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Theorem rdivmuldivd 24180
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 rdivmuldivd.a . . . 4  |-  ( ph  ->  X  e.  B )
2 rdivmuldivd.b . . . 4  |-  ( ph  ->  Y  e.  U )
3 dvrdir.b . . . . . 6  |-  B  =  ( Base `  R
)
4 rdivmuldivd.p . . . . . 6  |-  .x.  =  ( .r `  R )
5 dvrdir.u . . . . . 6  |-  U  =  (Unit `  R )
6 eqid 2404 . . . . . 6  |-  ( invr `  R )  =  (
invr `  R )
7 dvrdir.t . . . . . 6  |-  ./  =  (/r
`  R )
83, 4, 5, 6, 7dvrval 15745 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  U )  ->  ( X  ./  Y
)  =  ( X 
.x.  ( ( invr `  R ) `  Y
) ) )
98oveq1d 6055 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  U )  ->  ( ( X  ./  Y )  .x.  ( Z  ./  W ) )  =  ( ( X 
.x.  ( ( invr `  R ) `  Y
) )  .x.  ( Z  ./  W ) ) )
101, 2, 9syl2anc 643 . . 3  |-  ( ph  ->  ( ( X  ./  Y )  .x.  ( Z  ./  W ) )  =  ( ( X 
.x.  ( ( invr `  R ) `  Y
) )  .x.  ( Z  ./  W ) ) )
11 rdivmuldivd.r . . . . 5  |-  ( ph  ->  R  e.  CRing )
12 crngrng 15629 . . . . 5  |-  ( R  e.  CRing  ->  R  e.  Ring )
1311, 12syl 16 . . . 4  |-  ( ph  ->  R  e.  Ring )
143, 5unitss 15720 . . . . 5  |-  U  C_  B
155, 6unitinvcl 15734 . . . . . 6  |-  ( ( R  e.  Ring  /\  Y  e.  U )  ->  (
( invr `  R ) `  Y )  e.  U
)
1613, 2, 15syl2anc 643 . . . . 5  |-  ( ph  ->  ( ( invr `  R
) `  Y )  e.  U )
1714, 16sseldi 3306 . . . 4  |-  ( ph  ->  ( ( invr `  R
) `  Y )  e.  B )
18 rdivmuldivd.c . . . . 5  |-  ( ph  ->  Z  e.  B )
19 rdivmuldivd.d . . . . 5  |-  ( ph  ->  W  e.  U )
203, 5, 7dvrcl 15746 . . . . 5  |-  ( ( R  e.  Ring  /\  Z  e.  B  /\  W  e.  U )  ->  ( Z  ./  W )  e.  B )
2113, 18, 19, 20syl3anc 1184 . . . 4  |-  ( ph  ->  ( Z  ./  W
)  e.  B )
223, 4rngass 15635 . . . 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
) ) ) )
2313, 1, 17, 21, 22syl13anc 1186 . . 3  |-  ( ph  ->  ( ( X  .x.  ( ( invr `  R
) `  Y )
)  .x.  ( Z  ./  W ) )  =  ( X  .x.  (
( ( invr `  R
) `  Y )  .x.  ( Z  ./  W
) ) ) )
243, 4crngcom 15633 . . . . 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 )
) )
2511, 17, 21, 24syl3anc 1184 . . . 4  |-  ( ph  ->  ( ( ( invr `  R ) `  Y
)  .x.  ( Z  ./  W ) )  =  ( ( Z  ./  W )  .x.  (
( invr `  R ) `  Y ) ) )
2625oveq2d 6056 . . 3  |-  ( ph  ->  ( X  .x.  (
( ( invr `  R
) `  Y )  .x.  ( Z  ./  W
) ) )  =  ( X  .x.  (
( Z  ./  W
)  .x.  ( ( invr `  R ) `  Y ) ) ) )
2710, 23, 263eqtrd 2440 . 2  |-  ( ph  ->  ( ( X  ./  Y )  .x.  ( Z  ./  W ) )  =  ( X  .x.  ( ( Z  ./  W )  .x.  (
( invr `  R ) `  Y ) ) ) )
28 eqid 2404 . . . . . . . 8  |-  ( (mulGrp `  R )s  U )  =  ( (mulGrp `  R )s  U
)
295, 28unitgrp 15727 . . . . . . 7  |-  ( R  e.  Ring  ->  ( (mulGrp `  R )s  U )  e.  Grp )
3013, 29syl 16 . . . . . 6  |-  ( ph  ->  ( (mulGrp `  R
)s 
U )  e.  Grp )
315, 28unitgrpbas 15726 . . . . . . 7  |-  U  =  ( Base `  (
(mulGrp `  R )s  U
) )
32 eqid 2404 . . . . . . 7  |-  ( +g  `  ( (mulGrp `  R
)s 
U ) )  =  ( +g  `  (
(mulGrp `  R )s  U
) )
335, 28, 6invrfval 15733 . . . . . . 7  |-  ( invr `  R )  =  ( inv g `  (
(mulGrp `  R )s  U
) )
3431, 32, 33grpinvadd 14822 . . . . . 6  |-  ( ( ( (mulGrp `  R
)s 
U )  e.  Grp  /\  Y  e.  U  /\  W  e.  U )  ->  ( ( invr `  R
) `  ( Y
( +g  `  ( (mulGrp `  R )s  U ) ) W ) )  =  ( ( ( invr `  R
) `  W )
( +g  `  ( (mulGrp `  R )s  U ) ) ( ( invr `  R
) `  Y )
) )
3530, 2, 19, 34syl3anc 1184 . . . . 5  |-  ( ph  ->  ( ( invr `  R
) `  ( Y
( +g  `  ( (mulGrp `  R )s  U ) ) W ) )  =  ( ( ( invr `  R
) `  W )
( +g  `  ( (mulGrp `  R )s  U ) ) ( ( invr `  R
) `  Y )
) )
36 fvex 5701 . . . . . . . . . . 11  |-  (Unit `  R )  e.  _V
375, 36eqeltri 2474 . . . . . . . . . 10  |-  U  e. 
