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Theorem rdivmuldivd 26112
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 2433 . . . . . 6  |-  ( invr `  R )  =  (
invr `  R )
7 dvrdir.t . . . . . 6  |-  ./  =  (/r
`  R )
83, 4, 5, 6, 7dvrval 16711 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  U )  ->  ( X  ./  Y
)  =  ( X 
.x.  ( ( invr `  R ) `  Y
) ) )
98oveq1d 6095 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  U )  ->  ( ( X  ./  Y )  .x.  ( Z  ./  W ) )  =  ( ( X 
.x.  ( ( invr `  R ) `  Y
) )  .x.  ( Z  ./  W ) ) )
101, 2, 9syl2anc 654 . . 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 16591 . . . . 5  |-  ( R  e.  CRing  ->  R  e.  Ring )
1311, 12syl 16 . . . 4  |-  ( ph  ->  R  e.  Ring )
143, 5unitss 16686 . . . . 5  |-  U  C_  B
155, 6unitinvcl 16700 . . . . . 6  |-  ( ( R  e.  Ring  /\  Y  e.  U )  ->  (
( invr `  R ) `  Y )  e.  U
)
1613, 2, 15syl2anc 654 . . . . 5  |-  ( ph  ->  ( ( invr `  R
) `  Y )  e.  U )
1714, 16sseldi 3342 . . . 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 16712 . . . . 5  |-  ( ( R  e.  Ring  /\  Z  e.  B  /\  W  e.  U )  ->  ( Z  ./  W )  e.  B )
2113, 18, 19, 20syl3anc 1211 . . . 4  |-  ( ph  ->  ( Z  ./  W
)  e.  B )
223, 4rngass 16597 . . . 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 1213 . . 3  |-  ( ph  ->  ( ( X  .x.  ( ( invr `  R
) `  Y )
)  .x.  ( Z  ./  W ) )  =  ( X  .x.  (
( ( invr `  R
) `  Y )  .x.  ( Z  ./  W
) ) ) )
243, 4crngcom 16595 . . . . 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 1211 . . . 4  |-  ( ph  ->  ( ( ( invr `  R ) `  Y
)  .x.  ( Z  ./  W ) )  =  ( ( Z  ./  W )  .x.  (
( invr `  R ) `  Y ) ) )
2625oveq2d 6096 . . 3  |-  ( ph  ->  ( X  .x.  (
( ( invr `  R
) `  Y )  .x.  ( Z  ./  W
) ) )  =  ( X  .x.  (
( Z  ./  W
)  .x.  ( ( invr `  R ) `  Y ) ) ) )
2710, 23, 263eqtrd 2469 . 2  |-  ( ph  ->  ( ( X  ./  Y )  .x.  ( Z  ./  W ) )  =  ( X  .x.  ( ( Z  ./  W )  .x.  (
( invr `  R ) `  Y ) ) ) )
28 eqid 2433 . . . . . . . 8  |-  ( (mulGrp `  R )s  U )  =  ( (mulGrp `  R )s  U
)
295, 28unitgrp 16693 . . . . . . 7  |-  ( R  e.  Ring  ->  ( (mulGrp `  R )s  U )  e.  Grp )
3013, 29syl 16 . . . . . 6  |-  ( ph  ->  ( (mulGrp `  R
)s 
U )  e.  Grp )
315, 28unitgrpbas 16692 . . . . . . 7  |-  U  =  ( Base `  (
(mulGrp `  R )s  U
) )
32 eqid 2433 . . . . . . 7  |-  ( +g  `  ( (mulGrp `  R
)s 
U ) )  =  ( +g  `  (
(mulGrp `  R )s  U
) )
335, 28, 6invrfval 16699 . . . . . . 7  |-  ( invr `  R )  =  ( invg `  (
(mulGrp `  R )s  U
) )
3431, 32, 33grpinvadd 15584 . . . . . 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 1211 . . . . 5  |-  ( ph  ->  ( ( invr `  R
) `  ( Y
( +g  `  ( (mulGrp `  R )s  U ) ) W ) )  =  ( ( ( invr `  R
) `  W )
( +g  `  ( (mulGrp `  R )s  U ) ) ( ( invr `  R
) `  Y )
) )
36 fvex 5689 . . . . . . . . . . 11  |-  (Unit `  R )  e.  _V
375, 36eqeltri 2503 . . . . . . . . . 10  |-  U  e. 
