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Theorem dvrcan5 27432
Description: Cancellation law for common factor in ratio. (divcan5 10235 analog.) (Contributed by Thierry Arnoux, 26-Oct-2016.)
Hypotheses
Ref Expression
dvrcan5.b  |-  B  =  ( Base `  R
)
dvrcan5.o  |-  U  =  (Unit `  R )
dvrcan5.d  |-  ./  =  (/r
`  R )
dvrcan5.t  |-  .x.  =  ( .r `  R )
Assertion
Ref Expression
dvrcan5  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( ( X  .x.  Z )  ./  ( Y  .x.  Z ) )  =  ( X 
./  Y ) )

Proof of Theorem dvrcan5
StepHypRef Expression
1 dvrcan5.b . . . . . . 7  |-  B  =  ( Base `  R
)
2 dvrcan5.o . . . . . . 7  |-  U  =  (Unit `  R )
31, 2unitss 17086 . . . . . 6  |-  U  C_  B
4 simpr3 999 . . . . . 6  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  Z  e.  U )
53, 4sseldi 3495 . . . . 5  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  Z  e.  B )
6 dvrcan5.t . . . . . . 7  |-  .x.  =  ( .r `  R )
72, 6unitmulcl 17090 . . . . . 6  |-  ( ( R  e.  Ring  /\  Y  e.  U  /\  Z  e.  U )  ->  ( Y  .x.  Z )  e.  U )
873adant3r1 1200 . . . . 5  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( Y  .x.  Z )  e.  U
)
9 eqid 2460 . . . . . 6  |-  ( invr `  R )  =  (
invr `  R )
10 dvrcan5.d . . . . . 6  |-  ./  =  (/r
`  R )
111, 6, 2, 9, 10dvrval 17111 . . . . 5  |-  ( ( Z  e.  B  /\  ( Y  .x.  Z )  e.  U )  -> 
( Z  ./  ( Y  .x.  Z ) )  =  ( Z  .x.  ( ( invr `  R
) `  ( Y  .x.  Z ) ) ) )
125, 8, 11syl2anc 661 . . . 4  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( Z  ./  ( Y  .x.  Z
) )  =  ( Z  .x.  ( (
invr `  R ) `  ( Y  .x.  Z
) ) ) )
13 simpl 457 . . . . . 6  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  R  e.  Ring )
14 eqid 2460 . . . . . . 7  |-  ( (mulGrp `  R )s  U )  =  ( (mulGrp `  R )s  U
)
152, 14unitgrp 17093 . . . . . 6  |-  ( R  e.  Ring  ->  ( (mulGrp `  R )s  U )  e.  Grp )
1613, 15syl 16 . . . . 5  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( (mulGrp `  R )s  U )  e.  Grp )
17 simpr2 998 . . . . 5  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  Y  e.  U )
182, 14unitgrpbas 17092 . . . . . . 7  |-  U  =  ( Base `  (
(mulGrp `  R )s  U
) )
19 fvex 5867 . . . . . . . . 9  |-  (Unit `  R )  e.  _V
202, 19eqeltri 2544 . . . . . . . 8  |-  U  e. 
_V
21 eqid 2460 . . . . . . . . . 10  |-  (mulGrp `  R )  =  (mulGrp `  R )
2221, 6mgpplusg 16928 . . . . . . . . 9  |-  .x.  =  ( +g  `  (mulGrp `  R ) )
2314, 22ressplusg 14586 . . . . . . . 8  |-  ( U  e.  _V  ->  .x.  =  ( +g  `  ( (mulGrp `  R )s  U ) ) )
2420, 23ax-mp 5 . . . . . . 7  |-  .x.  =  ( +g  `  ( (mulGrp `  R )s  U ) )
252, 14, 9invrfval 17099 . . . . . . 7  |-  ( invr `  R )  =  ( invg `  (
(mulGrp `  R )s  U
) )
2618, 24, 25grpinvadd 15910 . . . . . 6  |-  ( ( ( (mulGrp `  R
)s 
U )  e.  Grp  /\  Y  e.  U  /\  Z  e.  U )  ->  ( ( invr `  R
) `  ( Y  .x.  Z ) )  =  ( ( ( invr `  R ) `  Z
)  .x.  ( ( invr `  R ) `  Y ) ) )
2726oveq2d 6291 . . . . 5  |-  ( ( ( (mulGrp `  R
)s 
U )  e.  Grp  /\  Y  e.  U  /\  Z  e.  U )  ->  ( Z  .x.  (
( invr `  R ) `  ( Y  .x.  Z
) ) )  =  ( Z  .x.  (
( ( invr `  R
) `  Z )  .x.  ( ( invr `  R
) `  Y )
) ) )
2816, 17, 4, 27syl3anc 1223 . . . 4  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( Z  .x.  ( ( invr `  R
) `  ( Y  .x.  Z ) ) )  =  ( Z  .x.  ( ( ( invr `  R ) `  Z
)  .x.  ( ( invr `  R ) `  Y ) ) ) )
29 eqid 2460 . . . . . . . 8  |-  ( 1r
`  R )  =  ( 1r `  R
)
302, 9, 6, 29unitrinv 17104 . . . . . . 7  |-  ( ( R  e.  Ring  /\  Z  e.  U )  ->  ( Z  .x.  ( ( invr `  R ) `  Z
) )  =  ( 1r `  R ) )
3130oveq1d 6290 . . . . . 6  |-  ( ( R  e.  Ring  /\  Z  e.  U )  ->  (
( Z  .x.  (
( invr `  R ) `  Z ) )  .x.  ( ( invr `  R
) `  Y )
)  =  ( ( 1r `  R ) 
.x.  ( ( invr `  R ) `  Y
) ) )