_V
38 eqid 2404 . . . . . . . . . . 11  |-  ( Rs  U )  =  ( Rs  U )
39 eqid 2404 . . . . . . . . . . 11  |-  (mulGrp `  R )  =  (mulGrp `  R )
4038, 39mgpress 15614 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  U  e.  _V )  ->  (
(mulGrp `  R )s  U
)  =  (mulGrp `  ( Rs  U ) ) )
4113, 37, 40sylancl 644 . . . . . . . . 9  |-  ( ph  ->  ( (mulGrp `  R
)s 
U )  =  (mulGrp `  ( Rs  U ) ) )
4241fveq2d 5691 . . . . . . . 8  |-  ( ph  ->  ( +g  `  (
(mulGrp `  R )s  U
) )  =  ( +g  `  (mulGrp `  ( Rs  U ) ) ) )
43 eqid 2404 . . . . . . . . 9  |-  (mulGrp `  ( Rs  U ) )  =  (mulGrp `  ( Rs  U
) )
4438, 4ressmulr 13537 . . . . . . . . . 10  |-  ( U  e.  _V  ->  .x.  =  ( .r `  ( Rs  U ) ) )
4537, 44ax-mp 8 . . . . . . . . 9  |-  .x.  =  ( .r `  ( Rs  U ) )
4643, 45mgpplusg 15607 . . . . . . . 8  |-  .x.  =  ( +g  `  (mulGrp `  ( Rs  U ) ) )
4742, 46syl6reqr 2455 . . . . . . 7  |-  ( ph  ->  .x.  =  ( +g  `  ( (mulGrp `  R
)s 
U ) ) )
4847oveqd 6057 . . . . . 6  |-  ( ph  ->  ( Y  .x.  W
)  =  ( Y ( +g  `  (
(mulGrp `  R )s  U
) ) W ) )
4948fveq2d 5691 . . . . 5  |-  ( ph  ->  ( ( invr `  R
) `  ( Y  .x.  W ) )  =  ( ( invr `  R
) `  ( Y
( +g  `  ( (mulGrp `  R )s  U ) ) W ) ) )
5047oveqd 6057 . . . . 5  |-  ( ph  ->  ( ( ( invr `  R ) `  W
)  .x.  ( ( invr `  R ) `  Y ) )  =  ( ( ( invr `  R ) `  W
) ( +g  `  (
(mulGrp `  R )s  U
) ) ( (
invr `  R ) `  Y ) ) )
5135, 49, 503eqtr4d 2446 . . . 4  |-  ( ph  ->  ( ( invr `  R
) `  ( Y  .x.  W ) )  =  ( ( ( invr `  R ) `  W
)  .x.  ( ( invr `  R ) `  Y ) ) )
5251oveq2d 6056 . . 3  |-  ( ph  ->  ( ( X  .x.  Z )  .x.  (
( invr `  R ) `  ( Y  .x.  W
) ) )  =  ( ( X  .x.  Z )  .x.  (
( ( invr `  R
) `  W )  .x.  ( ( invr `  R
) `  Y )
) ) )
533, 4rngcl 15632 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .x.  Z )  e.  B )
5413, 1, 18, 53syl3anc 1184 . . . 4  |-  ( ph  ->  ( X  .x.  Z
)  e.  B )
555, 4unitmulcl 15724 . . . . 5  |-  ( ( R  e.  Ring  /\  Y  e.  U  /\  W  e.  U )  ->  ( Y  .x.  W )  e.  U )
5613, 2, 19, 55syl3anc 1184 . . . 4  |-  ( ph  ->  ( Y  .x.  W
)  e.  U )
573, 4, 5, 6, 7dvrval 15745 . . . 4  |-  ( ( ( X  .x.  Z
)  e.  B  /\  ( Y  .x.  W )  e.  U )  -> 
( ( X  .x.  Z )  ./  ( Y  .x.  W ) )  =  ( ( X 
.x.  Z )  .x.  ( ( invr `  R
) `  ( Y  .x.  W ) ) ) )
5854, 56, 57syl2anc 643 . . 3  |-  ( ph  ->  ( ( X  .x.  Z )  ./  ( Y  .x.  W ) )  =  ( ( X 
.x.  Z )  .x.  ( ( invr `  R
) `  ( Y  .x.  W ) ) ) )
595, 6unitinvcl 15734 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  W  e.  U )  ->  (
( invr `  R ) `  W )  e.  U
)
6013, 19, 59syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( ( invr `  R
