Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dvrcan5 Structured version   Unicode version

Theorem dvrcan5 26264
Description: Cancellation law for common factor in ratio. (divcan5 10036 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 16755 . . . . . 6  |-  U  C_  B
4 simpr3 996 . . . . . 6  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  Z  e.  U )
53, 4sseldi 3357 . . . . 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 16759 . . . . . 6  |-  ( ( R  e.  Ring  /\  Y  e.  U  /\  Z  e.  U )  ->  ( Y  .x.  Z )  e.  U )
873adant3r1 1196 . . . . 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 16780 . . . . 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 16762 . . . . . 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 995 . . . . 5  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  Y  e.  U )
182, 14unitgrpbas 16761 . . . . . . 7  |-  U  =  ( Base `  (
(mulGrp `  R )s  U
) )
19 fvex 5704 . . . . . . . . 9  |-  (Unit `  R )  e.  _V
202, 19eqeltri 2513 . . . . . . . 8  |-  U  e. 
_V
21 eqid 2443 . . . . . . . . . 10  |-  (mulGrp `  R )  =  (mulGrp `  R )
2221, 6mgpplusg 16598 . . . . . . . . 9  |-  .x.  =  ( +g  `  (mulGrp `  R ) )
2314, 22ressplusg 14283 . . . . . . . 8  |-  ( U  e.  _V  ->  .x.  =  ( +g  `  ( (mulGrp `  R )s  U ) ) )
2420, 23ax-mp 5 . . . . . . 7  |-  .x.  =  ( +g  `  ( (mulGrp `  R )s  U ) )
252, 14, 9invrfval 16768 . . . . . . 7  |-  ( invr `  R )  =  ( invg `  (
(mulGrp `  R )s  U
) )
2618, 24, 25grpinvadd 15607 . . . . . 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 6110 . . . . 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 1218 . . . 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 16773 . . . . . . 7  |-  ( ( R  e.  Ring  /\  Z  e.  U )  ->  ( Z  .x.  ( ( invr `  R ) `  Z
) )  =  ( 1r `  R ) )
3130oveq1d 6109 . . . . . 6  |-  ( ( R  e.  Ring  /\  Z  e.  U )  ->  (
( Z  .x.  (
( invr `  R ) `  Z ) )  .x.  ( ( invr `  R
) `  Y )
)  =  ( ( 1r `  R ) 
.x.  ( ( invr `  R ) `  Y
) ) )
32313ad2antr3 1155 . . . . 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 16769 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  Z  e.  U )  ->  (
( invr `  R ) `  Z )  e.  U
)
34333ad2antr3 1155 . . . . . . 7  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( ( invr `  R ) `  Z )  e.  U
)
353, 34sseldi 3357 . . . . . 6  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( ( invr `  R ) `  Z )  e.  B
)
362, 9unitinvcl 16769 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  Y  e.  U )  ->  (
( invr `  R ) `  Y )  e.  U
)
37363ad2antr2 1154 . . . . . . 7  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( ( invr `  R ) `  Y )  e.  U
)
383, 37sseldi 3357 . . . . . 6  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( ( invr `  R ) `  Y )  e.  B
)
391, 6rngass 16664 . . . . . 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 1220 . . . . 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 16671 . . . . . 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 2483 . . . 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 2479 . . 3  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( Z  ./  ( Y  .x.  Z
) )  =  ( ( invr `  R
) `  Y )
)
4544oveq2d 6110 . 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 994 . . 3  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  X  e.  B )
471, 2, 10, 6dvrass 16785 . . 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 1220 . 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 16780 . . 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 2485 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 965    = wceq 1369    e. wcel 1756   _Vcvv 2975   ` cfv 5421  (class class class)co 6094   Basecbs 14177   ↾s cress 14178   +g cplusg 14241   .rcmulr 14242   Grpcgrp 15413  mulGrpcmgp 16594   1rcur 16606   Ringcrg 16648  Unitcui 16734   invrcinvr 16766  /rcdvr 16777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-om 6480  df-1st 6580  df-2nd 6581  df-tpos 6748  df-recs 6835  df-rdg 6869  df-er 7104  df-en 7314  df-dom 7315  df-sdom 7316  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-nn 10326  df-2 10383  df-3 10384  df-ndx 14180  df-slot 14181  df-base 14182  df-sets 14183  df-ress 14184  df-plusg 14254  df-mulr 14255  df-0g 14383  df-mnd 15418  df-grp 15548  df-minusg 15549  df-mgp 16595  df-ur 16607  df-rng 16650  df-oppr 16718  df-dvdsr 16736  df-unit 16737  df-invr 16767  df-dvr 16778
This theorem is referenced by:  rhmdvd  26292
  Copyright terms: Public domain W3C validator