MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  unitmulcl Unicode version

Theorem unitmulcl 15462
Description: The product of units is a unit. (Contributed by Mario Carneiro, 2-Dec-2014.)
Hypotheses
Ref Expression
unitmulcl.1  |-  U  =  (Unit `  R )
unitmulcl.2  |-  .x.  =  ( .r `  R )
Assertion
Ref Expression
unitmulcl  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  .x.  Y )  e.  U )

Proof of Theorem unitmulcl
StepHypRef Expression
1 simp1 955 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  R  e.  Ring )
2 simp3 957 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  Y  e.  U )
3 eqid 2296 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
4 unitmulcl.1 . . . . . . 7  |-  U  =  (Unit `  R )
53, 4unitcl 15457 . . . . . 6  |-  ( Y  e.  U  ->  Y  e.  ( Base `  R
) )
62, 5syl 15 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  Y  e.  ( Base `  R
) )
7 simp2 956 . . . . . . 7  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  X  e.  U )
8 eqid 2296 . . . . . . . 8  |-  ( 1r
`  R )  =  ( 1r `  R
)
9 eqid 2296 . . . . . . . 8  |-  ( ||r `  R
)  =  ( ||r `  R
)
10 eqid 2296 . . . . . . . 8  |-  (oppr `  R
)  =  (oppr `  R
)
11 eqid 2296 . . . . . . . 8  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
124, 8, 9, 10, 11isunit 15455 . . . . . . 7  |-  ( X  e.  U  <->  ( X
( ||r `
 R ) ( 1r `  R )  /\  X ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
137, 12sylib 188 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X ( ||r `
 R ) ( 1r `  R )  /\  X ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
1413simpld 445 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  X
( ||r `
 R ) ( 1r `  R ) )
15 unitmulcl.2 . . . . . 6  |-  .x.  =  ( .r `  R )
163, 9, 15dvdsrmul1 15451 . . . . 5  |-  ( ( R  e.  Ring  /\  Y  e.  ( Base `  R
)  /\  X ( ||r `  R ) ( 1r
`  R ) )  ->  ( X  .x.  Y ) ( ||r `  R
) ( ( 1r
`  R )  .x.  Y ) )
171, 6, 14, 16syl3anc 1182 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  .x.  Y ) (
||r `  R ) ( ( 1r `  R ) 
.x.  Y ) )
183, 15, 8rnglidm 15380 . . . . 5  |-  ( ( R  e.  Ring  /\  Y  e.  ( Base `  R
) )  ->  (
( 1r `  R
)  .x.  Y )  =  Y )
191, 6, 18syl2anc 642 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  (
( 1r `  R
)  .x.  Y )  =  Y )
2017, 19breqtrd 4063 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  .x.  Y ) (
||r `  R ) Y )
214, 8, 9, 10, 11isunit 15455 . . . . 5  |-  ( Y  e.  U  <->  ( Y
( ||r `
 R ) ( 1r `  R )  /\  Y ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
222, 21sylib 188 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( Y ( ||r `
 R ) ( 1r `  R )  /\  Y ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
2322simpld 445 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  Y
( ||r `
 R ) ( 1r `  R ) )
243, 9dvdsrtr 15450 . . 3  |-  ( ( R  e.  Ring  /\  ( X  .x.  Y ) (
||r `  R ) Y  /\  Y ( ||r `
 R ) ( 1r `  R ) )  ->  ( X  .x.  Y ) ( ||r `  R
) ( 1r `  R ) )
251, 20, 23, 24syl3anc 1182 . 2  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  .x.  Y ) (
||r `  R ) ( 1r
`  R ) )
2610opprrng 15429 . . . 4  |-  ( R  e.  Ring  ->  (oppr `  R
)  e.  Ring )
271, 26syl 15 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  (oppr `  R
)  e.  Ring )
28 eqid 2296 . . . . 5  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
293, 15, 10, 28opprmul 15424 . . . 4  |-  ( Y ( .r `  (oppr `  R
) ) X )  =  ( X  .x.  Y )
303, 4unitcl 15457 . . . . . . 7  |-  ( X  e.  U  ->  X  e.  ( Base `  R
) )
317, 30syl 15 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  X  e.  ( Base `  R
) )
3222simprd 449 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  Y
( ||r `
 (oppr
`  R ) ) ( 1r `  R
) )
3310, 3opprbas 15427 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
3433, 11, 28dvdsrmul1 15451 . . . . . 6  |-  ( ( (oppr
`  R )  e. 
Ring  /\  X  e.  (
Base `  R )  /\  Y ( ||r `
 (oppr
`  R ) ) ( 1r `  R
) )  ->  ( Y ( .r `  (oppr `  R ) ) X ) ( ||r `
 (oppr
`  R ) ) ( ( 1r `  R ) ( .r
`  (oppr
`  R ) ) X ) )
3527, 31, 32, 34syl3anc 1182 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( Y ( .r `  (oppr `  R ) ) X ) ( ||r `
 (oppr
`  R ) ) ( ( 1r `  R ) ( .r
`  (oppr
`  R ) ) X ) )
363, 15, 10, 28opprmul 15424 . . . . . 6  |-  ( ( 1r `  R ) ( .r `  (oppr `  R
) ) X )  =  ( X  .x.  ( 1r `  R ) )
373, 15, 8rngridm 15381 . . . . . . 7  |-  ( ( R  e.  Ring  /\  X  e.  ( Base `  R
) )  ->  ( X  .x.  ( 1r `  R ) )  =  X )
381, 31, 37syl2anc 642 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  .x.  ( 1r `  R ) )  =  X )
3936, 38syl5eq 2340 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  (
( 1r `  R
) ( .r `  (oppr `  R ) ) X )  =  X )
4035, 39breqtrd 4063 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( Y ( .r `  (oppr `  R ) ) X ) ( ||r `
 (oppr
`  R ) ) X )
4129, 40syl5eqbrr 4073 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  .x.  Y ) (
||r `  (oppr
`  R ) ) X )
4213simprd 449 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  X
( ||r `
 (oppr
`  R ) ) ( 1r `  R
) )
4333, 11dvdsrtr 15450 . . 3  |-  ( ( (oppr
`  R )  e. 
Ring  /\  ( X  .x.  Y ) ( ||r `  (oppr `  R
) ) X  /\  X ( ||r `
 (oppr
`  R ) ) ( 1r `  R
) )  ->  ( X  .x.  Y ) (
||r `  (oppr
`  R ) ) ( 1r `  R
) )
4427, 41, 42, 43syl3anc 1182 . 2  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  .x.  Y ) (
||r `  (oppr
`  R ) ) ( 1r `  R
) )
454, 8, 9, 10, 11isunit 15455 . 2  |-  ( ( X  .x.  Y )  e.  U  <->  ( ( X  .x.  Y ) (
||r `  R ) ( 1r
`  R )  /\  ( X  .x.  Y ) ( ||r `
 (oppr
`  R ) ) ( 1r `  R
) ) )
4625, 44, 45sylanbrc 645 1  |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  .x.  Y )  e.  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   .rcmulr 13225   Ringcrg 15353   1rcur 15355  opprcoppr 15420   ||rcdsr 15436  Unitcui 15437
This theorem is referenced by:  unitmulclb  15463  unitgrp  15465  unitdvcl  15485  irredrmul  15505  subrgugrp  15580  dchrelbasd  20494  dchrptlem2  20520
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-plusg 13237  df-mulr 13238  df-0g 13420  df-mnd 14383  df-grp 14505  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440
  Copyright terms: Public domain W3C validator