) `  W )  e.  U )
6114, 60sseldi 3306 . . . . . . 7  |-  ( ph  ->  ( ( invr `  R
) `  W )  e.  B )
623, 4rngass 15635 . . . . . . 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
) ) ) )
6313, 1, 18, 61, 62syl13anc 1186 . . . . . 6  |-  ( ph  ->  ( ( X  .x.  Z )  .x.  (
( invr `  R ) `  W ) )  =  ( X  .x.  ( Z  .x.  ( ( invr `  R ) `  W
) ) ) )
643, 4, 5, 6, 7dvrval 15745 . . . . . . . 8  |-  ( ( Z  e.  B  /\  W  e.  U )  ->  ( Z  ./  W
)  =  ( Z 
.x.  ( ( invr `  R ) `  W
) ) )
6518, 19, 64syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( Z  ./  W
)  =  ( Z 
.x.  ( ( invr `  R ) `  W
) ) )
6665oveq2d 6056 . . . . . 6  |-  ( ph  ->  ( X  .x.  ( Z  ./  W ) )  =  ( X  .x.  ( Z  .x.  ( (
invr `  R ) `  W ) ) ) )
6763, 66eqtr4d 2439 . . . . 5  |-  ( ph  ->  ( ( X  .x.  Z )  .x.  (
( invr `  R ) `  W ) )  =  ( X  .x.  ( Z  ./  W ) ) )
6867oveq1d 6055 . . . 4  |-  ( ph  ->  ( ( ( X 
.x.  Z )  .x.  ( ( invr `  R
) `  W )
)  .x.  ( ( invr `  R ) `  Y ) )  =  ( ( X  .x.  ( Z  ./  W ) )  .x.  ( (
invr `  R ) `  Y ) ) )
693, 4rngass 15635 . . . . 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 ) ) ) )
7013, 54, 61, 17, 69syl13anc 1186 . . . 4  |-  ( ph  ->  ( ( ( X 
.x.  Z )  .x.  ( ( invr `  R
) `  W )
)  .x.  ( ( invr `  R ) `  Y ) )  =  ( ( X  .x.  Z )  .x.  (
( ( invr `  R
) `  W )  .x.  ( ( invr `  R
) `  Y )
) ) )
713, 4rngass 15635 . . . . 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 )
) ) )
7213, 1, 21, 17, 71syl13anc 1186 . . . 4  |-  ( ph  ->  ( ( X  .x.  ( Z  ./  W ) )  .x.  ( (
invr `  R ) `  Y ) )  =  ( X  .x.  (
( Z  ./  W
)  .x.  ( ( invr `  R ) `  Y ) ) ) )
7368, 70, 723eqtr3rd 2445 . . 3  |-  ( ph  ->  ( X  .x.  (
( Z  ./  W
)  .x.  ( ( invr `  R ) `  Y ) ) )  =  ( ( X 
.x.  Z )  .x.  ( ( ( invr `  R ) `  W
)  .x.  ( ( invr `  R ) `  Y ) ) ) )
7452, 58, 733eqtr4rd 2447 . 2  |-  ( ph  ->  ( X  .x.  (
( Z  ./  W
)  .x.  ( ( invr `  R ) `  Y ) ) )  =  ( ( X 
.x.  Z )  ./  ( Y  .x.  W ) ) )
7527, 74eqtrd 2436 1  |-  ( ph  ->  ( ( X  ./  Y )  .x.  ( Z  ./  W ) )  =  ( ( X 
.x.  Z )  ./  ( Y  .x.  W ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916   ` cfv 5413  (class class class)co 6040   Basecbs 13424   ↾s cress 13425   +g cplusg 13484   .rcmulr 13485   Grpcgrp 14640  mulGrpcmgp 15603   Ringcrg 15615   CRingccrg 15616  Unitcui 15699   invrcinvr 15731  /rcdvr 15742
This theorem is referenced by:  qqhrhm  24326
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-0g 13682  df-mnd 14645  df-grp 14767  df-minusg 14768  df-cmn 15369  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-dvr 15743
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