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Theorem rdivmuldivd 26403
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 2454 . . . . . 6  |-  ( invr `  R )  =  (
invr `  R )
7 dvrdir.t . . . . . 6  |-  ./  =  (/r
`  R )
83, 4, 5, 6, 7dvrval 16899 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  U )  ->  ( X  ./  Y
)  =  ( X 
.x.  ( ( invr `  R ) `  Y
) ) )
98oveq1d 6214 . . . 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 16777 . . . . 5  |-  ( R  e.  CRing  ->  R  e.  Ring )
1311, 12syl 16 . . . 4  |-  ( ph  ->  R  e.  Ring )
143, 5unitss 16874 . . . . 5  |-  U  C_  B
155, 6unitinvcl 16888 . . . . . 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 3461 . . . 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 16900 . . . . 5  |-  ( ( R  e.  Ring  /\  Z  e.  B  /\  W  e.  U )  ->  ( Z  ./  W )  e.  B )
2113, 18, 19, 20syl3anc 1219 . . . 4  |-  ( ph  ->  ( Z  ./  W
)  e.  B )
223, 4rngass 16783 . . . 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 1221 . . 3  |-  ( ph  ->  ( ( X  .x.  ( ( invr `  R
) `  Y )
)  .x.  ( Z  ./  W ) )  =  ( X  .x.  (
( ( invr `  R
) `  Y )  .x.  ( Z  ./  W
) ) ) )
243, 4crngcom 16781 . . . . 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 1219 . . . 4  |-  ( ph  ->  ( ( ( invr `  R ) `  Y
)  .x.  ( Z  ./  W ) )  =  ( ( Z  ./  W )  .x.  (
( invr `  R ) `  Y ) ) )
2625oveq2d 6215 . . 3  |-  ( ph  ->  ( X  .x.  (
( ( invr `  R
) `  Y )  .x.  ( Z  ./  W
) ) )  =  ( X  .x.  (
( Z  ./  W
)  .x.  ( ( invr `  R ) `  Y ) ) ) )
2710, 23, 263eqtrd 2499 . 2  |-  ( ph  ->  ( ( X  ./  Y )  .x.  ( Z  ./  W ) )  =  ( X  .x.  ( ( Z  ./  W )  .x.  (
( invr `  R ) `  Y ) ) ) )
28 eqid 2454 . . . . . . . 8  |-  ( (mulGrp `  R )s  U )  =  ( (mulGrp `  R )s  U
)
295, 28unitgrp 16881 . . . . . . 7  |-  ( R  e.  Ring  ->  ( (mulGrp `  R )s  U )  e.  Grp )
3013, 29syl 16 . . . . . 6  |-  ( ph  ->  ( (mulGrp `  R
)s 
U )  e.  Grp )
315, 28unitgrpbas 16880 . . . . . . 7  |-  U  =  ( Base `  (
(mulGrp `  R )s  U
) )
32 eqid 2454 . . . . . . 7  |-  ( +g  `  ( (mulGrp `  R
)s 
U ) )  =  ( +g  `  (
(mulGrp `  R )s  U
) )
335, 28, 6invrfval 16887 . . . . . . 7  |-  ( invr `  R )  =  ( invg `  (
(mulGrp `  R )s  U
) )
3431, 32, 33grpinvadd 15722 . . . . . 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 1219 . . . . 5  |-  ( ph  ->  ( ( invr `  R
) `  ( Y
( +g  `  ( (mulGrp `  R )s  U ) ) W ) )  =  ( ( ( invr `  R
) `  W )
( +g  `  ( (mulGrp `  R )s  U ) ) ( ( invr `  R
) `  Y )
) )
36 fvex 5808 . . . . . . . . . . 11  |-  (Unit `  R )  e.  _V
375, 36eqeltri 2538 . . . . . . . . . 10  |-  U  e. 