_V
38 eqid 2433 . . . . . . . . . . 11  |-  ( Rs  U )  =  ( Rs  U )
39 eqid 2433 . . . . . . . . . . 11  |-  (mulGrp `  R )  =  (mulGrp `  R )
4038, 39mgpress 16576 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  U  e.  _V )  ->  (
(mulGrp `  R )s  U
)  =  (mulGrp `  ( Rs  U ) ) )
4113, 37, 40sylancl 655 . . . . . . . . 9  |-  ( ph  ->  ( (mulGrp `  R
)s 
U )  =  (mulGrp `  ( Rs  U ) ) )
4241fveq2d 5683 . . . . . . . 8  |-  ( ph  ->  ( +g  `  (
(mulGrp `  R )s  U
) )  =  ( +g  `  (mulGrp `  ( Rs  U ) ) ) )
43 eqid 2433 . . . . . . . . 9  |-  (mulGrp `  ( Rs  U ) )  =  (mulGrp `  ( Rs  U
) )
4438, 4ressmulr 14274 . . . . . . . . . 10  |-  ( U  e.  _V  ->  .x.  =  ( .r `  ( Rs  U ) ) )
4537, 44ax-mp 5 . . . . . . . . 9  |-  .x.  =  ( .r `  ( Rs  U ) )
4643, 45mgpplusg 16569 . . . . . . . 8  |-  .x.  =  ( +g  `  (mulGrp `  ( Rs  U ) ) )
4742, 46syl6reqr 2484 . . . . . . 7  |-  ( ph  ->  .x.  =  ( +g  `  ( (mulGrp `  R
)s 
U ) ) )
4847oveqd 6097 . . . . . 6  |-  ( ph  ->  ( Y  .x.  W
)  =  ( Y ( +g  `  (
(mulGrp `  R )s  U
) ) W ) )
4948fveq2d 5683 . . . . 5  |-  ( ph  ->  ( ( invr `  R
) `  ( Y  .x.  W ) )  =  ( ( invr `  R
) `  ( Y
( +g  `  ( (mulGrp `  R )s  U ) ) W ) ) )
5047oveqd 6097 . . . . 5  |-  ( ph  ->  ( ( ( invr `  R ) `  W
)  .x.  ( ( invr `  R ) `  Y ) )  =  ( ( ( invr `  R ) `  W
) ( +g  `  (
(mulGrp `  R )s  U
) ) ( (
invr `  R ) `  Y ) ) )
5135, 49, 503eqtr4d 2475 . . . 4  |-  ( ph  ->  ( ( invr `  R
) `  ( Y  .x.  W ) )  =  ( ( ( invr `  R ) `  W
)  .x.  ( ( invr `  R ) `  Y ) ) )
5251oveq2d 6096 . . 3  |-  ( ph  ->  ( ( X  .x.  Z )  .x.  (
( invr `  R ) `  ( Y  .x.  W
) ) )  =  ( ( X  .x.  Z )  .x.  (
( ( invr `  R
) `  W )  .x.  ( ( invr `  R
) `  Y )
) ) )
533, 4rngcl 16594 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .x.  Z )  e.  B )
5413, 1, 18, 53syl3anc 1211 . . . 4  |-  ( ph  ->  ( X  .x.  Z
)  e.  B )
555, 4unitmulcl 16690 . . . . 5  |-  ( ( R  e.  Ring  /\  Y  e.  U  /\  W  e.  U )  ->  ( Y  .x.  W )  e.  U )
5613, 2, 19, 55syl3anc 1211 . . . 4  |-  ( ph  ->  ( Y  .x.  W
)  e.  U )
573, 4, 5, 6, 7dvrval 16711 . . . 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 654 . . 3  |-  ( ph  ->  ( ( X  .x.  Z )  ./  ( Y  .x.  W ) )  =  ( ( X 
.x.  Z )  .x.  ( ( invr `  R
) `  ( Y  .x.  W ) ) ) )
595, 6unitinvcl 16700 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  W  e.  U )  ->  (
( invr `  R ) `  W )  e.  U
)
6013, 19, 59syl2anc 654 . . . . . . . 8  |-  ( ph  ->  ( ( invr `  R
) `  W )  e.  U )
