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Theorem dvrcan5 27656
Description: Cancellation law for common factor in ratio. (divcan5 10252 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 17183 . . . . . 6  |-  U  C_  B
4 simpr3 1005 . . . . . 6  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  Z  e.  U )
53, 4sseldi 3487 . . . . 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 17187 . . . . . 6  |-  ( ( R  e.  Ring  /\  Y  e.  U  /\  Z  e.  U )  ->  ( Y  .x.  Z )  e.  U )
873adant3r1 1206 . . . . 5  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( Y  .x.  Z )  e.  U
)
9 eqid 2443 . . . . . 6  |-  ( invr `  R )  =  (
invr `  R )
10 dvrcan5.d . . . . . 6  |-  ./  =  (/r
`  R )
111, 6, 2, 9, 10dvrval 17208 . . . . 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 2443 . . . . . . 7  |-  ( (mulGrp `  R )s  U )  =  ( (mulGrp `  R )s  U
)
152, 14unitgrp 17190 . . . . . 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 1004 . . . . 5  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  Y  e.  U )
182, 14unitgrpbas 17189 . . . . . . 7  |-  U  =  ( Base `  (
(mulGrp `  R )s  U
) )
19 fvex 5866 . . . . . . . . 9  |-  (Unit `  R )  e.  _V
202, 19eqeltri 2527 . . . . . . . 8  |-  U  e. 
_V
21 eqid 2443 . . . . . . . . . 10  |-  (mulGrp `  R )  =  (mulGrp `  R )
2221, 6mgpplusg 17019 . . . . . . . . 9  |-  .x.  =  ( +g  `  (mulGrp `  R ) )
2314, 22ressplusg 14616 . . . . . . . 8  |-  ( U  e.  _V  ->  .x.  =  ( +g  `  ( (mulGrp `  R )s  U ) ) )
2420, 23ax-mp 5 . . . . . . 7  |-  .x.  =  ( +g  `  ( (mulGrp `  R )s  U ) )
252, 14, 9invrfval 17196 . . . . . . 7  |-  ( invr `  R )  =  ( invg `  (
(mulGrp `  R )s  U
) )
2618, 24, 25grpinvadd 15990 . . . . . 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 6297 . . . . 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 1229 . . . 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 2443 . . . . . . . 8  |-  ( 1r
`  R )  =  ( 1r `  R
)
302, 9, 6, 29unitrinv 17201 . . . . . . 7  |-  ( ( R  e.  Ring  /\  Z  e.  U )  ->  ( Z  .x.  ( ( invr `  R ) `  Z
) )  =  ( 1r `  R ) )
3130oveq1d 6296 . . . . . 6  |-  ( ( R  e.  Ring  /\  Z  e.  U )  ->  (
( Z  .x.  (
( invr `  R ) `  Z ) )  .x.  ( ( invr `  R
) `  Y )
)  =  ( ( 1r `  R ) 
.x.  ( ( invr `  R ) `  Y
) ) )
32313ad2antr3 1164 . . . . 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 17197 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  Z  e.  U )  ->  (
( invr `  R ) `  Z )  e.  U
)
34333ad2antr3 1164 . . . . . . 7  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( ( invr `  R ) `  Z )  e.  U
)
353, 34sseldi 3487 . . . . . 6  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( ( invr `  R ) `  Z )  e.  B
)
362, 9unitinvcl 17197 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  Y  e.  U )  ->  (
( invr `  R ) `  Y )  e.  U
)
37363ad2antr2 1163 . . . . . . 7  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( ( invr `  R ) `  Y )  e.  U
)
383, 37sseldi 3487 . . . . . 6  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( ( invr `  R ) `  Y )  e.  B
)
391, 6ringass 17089 . . . . . 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 1231 . . . . 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, 29ringlidm 17096 . . . . . 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 2492 . . . 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 2488 . . 3  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( Z  ./  ( Y  .x.  Z
) )  =  ( ( invr `  R
) `  Y )
)
4544oveq2d 6297 . 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 1003 . . 3  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  X  e.  B )
471, 2, 10, 6dvrass 17213 . . 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 1231 . 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 17208 . . 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 2494 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 974    = wceq 1383    e. wcel 1804   _Vcvv 3095   ` cfv 5578  (class class class)co 6281   Basecbs 14509   ↾s cress 14510   +g cplusg 14574   .rcmulr 14575   Grpcgrp 15927  mulGrpcmgp 17015   1rcur 17027   Ringcrg 17072  Unitcui 17162   invrcinvr 17194  /rcdvr 17205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-tpos 6957  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-3 10601  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-mulr 14588  df-0g 14716  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15931  df-minusg 15932  df-mgp 17016  df-ur 17028  df-ring 17074  df-oppr 17146  df-dvdsr 17164  df-unit 17165  df-invr 17195  df-dvr 17206
This theorem is referenced by:  rhmdvd  27684
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