-slot 12479  df-base 12481  df-sets 12482  df-iress 12483  df-plusg 12563  df-mulr 12564  df-0g 12724  df-mgm 12793  df-sgrp 12826  df-mnd 12839  df-grp 12901  df-minusg 12902  df-cmn 13122  df-abl 13123  df-mgp 13171  df-ur 13207  df-srg 13211  df-ring 13245  df-cring 13246  df-oppr 13311  df-dvdsr 13332  df-unit 13333  df-invr 13364  df-dvr 13375
This theorem is referenced by: (None)
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