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Theorem dvrcan5 27937
Description: Cancellation law for common factor in ratio. (divcan5 10163 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 17422 . . . . . 6  |-  U  C_  B
4 simpr3 1002 . . . . . 6  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  Z  e.  U )
53, 4sseldi 3415 . . . . 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 17426 . . . . . 6  |-  ( ( R  e.  Ring  /\  Y  e.  U  /\  Z  e.  U )  ->  ( Y  .x.  Z )  e.  U )
873adant3r1 1203 . . . . 5  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( Y  .x.  Z )  e.  U
)
9 eqid 2382 . . . . . 6  |-  ( invr `  R )  =  (
invr `  R )
10 dvrcan5.d . . . . . 6  |-  ./  =  (/r
`  R )
111, 6, 2, 9, 10dvrval 17447 . . . . 5  |-  ( ( Z  e.  B  /\  ( Y  .x.  Z )  e.  U )  -> 
( Z  ./  ( Y  .x.  Z ) )  =  ( Z  .x.  ( ( invr `  R
) `  ( Y  .x.  Z ) ) ) )
125, 8, 11syl2anc 659 . . . 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 455 . . . . . 6  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  R  e.  Ring )
14 eqid 2382 . . . . . . 7  |-  ( (mulGrp `  R )s  U )  =  ( (mulGrp `  R )s  U
)
152, 14unitgrp 17429 . . . . . 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 1001 . . . . 5  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  Y  e.  U )
182, 14unitgrpbas 17428 . . . . . . 7  |-  U  =  ( Base `  (
(mulGrp `  R )s  U
) )
19 fvex 5784 . . . . . . . . 9  |-  (Unit `  R )  e.  _V
202, 19eqeltri 2466 . . . . . . . 8  |-  U  e. 
_V
21 eqid 2382 . . . . . . . . . 10  |-  (mulGrp `  R )  =  (mulGrp `  R )
2221, 6mgpplusg 17258 . . . . . . . . 9  |-  .x.  =  ( +g  `  (mulGrp `  R ) )
2314, 22ressplusg 14748 . . . . . . . 8  |-  ( U  e.  _V  ->  .x.  =  ( +g  `  ( (mulGrp `  R )s  U ) ) )
2420, 23ax-mp 5 . . . . . . 7  |-  .x.  =  ( +g  `  ( (mulGrp `  R )s  U ) )
252, 14, 9invrfval 17435 . . . . . . 7  |-  ( invr `  R )  =  ( invg `  (
(mulGrp `  R )s  U
) )
2618, 24, 25grpinvadd 16233 . . . . . 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 6212 . . . . 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 1226 . . . 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 2382 . . . . . . . 8  |-  ( 1r
`  R )  =  ( 1r `  R
)
302, 9, 6, 29unitrinv 17440 . . . . . . 7  |-  ( ( R  e.  Ring  /\  Z  e.  U )  ->  ( Z  .x.  ( ( invr `  R ) `  Z
) )  =  ( 1r `  R ) )
3130oveq1d 6211 . . . . . 6  |-  ( ( R  e.  Ring  /\  Z  e.  U )  ->  (
( Z  .x.  (
( invr `  R ) `  Z ) )  .x.  ( ( invr `  R
) `  Y )
)  =  ( ( 1r `  R ) 
.x.  ( ( invr `  R ) `  Y
) ) )
32313ad2antr3 1161 . . . . 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 17436 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  Z  e.  U )  ->  (
( invr `  R ) `  Z )  e.  U
)
34333ad2antr3 1161 . . . . . . 7  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( ( invr `  R ) `  Z )  e.  U
)
353, 34sseldi 3415 . . . . . 6  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( ( invr `  R ) `  Z )  e.  B
)
362, 9unitinvcl 17436 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  Y  e.  U )  ->  (
( invr `  R ) `  Y )  e.  U
)
37363ad2antr2 1160 . . . . . . 7  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( ( invr `  R ) `  Y )  e.  U
)
383, 37sseldi 3415 . . . . . 6  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( ( invr `  R ) `  Y )  e.  B
)
391, 6ringass 17328 . . . . . 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 1228 . . . . 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 17335 . . . . . 6  |-  ( ( R  e.  Ring  /\  (
( invr `  R ) `  Y )  e.  B
)  ->  ( ( 1r `  R )  .x.  ( ( invr `  R
) `  Y )
)  =  ( (
invr `  R ) `  Y ) )
4213, 38, 41syl2anc 659 . . . . 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 2431 . . . 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 2427 . . 3  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( Z  ./  ( Y  .x.  Z
) )  =  ( ( invr `  R
) `  Y )
)
4544oveq2d 6212 . 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 1000 . . 3  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  X  e.  B )
471, 2, 10, 6dvrass 17452 . . 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 1228 . 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 17447 . . 3  |-  ( ( X  e.  B  /\  Y  e.  U )  ->  ( X  ./  Y
)  =  ( X 
.x.  ( ( invr `  R ) `  Y
) ) )
5046, 17, 49syl2anc 659 . 2  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U )
)  ->  ( X  ./  Y )  =  ( X  .x.  ( (
invr `  R ) `  Y ) ) )
5145, 48, 503eqtr4d 2433 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826   _Vcvv 3034   ` cfv 5496  (class class class)co 6196   Basecbs 14634   ↾s cress 14635   +g cplusg 14702   .rcmulr 14703   Grpcgrp 16170  mulGrpcmgp 17254   1rcur 17266   Ringcrg 17311  Unitcui 17401   invrcinvr 17433  /rcdvr 17444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-tpos 6873  df-recs 6960  df-rdg 6994  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-3 10512  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-mulr 14716  df-0g 14849  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-grp 16174  df-minusg 16175  df-mgp 17255  df-ur 17267  df-ring 17313  df-oppr 17385  df-dvdsr 17403  df-unit 17404  df-invr 17434  df-dvr 17445
This theorem is referenced by:  rhmdvd  27965
  Copyright terms: Public domain W3C validator