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Theorem rdivmuldivd 26210
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 2438 . . . . . 6  |-  ( invr `  R )  =  (
invr `  R )
7 dvrdir.t . . . . . 6  |-  ./  =  (/r
`  R )
83, 4, 5, 6, 7dvrval 16765 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  U )  ->  ( X  ./  Y
)  =  ( X 
.x.  ( ( invr `  R ) `  Y
) ) )
98oveq1d 6101 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  U )  ->  ( ( X  ./  Y )  .x.  ( Z  ./  W ) )  =  ( ( X 
.x.  ( ( invr `  R ) `  Y
) )  .x.  ( Z  ./  W ) ) )
101, 2, 9syl2anc 661 . . 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 16643 . . . . 5  |-  ( R  e.  CRing  ->  R  e.  Ring )
1311, 12syl 16 . . . 4  |-  ( ph  ->  R  e.  Ring )
143, 5unitss 16740 . . . . 5  |-  U  C_  B
155, 6unitinvcl 16754 . . . . . 6  |-  ( ( R  e.  Ring  /\  Y  e.  U )  ->  (
( invr `  R ) `  Y )  e.  U
)
1613, 2, 15syl2anc 661 . . . . 5  |-  ( ph  ->  ( ( invr `  R
) `  Y )  e.  U )
1714, 16sseldi 3349 . . . 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 16766 . . . . 5  |-  ( ( R  e.  Ring  /\  Z  e.  B  /\  W  e.  U )  ->  ( Z  ./  W )  e.  B )
2113, 18, 19, 20syl3anc 1218 . . . 4  |-  ( ph  ->  ( Z  ./  W
)  e.  B )
223, 4rngass 16649 . . . 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 1220 . . 3  |-  ( ph  ->  ( ( X  .x.  ( ( invr `  R
) `  Y )
)  .x.  ( Z  ./  W ) )  =  ( X  .x.  (
( ( invr `  R
) `  Y )  .x.  ( Z  ./  W
) ) ) )
243, 4crngcom 16647 . . . . 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 1218 . . . 4  |-  ( ph  ->  ( ( ( invr `  R ) `  Y
)  .x.  ( Z  ./  W ) )  =  ( ( Z  ./  W )  .x.  (
( invr `  R ) `  Y ) ) )
2625oveq2d 6102 . . 3  |-  ( ph  ->  ( X  .x.  (
( ( invr `  R
) `  Y )  .x.  ( Z  ./  W
) ) )  =  ( X  .x.  (
( Z  ./  W
)  .x.  ( ( invr `  R ) `  Y ) ) ) )
2710, 23, 263eqtrd 2474 . 2  |-  ( ph  ->  ( ( X  ./  Y )  .x.  ( Z  ./  W ) )  =  ( X  .x.  ( ( Z  ./  W )  .x.  (
( invr `  R ) `  Y ) ) ) )
28 eqid 2438 . . . . . . . 8  |-  ( (mulGrp `  R )s  U )  =  ( (mulGrp `  R )s  U
)
295, 28unitgrp 16747 . . . . . . 7  |-  ( R  e.  Ring  ->  ( (mulGrp `  R )s  U )  e.  Grp )
3013, 29syl 16 . . . . . 6  |-  ( ph  ->  ( (mulGrp `  R
)s 
U )  e.  Grp )
315, 28unitgrpbas 16746 . . . . . . 7  |-  U  =  ( Base `  (
(mulGrp `  R )s  U
) )
32 eqid 2438 . . . . . . 7  |-  ( +g  `  ( (mulGrp `  R
)s 
U ) )  =  ( +g  `  (
(mulGrp `  R )s  U
) )
335, 28, 6invrfval 16753 . . . . . . 7  |-  ( invr `  R )  =  ( invg `  (
(mulGrp `  R )s  U
) )
3431, 32, 33grpinvadd 15595 . . . . . 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 1218 . . . . 5  |-  ( ph  ->  ( ( invr `  R
) `  ( Y
( +g  `  ( (mulGrp `  R )s  U ) ) W ) )  =  ( ( ( invr `  R
) `  W )
( +g  `  ( (mulGrp `  R )s  U ) ) ( ( invr `  R
) `  Y )
) )
36 fvex 5696 . . . . . . . . . . 11  |-  (Unit `  R )  e.  _V
375, 36eqeltri 2508 . . . . . . . . . 10  |-  U  e. 
