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Theorem dvrcan5 26114
Description: Cancellation law for common factor in ratio. (divcan5 10021 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 16686 . . . . . 6  |-  U  C_  B
4 simpr3 989 . . . . . 6  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  Z  e.  U )
53, 4sseldi 3342 . . . . 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 16690 . . . . . 6  |-  ( ( R  e.  Ring  /\  Y  e.  U  /\  Z  e.  U )  ->  ( Y  .x.  Z )  e.  U )
873adant3r1 1189 . . . . 5  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( Y  .x.  Z )  e.  U
)
9 eqid 2433 . . . . . 6  |-  ( invr `  R )  =  (
invr `  R )
10 dvrcan5.d . . . . . 6  |-  ./  =  (/r
`  R )
111, 6, 2, 9, 10dvrval 16711 . . . . 5  |-  ( ( Z  e.  B  /\  ( Y  .x.  Z )  e.  U )  -> 
( Z  ./  ( Y  .x.  Z ) )  =  ( Z  .x.  ( ( invr `  R
) `  ( Y  .x.  Z ) ) ) )
125, 8, 11syl2anc 654 . . . 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 454 . . . . . 6  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  R  e.  Ring )
14 eqid 2433 . . . . . . 7  |-  ( (mulGrp `  R )s  U )  =  ( (mulGrp `  R )s  U
)
152, 14unitgrp 16693 . . . . . 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 988 . . . . 5  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  Y  e.  U )
182, 14unitgrpbas 16692 . . . . . . 7  |-  U  =  ( Base `  (
(mulGrp `  R )s  U
) )
19 fvex 5689 . . . . . . . . 9  |-  (Unit `  R )  e.  _V
202, 19eqeltri 2503 . . . . . . . 8  |-  U  e. 
_V
21 eqid 2433 . . . . . . . . . 10  |-  (mulGrp `  R )  =  (mulGrp `  R )
2221, 6mgpplusg 16569 . . . . . . . . 9  |-  .x.  =  ( +g  `  (mulGrp `  R ) )
2314, 22ressplusg 14263 . . . . . . . 8  |-  ( U  e.  _V  ->  .x.  =  ( +g  `  ( (mulGrp `  R )s  U ) ) )
2420, 23ax-mp 5 . . . . . . 7  |-  .x.  =  ( +g  `  ( (mulGrp `  R )s  U ) )
252, 14, 9invrfval 16699 . . . . . . 7  |-  ( invr `  R )  =  ( invg `  (
(mulGrp `  R )s  U
) )
2618, 24, 25grpinvadd 15584 . . . . . 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 6096 . . . . 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 1211 . . . 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 2433 . . . . . . . 8  |-  ( 1r
`  R )  =  ( 1r `  R
)
302, 9, 6, 29unitrinv 16704 . . . . . . 7  |-  ( ( R  e.  Ring  /\  Z  e.  U )  ->  ( Z  .x.  ( ( invr `  R ) `  Z
) )  =  ( 1r `  R ) )
3130oveq1d 6095 . . . . . 6  |-  ( ( R  e.  Ring  /\  Z  e.  U )  ->  (
( Z  .x.  (
( invr `  R ) `  Z ) )  .x.  ( ( invr `  R
) `  Y )
)  =  ( ( 1r `  R ) 
.x.  ( ( invr `  R ) `  Y
) ) )
32313ad2antr3 1148 . . . . 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 16700 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  Z  e.  U )  ->  (
( invr `  R ) `  Z )  e.  U
)
34333ad2antr3 1148 . . . . . . 7  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( ( invr `  R ) `  Z )  e.  U
)
353, 34sseldi 3342 . . . . . 6  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( ( invr `  R ) `  Z )  e.  B
)
362, 9unitinvcl 16700 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  Y  e.  U )  ->  (
( invr `  R ) `  Y )  e.  U
)
37363ad2antr2 1147 . . . . . . 7  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( ( invr `  R ) `  Y )  e.  U
)
383, 37sseldi 3342 . . . . . 6  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( ( invr `  R ) `  Y )  e.  B
)
391, 6rngass 16597 . . . . . 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 1213 . . . . 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 16604 . . . . . 6  |-  ( ( R  e.  Ring  /\  (
( invr `  R ) `  Y )  e.  B
)  ->  ( ( 1r `  R )  .x.  ( ( invr `  R
) `  Y )
)  =  ( (
invr `  R ) `  Y ) )
4213, 38, 41syl2anc 654 . . . . 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 2473 . . . 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 2469 . . 3  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( Z  ./  ( Y  .x.  Z
) )  =  ( ( invr `  R
) `  Y )
)
4544oveq2d 6096 . 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 987 . . 3  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  X  e.  B )
471, 2, 10, 6dvrass 16716 . . 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 1213 . 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 16711 . . 3  |-  ( ( X  e.  B  /\  Y  e.  U )  ->  ( X  ./  Y
)  =  ( X 
.x.  ( ( invr `  R ) `  Y
) ) )
5046, 17, 49syl2anc 654 . 2  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( X  ./  Y )  =  ( X  .x.  ( (
invr `  R ) `  Y ) ) )
5145, 48, 503eqtr4d 2475 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 958    = 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   1rcur 16579  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-mgp 16566  df-rng 16580  df-ur 16582  df-oppr 16649  df-dvdsr 16667  df-unit 16668  df-invr 16698  df-dvr 16709
This theorem is referenced by:  rhmdvd  26142
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