6114, 60sseldi 3342 . . . . . . 7  |-  ( ph  ->  ( ( invr `  R
) `  W )  e.  B )
623, 4rngass 16597 . . . . . . 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 1213 . . . . . 6  |-  ( ph  ->  ( ( X  .x.  Z )  .x.  (
( invr `  R ) `  W ) )  =  ( X  .x.  ( Z  .x.  ( ( invr `  R ) `  W
) ) ) )
643, 4, 5, 6, 7dvrval 16711 . . . . . . . 8  |-  ( ( Z  e.  B  /\  W  e.  U )  ->  ( Z  ./  W
)  =  ( Z 
.x.  ( ( invr `  R ) `  W
) ) )
6518, 19, 64syl2anc 654 . . . . . . 7  |-  ( ph  ->  ( Z  ./  W
)  =  ( Z 
.x.  ( ( invr `  R ) `  W
) ) )
6665oveq2d 6096 . . . . . 6  |-  ( ph  ->  ( X  .x.  ( Z  ./  W ) )  =  ( X  .x.  ( Z  .x.  ( (
invr `  R ) `  W ) ) ) )
6763, 66eqtr4d 2468 . . . . 5  |-  ( ph  ->  ( ( X  .x.  Z )  .x.  (
( invr `  R ) `  W ) )  =  ( X  .x.  ( Z  ./  W ) ) )
6867oveq1d 6095 . . . 4  |-  ( ph  ->  ( ( ( X 
.x.  Z )  .x.  ( ( invr `  R
) `  W )
)  .x.  ( ( invr `  R ) `  Y ) )  =  ( ( X  .x.  ( Z  ./  W ) )  .x.  ( (
invr `  R ) `  Y ) ) )
693, 4rngass 16597 . . . . 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 1213 . . . 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 16597 . . . . 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 1213 . . . 4  |-  ( ph  ->  ( ( X  .x.  ( Z  ./  W ) )  .x.  ( (
invr `  R ) `  Y ) )  =  ( X  .x.  (
( Z  ./  W
)  .x.  ( ( invr `  R ) `  Y ) ) ) )
7368, 70, 723eqtr3rd 2474 . . 3  |-  ( ph  ->  ( X  .x.  (
( Z  ./  W
)  .x.  ( ( invr `  R ) `  Y ) ) )  =  ( ( X 
.x.  Z )  .x.  ( ( ( invr `  R ) `  W
)  .x.  ( ( invr `  R ) `  Y ) ) ) )
7452, 58, 733eqtr4rd 2476 . 2  |-  ( ph  ->  ( X  .x.  (
( Z  ./  W
)  .x.  ( ( invr `  R ) `  Y ) ) )  =  ( ( X 
.x.  Z )  ./  ( Y  .x.  W ) ) )
7527, 74eqtrd 2465 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 1362    e. wcel 1755   _Vcvv 2962   ` cfv 5406  (class class class)co 6080   Basecbs 14157   ↾s cress 14158   +g cplusg 14221   .rcmulr 14222   Grpcgrp 15393  mulGrpcmgp 16565   Ringcrg 16577   CRingccrg 16578  Unitcui 16665   invrcinvr 16697  /rcdvr 16708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-tpos 6734  df-recs 6818  df-rdg 6852  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-2 10368  df-3 10369  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-0g 14363  df-mnd 15398  df-grp 15525  df-minusg 15526  df-cmn 16259  df-mgp 16566  df-rng 16580  df-cring 16581  df-ur 16582  df-oppr 16649  df-dvdsr 16667  df-unit 16668  df-invr 16698  df-dvr 16709
This theorem is referenced by:  qqhrhm  26272
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