_V
38 eqid 2438 . . . . . . . . . . 11  |-  ( Rs  U )  =  ( Rs  U )
39 eqid 2438 . . . . . . . . . . 11  |-  (mulGrp `  R )  =  (mulGrp `  R )
4038, 39mgpress 16590 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  U  e.  _V )  ->  (
(mulGrp `  R )s  U
)  =  (mulGrp `  ( Rs  U ) ) )
4113, 37, 40sylancl 662 . . . . . . . . 9  |-  ( ph  ->  ( (mulGrp `  R
)s 
U )  =  (mulGrp `  ( Rs  U ) ) )
4241fveq2d 5690 . . . . . . . 8  |-  ( ph  ->  ( +g  `  (
(mulGrp `  R )s  U
) )  =  ( +g  `  (mulGrp `  ( Rs  U ) ) ) )
43 eqid 2438 . . . . . . . . 9  |-  (mulGrp `  ( Rs  U ) )  =  (mulGrp `  ( Rs  U
) )
4438, 4ressmulr 14283 . . . . . . . . . 10  |-  ( U  e.  _V  ->  .x.  =  ( .r `  ( Rs  U ) ) )
4537, 44ax-mp 5 . . . . . . . . 9  |-  .x.  =  ( .r `  ( Rs  U ) )
4643, 45mgpplusg 16583 . . . . . . . 8  |-  .x.  =  ( +g  `  (mulGrp `  ( Rs  U ) ) )
4742, 46syl6reqr 2489 . . . . . . 7  |-  ( ph  ->  .x.  =  ( +g  `  ( (mulGrp `  R
)s 
U ) ) )
4847oveqd 6103 . . . . . 6  |-  ( ph  ->  ( Y  .x.  W
)  =  ( Y ( +g  `  (
(mulGrp `  R )s  U
) ) W ) )
4948fveq2d 5690 . . . . 5  |-  ( ph  ->  ( ( invr `  R
) `  ( Y  .x.  W ) )  =  ( ( invr `  R
) `  ( Y
( +g  `  ( (mulGrp `  R )s  U ) ) W ) ) )
5047oveqd 6103 . . . . 5  |-  ( ph  ->  ( ( ( invr `  R ) `  W
)  .x.  ( ( invr `  R ) `  Y ) )  =  ( ( ( invr `  R ) `  W
) ( +g  `  (
(mulGrp `  R )s  U
) ) ( (
invr `  R ) `  Y ) ) )
5135, 49, 503eqtr4d 2480 . . . 4  |-  ( ph  ->  ( ( invr `  R
) `  ( Y  .x.  W ) )  =  ( ( ( invr `  R ) `  W
)  .x.  ( ( invr `  R ) `  Y ) ) )
5251oveq2d 6102 . . 3  |-  ( ph  ->  ( ( X  .x.  Z )  .x.  (
( invr `  R ) `  ( Y  .x.  W
) ) )  =  ( ( X  .x.  Z )  .x.  (
( ( invr `  R
) `  W )  .x.  ( ( invr `  R
) `  Y )
) ) )
533, 4rngcl 16646 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .x.  Z )  e.  B )
5413, 1, 18, 53syl3anc 1218 . . . 4  |-  ( ph  ->  ( X  .x.  Z
)  e.  B )
555, 4unitmulcl 16744 . . . . 5  |-  ( ( R  e.  Ring  /\  Y  e.  U  /\  W  e.  U )  ->  ( Y  .x.  W )  e.  U )
5613, 2, 19, 55syl3anc 1218 . . . 4  |-  ( ph  ->  ( Y  .x.  W
)  e.  U )
573, 4, 5, 6, 7dvrval 16765 . . . 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 661 . . 3  |-  ( ph  ->  ( ( X  .x.  Z )  ./  ( Y  .x.  W ) )  =  ( ( X 
.x.  Z )  .x.  ( ( invr `  R
) `  ( Y  .x.  W ) ) ) )
595, 6unitinvcl 16754 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  W  e.  U )  ->  (
( invr `  R ) `  W )  e.  U
)
6013, 19, 59syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( ( invr `  R
) `  W )  e.  U )