32313ad2antr3 1158 . . . . 5  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( ( Z  .x.  ( ( invr `  R ) `  Z
) )  .x.  (
( invr `  R ) `  Y ) )  =  ( ( 1r `  R )  .x.  (
( invr `  R ) `  Y ) ) )
332, 9unitinvcl 17100 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  Z  e.  U )  ->  (
( invr `  R ) `  Z )  e.  U
)
34333ad2antr3 1158 . . . . . . 7  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( ( invr `  R ) `  Z )  e.  U
)
353, 34sseldi 3495 . . . . . 6  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( ( invr `  R ) `  Z )  e.  B
)
362, 9unitinvcl 17100 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  Y  e.  U )  ->  (
( invr `  R ) `  Y )  e.  U
)
37363ad2antr2 1157 . . . . . . 7  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( ( invr `  R ) `  Y )  e.  U
)
383, 37sseldi 3495 . . . . . 6  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( ( invr `  R ) `  Y )  e.  B
)
391, 6rngass 16995 . . . . . 6  |-  ( ( R  e.  Ring  /\  ( Z  e.  B  /\  ( ( invr `  R
) `  Z )  e.  B  /\  (
( invr `  R ) `  Y )  e.  B
) )  ->  (
( Z  .x.  (
( invr `  R ) `  Z ) )  .x.  ( ( invr `  R
) `  Y )
)  =  ( Z 
.x.  ( ( (
invr `  R ) `  Z )  .x.  (
( invr `  R ) `  Y ) ) ) )
4013, 5, 35, 38, 39syl13anc 1225 . . . . 5  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( ( Z  .x.  ( ( invr `  R ) `  Z
) )  .x.  (
( invr `  R ) `  Y ) )  =  ( Z  .x.  (
( ( invr `  R
) `  Z )  .x.  ( ( invr `  R
) `  Y )
) ) )
411, 6, 29rnglidm 17002 . . . . . 6  |-  ( ( R  e.  Ring  /\  (
( invr `  R ) `  Y )  e.  B
)  ->  ( ( 1r `  R )  .x.  ( ( invr `  R
) `  Y )
)  =  ( (
invr `  R ) `  Y ) )
4213, 38, 41syl2anc 661 . . . . 5  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( ( 1r `  R )  .x.  ( ( invr `  R
) `  Y )
)  =  ( (
invr `  R ) `  Y ) )
4332, 40, 423eqtr3d 2509 . . . 4  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( Z  .x.  ( ( ( invr `  R ) `  Z
)  .x.  ( ( invr `  R ) `  Y ) ) )  =  ( ( invr `  R ) `  Y
) )
4412, 28, 433eqtrd 2505 . . 3  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( Z  ./  ( Y  .x.  Z
) )  =  ( ( invr `  R
) `  Y )
)
4544oveq2d 6291 . 2  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( X  .x.  ( Z  ./  ( Y  .x.  Z ) ) )  =  ( X 
.x.  ( ( invr `  R ) `  Y
) ) )
46 simpr1 997 . . 3  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  X  e.  B )
471, 2, 10, 6dvrass 17116 . . 3  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Z  e.  B  /\  ( Y  .x.  Z )  e.  U ) )  ->  ( ( X 
.x.  Z )  ./  ( Y  .x.  Z ) )  =  ( X 
.x.  ( Z  ./  ( Y  .x.  Z ) ) ) )
4813, 46, 5, 8, 47syl13anc 1225 . 2  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( ( X  .x.  Z )  ./  ( Y  .x.  Z ) )  =  ( X 
.x.  ( Z  ./  ( Y  .x.  Z ) ) ) )
491, 6, 2, 9, 10dvrval 17111 . . 3  |-  ( ( X  e.  B  /\  Y  e.  U )  ->  ( X  ./  Y
)  =  ( X 
.x.  ( ( invr `  R ) `  Y
) ) )
5046, 17, 49syl2anc 661 . 2  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( X  ./  Y )  =  ( X  .x.  ( (
invr `  R ) `  Y ) ) )
5145, 48, 503eqtr4d 2511 1  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( ( X  .x.  Z )  ./  ( Y  .x.  Z ) )  =  ( X 
./  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   _Vcvv 3106   ` cfv 5579  (class class class)co 6275   Basecbs 14479   ↾s cress 14480   +g cplusg 14544   .rcmulr 14545   Grpcgrp 15716  mulGrpcmgp 16924   1rcur 16936   Ringcrg 16979  Unitcui 17065   invrcinvr 17097  /rcdvr 17108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-tpos 6945  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-0g 14686  df-mnd 15721  df-grp 15851  df-minusg 15852  df-mgp 16925  df-ur 16937  df-rng 16981  df-oppr 17049  df-dvdsr 17067  df-unit 17068  df-invr 17098  df-dvr 17109
This theorem is referenced by:  rhmdvd  27460
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