_V
38 eqid 2454 . . . . . . . . . . 11  |-  ( Rs  U )  =  ( Rs  U )
39 eqid 2454 . . . . . . . . . . 11  |-  (mulGrp `  R )  =  (mulGrp `  R )
4038, 39mgpress 16723 . . . . . . . . . 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 5802 . . . . . . . 8  |-  ( ph  ->  ( +g  `  (
(mulGrp `  R )s  U
) )  =  ( +g  `  (mulGrp `  ( Rs  U ) ) ) )
43 eqid 2454 . . . . . . . . 9  |-  (mulGrp `  ( Rs  U ) )  =  (mulGrp `  ( Rs  U
) )
4438, 4ressmulr 14409 . . . . . . . . . 10  |-  ( U  e.  _V  ->  .x.  =  ( .r `  ( Rs  U ) ) )
4537, 44ax-mp 5 . . . . . . . . 9  |-  .x.  =  ( .r `  ( Rs  U ) )
4643, 45mgpplusg 16716 . . . . . . . 8  |-  .x.  =  ( +g  `  (mulGrp `  ( Rs  U ) ) )
4742, 46syl6reqr 2514 . . . . . . 7  |-  ( ph  ->  .x.  =  ( +g  `  ( (mulGrp `  R
)s 
U ) ) )
4847oveqd 6216 . . . . . 6  |-  ( ph  ->  ( Y  .x.  W
)  =  ( Y ( +g  `  (
(mulGrp `  R )s  U
) ) W ) )
4948fveq2d 5802 . . . . 5  |-  ( ph  ->  ( ( invr `  R
) `  ( Y  .x.  W ) )  =  ( ( invr `  R
) `  ( Y
( +g  `  ( (mulGrp `  R )s  U ) ) W ) ) )
5047oveqd 6216 . . . . 5  |-  ( ph  ->  ( ( ( invr `  R ) `  W
)  .x.  ( ( invr `  R ) `  Y ) )  =  ( ( ( invr `  R ) `  W
) ( +g  `  (
(mulGrp `  R )s  U
) ) ( (
invr `  R ) `  Y ) ) )
5135, 49, 503eqtr4d 2505 . . . 4  |-  ( ph  ->  ( ( invr `  R
) `  ( Y  .x.  W ) )  =  ( ( ( invr `  R ) `  W
)  .x.  ( ( invr `  R ) `  Y ) ) )
5251oveq2d 6215 . . 3  |-  ( ph  ->  ( ( X  .x.  Z )  .x.  (
( invr `  R ) `  ( Y  .x.  W
) ) )  =  ( ( X  .x.  Z )  .x.  (
( ( invr `  R
) `  W )  .x.  ( ( invr `  R
) `  Y )
) ) )
533, 4rngcl 16780 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .x.  Z )  e.  B )
5413, 1, 18, 53syl3anc 1219 . . . 4  |-  ( ph  ->  ( X  .x.  Z
)  e.  B )
555, 4unitmulcl 16878 . . . . 5  |-  ( ( R  e.  Ring  /\  Y  e.  U  /\  W  e.  U )  ->  ( Y  .x.  W )  e.  U )
5613, 2, 19, 55syl3anc 1219 . . . 4  |-  ( ph  ->  ( Y  .x.  W
)  e.  U )
573, 4, 5, 6, 7dvrval 16899 . . . 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 16888 . . . . . . . . 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 3461 . . . . . . 7  |-  ( ph  ->  ( ( invr `  R
) `  W )  e.  B )
623, 4rngass 16783 . . . . . . 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 1221 . . . . . 6  |-  ( ph  ->  ( ( X  .x.  Z )  .x.  (
( invr `  R ) `  W ) )  =  ( X  .x.  ( Z  .x.  ( ( invr `  R ) `  W
) ) ) )
643, 4, 5, 6, 7dvrval 16899 . . . . . . . 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 6215 . . . . . 6  |-  ( ph  ->  ( X  .x.  ( Z  ./  W ) )  =  ( X  .x.  ( Z  .x.  ( (
invr `  R ) `  W ) ) ) )
6763, 66eqtr4d 2498 . . . . 5  |-  ( ph  ->  ( ( X  .x.  Z )  .x.  (
( invr `  R ) `  W ) )  =  ( X  .x.  ( Z  ./  W ) ) )
6867oveq1d 6214 . . . 4  |-  ( ph  ->  ( ( ( X 
.x.  Z )  .x.  ( ( invr `  R
) `  W )
)  .x.  ( ( invr `  R ) `  Y ) )  =  ( ( X  .x.  ( Z  ./  W ) )  .x.  ( (
invr `  R ) `  Y ) ) )
693, 4rngass 16783 . . . . 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 1221 . . . 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 16783 . . . . 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 1221 . . . 4  |-  ( ph  ->  ( ( X  .x.  ( Z  ./  W ) )  .x.  ( (
invr `  R ) `  Y ) )  =  ( X  .x.  (
( Z  ./  W
)  .x.  ( ( invr `  R ) `  Y ) ) ) )
7368, 70, 723eqtr3rd 2504 . . 3  |-  ( ph  ->  ( X  .x.  (
( Z  ./  W
)  .x.  ( ( invr `  R ) `  Y ) ) )  =  ( ( X 
.x.  Z )  .x.  ( ( ( invr `  R ) `  W
)  .x.  ( ( invr `  R ) `  Y ) ) ) )
7452, 58, 733eqtr4rd 2506 . 2  |-  ( ph  ->  ( X  .x.  (
( Z  ./  W
)  .x.  ( ( invr `  R ) `  Y ) ) )  =  ( ( X 
.x.  Z )  ./  ( Y  .x.  W ) ) )
7527, 74eqtrd 2495 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 1370    e. wcel 1758   _Vcvv 3076   ` cfv 5525  (class class class)co 6199   Basecbs 14291   ↾s cress 14292   +g cplusg 14356   .rcmulr 14357   Grpcgrp 15528  mulGrpcmgp 16712   Ringcrg 16767   CRingccrg 16768  Unitcui 16853   invrcinvr 16885  /rcdvr 16896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-tpos 6854  df-recs 6941  df-rdg 6975  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-nn 10433  df-2 10490  df-3 10491  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-plusg 14369  df-mulr 14370  df-0g 14498  df-mnd 15533  df-grp 15663  df-minusg 15664  df-cmn 16399  df-mgp 16713  df-ur 16725  df-rng 16769  df-cring 16770  df-oppr 16837  df-dvdsr 16855  df-unit 16856  df-invr 16886  df-dvr 16897
This theorem is referenced by:  qqhrhm  26562
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