6114, 60sseldi 3349 . . . . . . 7  |-  ( ph  ->  ( ( invr `  R
) `  W )  e.  B )
623, 4rngass 16649 . . . . . . 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 1220 . . . . . 6  |-  ( ph  ->  ( ( X  .x.  Z )  .x.  (
( invr `  R ) `  W ) )  =  ( X  .x.  ( Z  .x.  ( ( invr `  R ) `  W
) ) ) )
643, 4, 5, 6, 7dvrval 16765 . . . . . . . 8  |-  ( ( Z  e.  B  /\  W  e.  U )  ->  ( Z  ./  W
)  =  ( Z 
.x.  ( ( invr `  R ) `  W
) ) )
6518, 19, 64syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( Z  ./  W
)  =  ( Z 
.x.  ( ( invr `  R ) `  W
) ) )
6665oveq2d 6102 . . . . . 6  |-  ( ph  ->  ( X  .x.  ( Z  ./  W ) )  =  ( X  .x.  ( Z  .x.  ( (
invr `  R ) `  W ) ) ) )
6763, 66eqtr4d 2473 . . . . 5  |-  ( ph  ->  ( ( X  .x.  Z )  .x.  (
( invr `  R ) `  W ) )  =  ( X  .x.  ( Z  ./  W ) ) )
6867oveq1d 6101 . . . 4  |-  ( ph  ->  ( ( ( X 
.x.  Z )  .x.  ( ( invr `  R
) `  W )
)  .x.  ( ( invr `  R ) `  Y ) )  =  ( ( X  .x.  ( Z  ./  W ) )  .x.  ( (
invr `  R ) `  Y ) ) )
693, 4rngass 16649 . . . . 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 1220 . . . 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 16649 . . . . 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 1220 . . . 4  |-  ( ph  ->  ( ( X  .x.  ( Z  ./  W ) )  .x.  ( (
invr `  R ) `  Y ) )  =  ( X  .x.  (
( Z  ./  W
)  .x.  ( ( invr `  R ) `  Y ) ) ) )
7368, 70, 723eqtr3rd 2479 . . 3  |-  ( ph  ->  ( X  .x.  (
( Z  ./  W
)  .x.  ( ( invr `  R ) `  Y ) ) )  =  ( ( X 
.x.  Z )  .x.  ( ( ( invr `  R ) `  W
)  .x.  ( ( invr `  R ) `  Y ) ) ) )
7452, 58, 733eqtr4rd 2481 . 2  |-  ( ph  ->  ( X  .x.  (
( Z  ./  W
)  .x.  ( ( invr `  R ) `  Y ) ) )  =  ( ( X 
.x.  Z )  ./  ( Y  .x.  W ) ) )
7527, 74eqtrd 2470 1  |-  ( ph  ->  ( ( X  ./  Y )  .x.  ( Z  ./  W ) )  =  ( ( X 
.x.  Z )  ./  ( Y  .x.  W ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2967   ` cfv 5413  (class class class)co 6086   Basecbs 14166   ↾s cress 14167   +g cplusg 14230   .rcmulr 14231   Grpcgrp 15402  mulGrpcmgp 16579   Ringcrg 16633   CRingccrg 16634  Unitcui 16719   invrcinvr 16751  /rcdvr 16762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-tpos 6740  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-0g 14372  df-mnd 15407  df-grp 15536  df-minusg 15537  df-cmn 16270  df-mgp 16580  df-ur 16592  df-rng 16635  df-cring 16636  df-oppr 16703  df-dvdsr 16721  df-unit 16722  df-invr 16752  df-dvr 16763
This theorem is referenced by:  qqhrhm  26370
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